Eigenvalues and Canonical Forms

Lec-26 Eigenvalues and Eigenvectors of Linear Operators

NPTEL · Linear Algebra

Lec-28 The Minimal Polynomial

NPTEL · Linear Algebra

The minimal polynomial is what decides diagonalisability — watch this one closely.

Lec-29 The Cayley-Hamilton Theorem

NPTEL · Linear Algebra

Characteristic vs minimal polynomial — what each tells you free

In plain words

An eigenvector is a direction the matrix only stretches, never turns. The characteristic polynomial says which stretch factors occur. The minimal polynomial says how badly the eigenvectors are missing.

Definitions and the three facts

For $A \in M_{n}(F): \chi_A(x) = \det($ xI − A) (degree n), m_A(x) = monic polynomial of least degree with m_A(A) = 0.

Cayley–Hamilton: $\chi_A(A) = 0$ , hence m_A | $\chi_A$ .
Same roots: every eigenvalue is a root of m_A (and vice versa). So $\chi_A$ and m_A have the same irreducible factors, possibly with different multiplicities.
Diagonalisable ⇔ m_A splits into distinct linear factors.

Reading the Jordan form off polynomials

For eigenvalue $\lambda$ : the exponent of $(x - \lambda)$ in $\chi_A =$ algebraic multiplicity = total size of $\lambda-$ blocks; the exponent in m_A = size of the largest $\lambda-$ block; the geometric multiplicity = number of $\lambda-$ blocks $= n - rank(A - \lambda{}I)$ .

Example: $\chi = (x-2)^{3}(x-3), m = (x-2)^{2}(x-3) \Rightarrow$ blocks for 2 have sizes {2, 1}; A is 4×4, not diagonalisable, geometric multiplicity of 2 is 2.

Similarity

Statement	Truth
Similar ⇒ same $\chi$ , same m, same rank, same trace, same det	✓
Same $\chi \Rightarrow$ similar	✗: 0 and [[0,1],[0,0]]
Same $\chi$ and same m ⇒ similar	✗ for $n \ge 4: \chi = x^{4}, m = x^{2}$ with block sizes {2,2} vs {2,1,1}
Same $\chi$ and same m ⇒ similar, n ≤ 3	✓
A similar to Aᵀ	✓ always
AB and BA have the same eigenvalues (with multiplicity)	✓; same $\chi$ . But $m_{AB}$ ≠ $m_{BA}$ possible

Cayley–Hamilton in practice

A invertible $\Leftrightarrow \chi_A(0) \ne 0$ ; then A⁻ $^{1}$ is a polynomial in A: if $\chi = x^{n} + \cdots + c_{1}x + c_{0}, A$ ⁻ $^{1} = -(A^{n}$ ⁻ $^{1} + \cdots + c_{1}I)/c_{0}$ . $- A^{100}$ for a 2×2: divide $x^{100}$ by $\chi_A$ , remainder $r(x) = \alpha{}x + \beta; A^{100} = \alpha{}A + \beta{}I$ . $- A^{2} = A \Rightarrow m$ | x(x−1) ⇒ diagonalisable with eigenvalues in {0, 1}; rank = trace.
Aᵏ = I ⇒ m | xᵏ − 1 ⇒ diagonalisable over $\mathbb{C} ($ distinct roots). Over $\mathbb{R}$ only if eigenvalues are ±1.
A nilpotent $\Rightarrow \chi = x^{n}$ , trace $= 0, A^{n} = 0$ ; diagonalisable only if A = 0.

Quick facts

trace $= \sum\lambda_{i}, \det = \prod\lambda_{i} ($ over the algebraic closure).
Real matrix, non-real eigenvalues come in conjugate pairs; odd n ⇒ at least one real eigenvalue.
Rank-1 matrix uvᵀ: eigenvalues vᵀu (once) and 0 (n−1 times); diagonalisable iff vᵀu ≠ 0.

Key takeaways

Minimal polynomial decides diagonalisability; characteristic decides multiplicities.
"Same polynomials" ≠ "similar" once n ≥ 4.
Any polynomial identity in A pins the minimal polynomial's factors — use it to classify A instantly.

What a matrix does to a vectorinteractive

This visual is part of Trap Analysis

Sweep a direction and watch Av swing — except on the eigendirections, where it only stretches. Det appears as the area of the image.

Exam focus: Read eigen-structure off the characteristic and minimal polynomials without computing: diagonalisable ⇔ minimal polynomial has distinct linear factors; similar matrices share both polynomials but the converse fails; Cayley–Hamilton lets you compute inverses and high powers.

Practice

Part B2025 Decembershow ▾

Let A be a 3×3 matrix over complex numbers with trace 1 and determinant 1. Suppose, further, that one of the eigenvalues of A is 1. Which of the following statements is necessarily true?

A.The characteristic polynomial of A has repeated roots.
B.Every eigenvalue of A has absolute value 1.✓ Correct
C.A does not have any eigenvalue on the imaginary axis.
D. $A^{2}$ is the identity matrix.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Let A be a non-zero 3×3 matrix with integer entries. Let

\lambda_{i}\in\mathbb{C}, 1\le{}i\le3

be all the eigenvalues of A (not necessarily distinct). Which of the following statements are necessarily true?

A.There exists a cubic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.✓ Correct
B.There exists a quadratic polynomial $f(X)\in\mathbb{Q}[X]$ such that $f(\lambda_{i})=0$ for all 1≤i≤3.
C.If $f(X)\in\mathbb{Q}[X]$ is such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.
D.If $f(X)\in\mathbb{Q}[X]$ is a cubic polynomial such that $f(\lambda_{i})=0$ for all 1≤i≤3, then f(A)=0.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Let A, B be distinct 2 × 2 real matrices. Which of the following statements are true?

A.If A is invertible, then AB and BA have the same minimal polynomial.✓ Correct
B.If 0 is an eigenvalue of A, then 0 is an eigenvalue of AB.✓ Correct
C.If 0 is the only eigenvalue of A and of B, then 0 is the only eigenvalue of AB.
D.If AB and BA have the same minimal polynomial, then either A or B is invertible.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Let

S, T : \mathbb{C}^{2} \to \mathbb{C}^{2}

\mathbb{C}-

linear transformations and I denote the identity transformation on

\mathbb{C}^{2}

. Which of the following statements are necessarily true?

A.(ST − TS $)^{2} = \lambda{}I$ for some $\lambda \in \mathbb{C}$ .✓ Correct
B.The characteristic polynomial of (ST − TS $)^{2}$ is $(x - \lambda)^{2}$ for some $\lambda \in \mathbb{C}$ .✓ Correct
C.If ST − TS has only one eigenvalue, then ST − TS $= \lambda{}I$ for some $\lambda \in \mathbb{C}$ .
D.If ST − TS has only one eigenvalue, then (ST − TS $)^{2}$ is the zero transformation.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let

M, N \in M_{3}(\mathbb{C})

be such that Trace(Mᵏ) = Trace(Nᵏ), 1 ≤ k ≤ 3. Which of the following statements is necessarily true?

A. $M^{2} = N^{2}$
B. $M^{3} = N^{3}$
C.The characteristic polynomials of M and N are the same.✓ Correct
D.The minimal polynomials of M and N are the same.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾

Let I denote the 3×3 identity matrix. Given any three distinct matrices

A, B, C \in M_{3}(\mathbb{R})

, which of the following statements is necessarily true?

A.There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = f(C) = 0 and f(D) = I.✓ Correct
B.There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = 0, f(C) = I and f(D) = I.
C.There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = 0, f(B) = f(C) = I and f(D) ≠ 0.
D.There exists $D \in M_{3}(\mathbb{R})$ and a polynomial $f \in \mathbb{R}[X]$ satisfying f(A) = f(B) = 0, f(C) = I and f(D) = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾

Consider the real matrix A whose rows are (29, 0, 55, 17), (1, 28, 46, 26), (17, 13, 33, 38) and (21, 67, 0, 13). What is the largest real eigenvalue of A?

A.101✓ Correct
B.67
C.103
D.113

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾

Which of the following polynomials is the characteristic polynomial of a real 2 × 2 matrix A such that trace(A) = 7 and trace

(A^{2}) = 29

A. $t^{2} + 7t + 10$
B. $t^{2} - 7t + 29$
C. $t^{2} - 7t - 10$
D. $t^{2} - 7t + 10$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Let V be a 4-dimensional complex vector space and A a linear operator on V. Which of the following statements are necessarily true?

A.There exist $\lambda\in\mathbb{C}$ and a non-zero v∈V such that Av $=\lambda{}v$ .✓ Correct
B.There exist $\lambda,\mu\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av $=\lambda{}v$ and Aw $=\mu{}w$ .
C.There exist $\lambda,\mu,\delta\in\mathbb{C}$ and linearly independent vectors v,w∈V such that Av $=\lambda{}v$ and Aw $=\mu{}v+\delta{}w$ .✓ Correct
D.There exists a three-dimensional subspace W such that Aw∈W for all w∈W.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾

Let

A \in M_{4}(\mathbb{C})

be such that

A^{2} = I

and Trace(A) = 0. Which of the following statements are necessarily true?

A.The set { $B \in M_{4}(\mathbb{C})$ ∣ AB+BA=0} is an 8-dimensional $\mathbb{C}-$ vector space.✓ Correct
B.The set { $B \in M_{4}(\mathbb{C})$ ∣ AB−BA=0} is an 8-dimensional $\mathbb{C}-$ vector space.✓ Correct
C.There exists $B \in M_{4}(\mathbb{C})$ such that AB+BA $=0, B^{2}=I$ and Trace(B)=0.✓ Correct
D.If $B \in M_{4}(\mathbb{C})$ is such that AB+BA=0, then B is diagonalisable.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

For each n > 1, let V denote the

\mathbb{C}-

vector space of all n × n complex matrices and A ∈ V. Which of the following statements are necessarily true?

A.The set { $I, A, \dots, A^{n}$ } is linearly independent but the set { $I, A, \dots, A^(n^{2})$ } is not linearly independent.
B.If A is a singular matrix, then the set {I, A, …, A^k} spans a (k + 1)-dimensional subspace of V for all k ≤ rank(A).
C.The sets { $I, A, \dots, A^{n}$ } and { $I, A, \dots, A^(n^{2})$ } both span the same subspace of V.✓ Correct
D.The set { $I, A, \dots, A^{n}$ } is linearly dependent.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

For each n > 1, let V be the

\mathbb{R}-

vector space of all n × n real matrices and A ∈ V be invertible. Consider the

\mathbb{R}-

linear transformation

\varphi : \mathbb{R}[x] \to V

such that

\varphi(x^k) = A^k

for all k ≥ 1 and

\varphi(1)

is the identity matrix of order n. Which of the following statements are necessarily true?

A. $\varphi$ is one-to-one but not onto.
B. $\varphi$ is onto but not one-to-one.
C.There exists $f \in \mathbb{R}[x]$ such that deg f ≤ n and $\ker \varphi =$ {fg | $g \in \mathbb{R}[x]$ }.✓ Correct
D.deg h ≥ n for every nonzero $h \in \ker \varphi$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

T : \mathbb{C}^{2} \to \mathbb{C}^{2}

a \mathbb{C}-

linear transformation. For any ordered basis ℬ of

\mathbb{C}^{2}

, let [T]ℬ denote the matrix of T with respect to ℬ. Suppose that ℬ $_{1}$ and ℬ $_{2}$ are two ordered bases of $\mathbb{C}^{2}$ such that the matrix [T](ℬ

_{1})

is upper-triangular and [T]_(ℬ

_{2}) = ([T]_(

ℬ

_{1}))^{2}

. Which of the following statements are FALSE?

A.The characteristic polynomial of T can be $x^{2} + x + 1$ .
B.The characteristic polynomial of T can be x(x − 1).
C.The minimal polynomial of T can be $x^{2}$ .✓ Correct
D.The characteristic polynomial of T can be $x(x - e^(2\pi{}i/3))$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let A be a 4 × 4 real matrix whose minimal polynomial is

x^{2} + x + 1

and let

B = A + I_{4}

. Which of the following statements are necessarily true?

A.The minimal polynomial of B is $x^{2} + x + 1$
B.The minimal polynomial of B is $x^{2} - x + 1$ ✓ Correct
C. $B^{3} = I_{4}$
D. $B^{3} + I_{4} = 0$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let

M_{5}(\mathbb{C})

be the complex vector space of 5 × 5 matrices with entries in

\mathbb{C}

. Let V be a non-zero subspace of

M_{5}(\mathbb{C})

such that every non-zero A ∈ V is invertible. Which among the following are possible values for the dimension of V?

A.1✓ Correct
B.2
C.3
D.5

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾

Let

T : \mathbb{R}^{4} \to \mathbb{R}^{4}

be a linear map with four distinct eigenvalues and satisfying

T^{4} - 15T^{2} + 10T + 24I = 0

. Which of the following statements are necessarily true?

A.There exists a non-zero vector $v_{1} \in \mathbb{R}^{4}$ such that Tv $_{1} = 2v_{1}$ ✓ Correct
B.There exists a non-zero vector $v_{2} \in \mathbb{R}^{4}$ such that Tv $_{2} = v_{2}$
C.For every non-zero vector $v \in \mathbb{R}^{4}$ , the set {2v, 3Tv} is linearly independent
D.T is a one-one function✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let

T : \mathbb{R}^{4} \to \mathbb{R}^{4}

be an

\mathbb{R}-

linear transformation. Assume that the characteristic polynomial of T has two distinct monic irreducible quadratic factors

q_{1}(x)

and

q_{2}(x)

. Which of the following statements are true?

A.There exists a nonzero $v \in \mathbb{R}^{4}$ such that $q_{1}(T)v = 0$ .✓ Correct
B.There exists a nonzero $v \in \mathbb{R}^{4}$ such that $q_{1}(T)v = 0$ and $q_{2}(T)v = 0$ .
C.For all nonzero $v \in \mathbb{R}^{4}, v$ and Tv are linearly independent.✓ Correct
D.If $q_{1}(T)v = 0$ for some nonzero $v \in \mathbb{R}^{4}$ , then {v, Tv, $T^{2}v, T^{3}v$ } is a basis of $\mathbb{R}^{4}$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let A = [[0, a, 0], [0, 0, b], [c, 0, 0]], where a, b, c are real numbers with abc = 1. If

B = A + A^{2} + A^{3}

, then which of the following statements is true?

A.det B = 1
B.det A = 0
C.rank(B) = 2
D. $rank(B^{2}) = 1$ ✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let

T : \mathbb{R}^{4} \to \mathbb{R}^{4}

be an

\mathbb{R}-

linear transformation such that

(T^{2} + T + I)(T - 2I) = 0

and

(T^{2} + T + 2I)(T^{2} - 4I) = 0

. Which of the following statements is FALSE?

A.T is diagonalizable over $\mathbb{R}$ .
B.The characteristic polynomial of T is $(x - 2)^{4}$ .
C.The characteristic polynomial of T is $(x^{2} + x + 2)(x^{2} - 4)$ .✓ Correct
D.For every $\mathbb{R}-$ linear transformation $S : \mathbb{R}^{4} \to \mathbb{R}^{4}$ , we have ST = TS.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾

Let ((2, a), (b, c)) be a 2 × 2 real matrix for which 6 is an eigenvalue. Which of the following statements is necessarily true?

A.24 − ab = 4c✓ Correct
B.a + b = 8
C.c = 6
D.ab = 0

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Let X, Y be two n × n real matrices such that XY

= X^{2} + X + I

. Which of the following statements are necessarily true?

A.X is invertible.✓ Correct
B.X + I is invertible.
C.XY = YX.✓ Correct
D.Y is invertible.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾

Let

A = (a_{ij})

be the n × n real matrix with

a_{ij} =

ij for all 1 ≤ i, j ≤ n. If n ≥ 3, which one of the following is an eigenvalue of A?

A.1
B.n
C.n(n+1)/2
D.n(n+1)(2n+1)/6✓ Correct

Solution

A = vvᵀ with v = (1, 2, …, n) has rank 1; its only non-zero eigenvalue is $v^{T}v = \sum{}i^{2} = n(n+1)(2n+1)/6$ .

Part C2023 Decembershow ▾

Let A be an n × n real symmetric matrix. Which of the following statements are necessarily true?

A.A is diagonalizable.✓ Correct
B.If Aᵏ = I for some positive integer k, then $A^{2} = I$ .✓ Correct
C.If Aᵏ = 0 for some positive integer k, then $A^{2} = 0$ .✓ Correct
D.All eigenvalues of A are real.✓ Correct

Solution

Spectral theorem: A = QDQᵀ with real D.

Aᵏ = I forces each real eigenvalue to satisfy $\lambda^{k} = 1$ , so $\lambda = \pm1$ and $A^{2} = I; A^{k} = 0$ forces all $\lambda = 0$ so A = 0.

Part C2023 Decembershow ▾

Consider A = [[1, 4], [2, 3]]. Suppose

A^{5} - 4A^{4} - 7A^{3} + 11A^{2} - A - 10I =

aA + bI for some

a, b \in \mathbb{Z}

. Which of the following statements are true?

A.a + b > 8.
B.a + b < 7.✓ Correct
C.a + b is divisible by 2.✓ Correct
D.a > b.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾

Let T be a linear operator on

\mathbb{R}^{3}

. Let

f(X) \in \mathbb{R}[X]

denote its characteristic polynomial. Consider the following statements. (a) Suppose T is non-zero and 0 is an eigenvalue of T. If we write f(X) = X·g(X) in

\mathbb{R}[X]

, then the linear operator g(T) is zero. (b) Suppose 0 is an eigenvalue of T with at least two linearly independent eigenvectors. If we write f(X) = X·g(X) in

\mathbb{R}[X]

, then the linear operator g(T) is zero. Which of the following is true?

A.Both (a) and (b) are true.
B.Both (a) and (b) are false.
C.(a) is true and (b) is false.
D.(a) is false and (b) is true.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾

Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by

T^{2}

. Which of the following statements is true?

A.The eigenspace of A for the eigenvalue 0 is two-dimensional.
B.All the eigenvalues of A are real.✓ Correct
C. $A^{3} = 0$ .
D.A is diagonalizable.

Solution

$p(T) = T^{2}(T - c)$ with c real (degree-3 real polynomial).

So all eigenvalues are real.

The geometric multiplicity of 0 may be 1 (Jordan block), so (1), (4) fail; A need not be nilpotent (c ≠ 0).

Part Cshow ▾

Let A be a 3 × 3 matrix with real entries satisfying

A^{3} = A

. Which of the following must be true?

A.A is diagonalisable over $\mathbb{R}$ .✓ Correct
B.A is invertible.
C.The trace of A is an integer.✓ Correct
D.A is symmetric.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Which of the following pairs of matrices are necessarily similar?

A.A and Aᵀ✓ Correct
B.Two 3×3 matrices with the same characteristic and minimal polynomials✓ Correct
C.Two 4×4 matrices with the same characteristic and minimal polynomials
D.AB and BA, for invertible A✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

Let A be a 3×3 real matrix satisfying

A^{3} = I

and A ≠ I. Then

A.A is diagonalisable over $\mathbb{R}$
B.A is diagonalisable over $\mathbb{C}$ but not over $\mathbb{R}$ ✓ Correct
C.A is not diagonalisable over $\mathbb{C}$
D.trace(A) = 3

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

Let A be a 3×3 real matrix with

A^{3} = A

. Which is necessarily true?

A.A is diagonalisable over $\mathbb{R}$ ✓ Correct
B.A is invertible
C.A is symmetric
D.trace(A) = 0

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

Let A be a 3 × 3 real matrix with characteristic polynomial

\lambda^{3} - \lambda

. Which statement is correct?

A.A is invertible.
B.trace A = 0 and A is diagonalisable.✓ Correct
C.det A = 1.
D.A is not diagonalisable.

Solution

The eigenvalues are the roots 0, 1, −1: distinct, so A is diagonalisable; the trace is their sum, 0, and the determinant their product, 0 — in particular A is singular.

Part Cshow ▾

Let A be a 4×4 complex matrix with characteristic polynomial

(x - 1)^{2}(x - 2)^{2}

and minimal polynomial

(x - 1)(x - 2)^{2}

. Which of the following are true?

A.A is diagonalisable
B.rank(A − I) = 2✓ Correct
C.rank(A − 2I) = 3✓ Correct
D.A is similar to diag(1, 1, 2, 2)

Solution

Minimal polynomial has a repeated factor ⇒ not diagonalisable.

Eigenvalue 1: single-size blocks, two of them ⇒ rank(A − I) = 4 − 2 = 2.

Eigenvalue 2: one block of size 2 ⇒ geometric multiplicity 1 ⇒ rank(A − 2I) = 3.

Part Bshow ▾

Let A be a square matrix satisfying

A^{2} = A + I

. Then A⁻

^{1}

equals

A.A + I
B.A − I✓ Correct
C.I − A
D.A

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

Let A = uvᵀ where

u, v \in \mathbb{R}^{n}

are non-zero and vᵀu = 0. Then

A.A is diagonalisable
B.A is nilpotent with $A^{2} = 0$ ✓ Correct
C.A is invertible
D.A has a non-zero eigenvalue

The worked solution and the per-option analysis are in Trap Analysis.

2. Diagonalisability criteria

Lec-27 Diagonalization of Linear Operators. A Characterization

NPTEL · Linear Algebra

Lec-31 Triangulability, Diagonalization in Terms of the Minimal Polynomial

NPTEL · Linear Algebra

The distinct-linear-factors criterion, stated precisely.

Diagonalisability — the criteria card free

In plain words

Diagonalisable means there are enough eigenvectors to build a whole coordinate system. Having distinct eigenvalues guarantees it and is not required, and the field you work over decides the answer.

The criterion

$A \in M_{n}(F)$ is diagonalisable over F ⇔ m_A splits over F into distinct linear factors ⇔ for every eigenvalue, geometric multiplicity = algebraic multiplicity $\Leftrightarrow F ^{n}$ has a basis of eigenvectors.

Instant verdicts

Property	Diagonalisable?	Field / reason
n distinct eigenvalues in F	✓	sufficient, not necessary (I has one eigenvalue)
$A^{2} = A ($ idempotent)	✓ over any field	m \| x(x − 1)
$A^{2} = I ($ involution), char F ≠ 2	✓	m \| (x−1)(x+1)
Aᵏ = I	✓ over $\mathbb{C}$ ; over $\mathbb{R}$ only if eigenvalues ±1	m \| xᵏ − 1, distinct roots in $\mathbb{C}$
A nilpotent, A ≠ 0	✗	m = xʳ, r ≥ 2
real symmetric	✓ over $\mathbb{R}$ , orthogonally	spectral theorem
Hermitian / unitary / normal	✓ over $\mathbb{C}$ , unitarily	spectral theorem
real orthogonal	✓ over $\mathbb{C}$ , usually ✗ over $\mathbb{R}$	rotation by 90° has eigenvalues ±i
real skew-symmetric	✓ over $\mathbb{C}$ ; over $\mathbb{R}$ only if A = 0	eigenvalues purely imaginary
upper triangular with distinct diagonal	✓	distinct eigenvalues
upper triangular with repeated diagonal	not necessarily	[[1,1],[0,1]]
rank 1, A = uvᵀ	✓ iff trace ≠ 0	eigenvalues vᵀu, 0, …, 0
A commutes with a matrix having distinct eigenvalues	✓	A is a polynomial in that matrix

Simultaneous diagonalisation

A and B diagonalisable and AB = BA ⇒ simultaneously diagonalisable. Commuting alone is not enough (take A = B nilpotent).

Over $\mathbb{R}$ vs $\mathbb{C}$ — the recurring trap

A real matrix with non-real eigenvalues is never diagonalisable over $\mathbb{R}$ but may be over $\mathbb{C}$ : rotation [[0,−1],[1,0]] is the canonical example. Conversely, a real matrix diagonalisable over $\mathbb{C}$ with all eigenvalues real is diagonalisable over $\mathbb{R} ($ the minimal polynomial is the same over both fields).

Key takeaways

One polynomial identity ⇒ minimal polynomial divides it ⇒ check for repeated roots.
Symmetric/normal ⇒ diagonalisable, with an orthonormal eigenbasis.
Nilpotent ≠ 0 is the universal non-diagonalisable example; rotation is the universal "over $\mathbb{C}$ but not $\mathbb{R}$ ".

Is this matrix diagonalisable? The decision procedureinteractive

This visual is part of Trap Analysis

The criterion that always settles it, and the two conditions that look sufficient but are not.

Exam focus: Decide diagonalisability from a single equation or property: idempotent, involution, Aᵏ = I, nilpotent, symmetric, normal, distinct eigenvalues. Know over which field.

Practice

Part C2025 Decembershow ▾

Let V be a finite-dimensional

\mathbb{R}-

vector space and T:V→V a linear operator such that

T^{2}

is diagonalizable over

\mathbb{R}

. Which of the following statements are necessarily true?

A.If T is not diagonalizable over $\mathbb{R}$ , then $T^{2}$ has an eigenvalue ≤0.✓ Correct
B.If $T^{2}$ has only negative eigenvalues, then dim V is an even integer.✓ Correct
C.If $T^{2}$ has only non-negative eigenvalues, then T is diagonalizable over $\mathbb{R}$ .
D.For each non-zero v∈V, {v,Tv, $T^{2}v$ } is linearly dependent.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

Let A be a 3 × 3 complex matrix such that

A^{3}

is the identity matrix. Which of the following statements is true?

A.A is diagonalizable.✓ Correct
B.A has at least two distinct eigenvalues.
C.The characteristic polynomial of A is $x^{3} - 1$ .
D.The minimal polynomial of A cannot have degree 2.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let A = [[0, 0, 1], [1, 0, 0], [0, 1, 0]] and B = [[0, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]]. Which of the following statements are true?

A.Both A and B are diagonalizable over $\mathbb{R}$ .
B.A is diagonalizable over $\mathbb{C}$ but not over $\mathbb{R}$ .✓ Correct
C.Neither A nor B is diagonalizable over $\mathbb{R}$ , but both A and B are diagonalizable over $\mathbb{C}$ .✓ Correct
D.Neither A nor B is diagonalizable over $\mathbb{C}$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

For every integer n ≥ 2, consider

a \mathbb{C}-

linear transformation

T : \mathbb{C}^{n} \to \mathbb{C}^{n}

. Let V be a subspace of

\mathbb{C}^{n}

such that T(V) ⊆ V. Which of the following statements are necessarily true?

A.There exists a subspace W of $\mathbb{C}^{n}$ such that $\mathbb{C}^{n} = V + W$ and V ∩ W = {0}.✓ Correct
B.There exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.
C.Suppose that there exists a positive integer k such that Tᵏ is the identity map. Then there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}.✓ Correct
D.Suppose that there exists a subspace W of $\mathbb{C}^{n}$ such that $T(W) \subseteq W, \mathbb{C}^{n} = V + W$ and V ∩ W = {0}. Then there exists a positive integer k such that Tᵏ is the identity map.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾

For any matrix P, the transpose of P is denoted by Pᵗ. Consider the real matrix A = [[1, 1, 0], [1, 2, 1], [1, 1, 2]]. Which of the following statements is true?

A.There exists a real invertible matrix P such that PAP⁻ $^{1}$ is a diagonal matrix and Pᵗ $P = I_{3}$ .
B.There exists a real invertible matrix P such that PAP⁻ $^{1}$ is a diagonal matrix and Pᵗ $P \ne I_{3}$ .✓ Correct
C.One of the eigenvalues of A is not real.
D.A has only real eigenvalues and it is not diagonalizable over $\mathbb{R}$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let A, B be n×n complex matrices. Which of the following are true?

A.If A and B are diagonalisable and AB = BA, they are simultaneously diagonalisable✓ Correct
B.If AB = BA then A and B are simultaneously diagonalisable
C.If A is diagonalisable then so is every polynomial in A✓ Correct
D.If A is diagonalisable and invertible then A⁻ $^{1}$ is diagonalisable✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let A be an n × n complex matrix. Which of the following conditions, each on its own, guarantee that A is diagonalisable?

A.A has n distinct eigenvalues.✓ Correct
B.The minimal polynomial of A has no repeated roots.✓ Correct
C.A is normal, i.e. AA* = A*A.✓ Correct
D.The characteristic polynomial of A splits into linear factors over $\mathbb{C}$ .

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let A be a real n×n matrix. Which of the following imply that A is diagonalisable over

\mathbb{R}

A. $A^{2} = A$ ✓ Correct
B. $A^{2} = -I$
C.A is symmetric✓ Correct
D.A has n distinct real eigenvalues✓ Correct

Solution

(b): eigenvalues ±i, not even real — not diagonalisable over $\mathbb{R} (e.g$ . the 90° rotation).

Part Bshow ▾

Let A be a real orthogonal 2×2 matrix with det A = 1 and A ≠ ±I. Then A is

A.diagonalisable over $\mathbb{R}$
B.diagonalisable over $\mathbb{C}$ but not over $\mathbb{R}$ ✓ Correct
C.not diagonalisable over $\mathbb{C}$
D.nilpotent

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let A be an n×n complex matrix. Which of the following are true?

A.If $A^{2} = A$ then A is diagonalisable✓ Correct
B.If A is nilpotent and non-zero then A is not diagonalisable✓ Correct
C.If A has n distinct eigenvalues then A is diagonalisable✓ Correct
D.If A is diagonalisable then A is invertible

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾

A 4×4 real matrix A has characteristic polynomial

(x - 2)^{2}(x - 3)^{2}

and minimal polynomial (x − 2)(x − 3). Then A is

A.diagonalisable over $\mathbb{R}$ ✓ Correct
B.nilpotent
C.not diagonalisable
D.singular

The worked solution and the per-option analysis are in Trap Analysis.

3. Jordan canonical form

Lec-35 The Primary Decomposition Theorem and Jordan Decomposition

NPTEL · Linear Algebra

How the block structure arises from the primary decomposition.

Jordan canonical form

NPTEL · Matrix Theory

Determining the Jordan form of a matrix

NPTEL · Matrix Theory

The practical recipe: read the blocks off ranks of (A − λI)^k.

Jordan canonical form free

In plain words

With too few eigenvectors the best you can reach is nearly diagonal: blocks carrying ones just above the diagonal. Every feature of that block picture is a rank you can compute.

Reading the blocks

For an eigenvalue $\lambda$ :

Quantity	Equals
algebraic multiplicity (exponent in $\chi)$	total size of the $\lambda-$ blocks
geometric multiplicity $= \dim \ker(A - \lambda{}I) = n - rank(A - \lambda{}I)$	number of $\lambda-$ blocks
exponent of $(x - \lambda)$ in the minimal polynomial	size of the largest $\lambda-$ block
$rank(A - \lambda{}I)^k$ drops	finer block structure

Number of blocks of size ≥ k is $rank(A-\lambda{}I)^{k-1}$ $- rank(A-\lambda{}I)^k$ .

Example. $\chi = (x-2)^{3}(x-3), m = (x-2)^{2}(x-3)$ : blocks for 2 are {2,1}, so geometric multiplicity 2, and A is 4×4, not diagonalisable.

Existence

Jordan form exists over an algebraically closed field (or whenever $\chi$ splits). Over $\mathbb{R}$ with complex eigenvalues use the real Jordan form with 2×2 rotation blocks.

Two matrices are similar ⇔ they have the same Jordan form up to reordering.

Nilpotent case

A nilpotent on $\mathbb{R}^{n}$ has $\chi = x^{n}$ . If it has k blocks (k = nullity) then:

the largest block has size ≤ n − k + 1, so $A^{n-k+1}$ = 0;
A is diagonalisable ⇔ A = 0.

For n = 5 with nullity 3: block sizes {3,1,1} or {2,2,1}, so $T^{3} = 0$ always but $T^{2} = 0$ not necessarily.

Functions of a matrix

Jordan form computes A^k, e^A, and shows $e^{At}$ has terms t^{j}e^{ $\lambda{}t$ } up to j = (largest block − 1). This is where degenerate nodes in ODE come from.

Key takeaways

Three numbers per eigenvalue determine the blocks: total, count, largest.
Nullity = number of blocks; that fixes the nilpotency index range.
Same Jordan form ⇔ similar; same $\chi$ and m is not enough for n ≥ 4.

Building a Jordan form from the data you are giveninteractive

This visual is part of Trap Analysis

How block sizes are forced by the minimal polynomial and the kernel dimensions.

Exam focus: Recover the block structure from three numbers per eigenvalue: algebraic multiplicity (total size), geometric multiplicity (number of blocks), minimal-polynomial exponent (largest block).

Practice

Part C2025 Juneshow ▾

Which of the following matrices are similar over

\mathbb{R}

to the matrix A whose rows are (−1, 1, 0, 0), (0, −1, 0, 0), (0, 0, 1, 1) and (0, 0, 0, 1)?

A.The matrix with rows (0, 0, 0, −1), (1, 0, 0, 0), (0, 1, 0, 2), (0, 0, 1, 0)✓ Correct
B.The matrix with rows (0, 0, 0, 1), (1, 0, 0, 0), (0, 1, 0, −2), (0, 0, 1, 0)
C.The matrix with rows (0, 1, 1, 0), (1, 0, 0, 1), (0, 0, 0, 1), (0, 0, 1, 0)✓ Correct
D.The matrix with rows (0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 0, 1), (0, 0, 1, 0)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾

Let

T : \mathbb{C}^{7} \to \mathbb{C}^{7}

a \mathbb{C}-

linear operator with eigenvalues 2, 3 and 5. Consider the subspace W := {

v \in \mathbb{C}^{7} : (T - 5I)^{k}v = 0

for some integer k > 0} of

\mathbb{C}^{7}

. Suppose that

(T - 2I)^{2}(T - 3I)^{2}(T - 5I)^{2} = 0

. Which of the following statements are necessarily true?

A.T has at least four linearly independent eigenvectors.✓ Correct
B.dim W ≥ 2.
C. $\ker((T - 2I)^{2025}) = \ker((T - 2I)^{2026})$ ✓ Correct
D.(T − 2I)(T − 3I) is a nilpotent operator.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Which of the following statements are true?

A.There exists a 5 × 5 real matrix whose minimal polynomial is $(x^{2} + x + 1)x^{3}$ .✓ Correct
B.There exists a 5 × 5 real matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}$ .
C.There exists a 5 × 5 complex matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}x$ .✓ Correct
D.There exists a 5 × 5 complex matrix whose minimal polynomial is $(x^{2} + x + 1)^{2}$ .✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾

Let n be a positive integer and A, B be n × n complex matrices such that the minimal polynomials of A and B are the same. Which of the following conditions ensure that A is similar to B?

A.n = 3 and the characteristic polynomials of A and B are the same.✓ Correct
B.n = 4 and A has two distinct eigenvalues.
C.n = 5 and A has some eigenvalue for which the dimensions of the eigenspaces of A and B are the same.
D.n = 6 and A has only one eigenvalue and for this eigenvalue the dimensions of the eigenspaces of A and B are the same.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾

Suppose a 7 × 7 block diagonal complex matrix A has blocks (0), (1), [[0, 1], [0, 0]] and the 3×3 block with

2\pi{}i

on the diagonal, a 1 in position (1,2) and zeros elsewhere, along the diagonal. Which of the following statements are true?

A.The characteristic polynomial of A is $x^{3}(x - 1)(x - 2\pi{}i)^{3}$ .✓ Correct
B.The minimal polynomial of A is $x^{2}(x - 1)(x - 2\pi{}i)^{3}$ .
C.The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 3 respectively.
D.The dimensions of the eigenspaces for $0, 1, 2\pi{}i$ are 2, 1, 2 respectively.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾

Let V be a finite dimensional real vector space and

T_{1}, T_{2}

be two nilpotent operators on V. Let

W_{1} =

{

v \in V : T_{1}(v) = 0

} and

W_{2} =

{

v \in V : T_{2}(v) = 0

}. Which of the following statements are FALSE?

A.If $T_{1}$ and $T_{2}$ are similar, then $W_{1}$ and $W_{2}$ are isomorphic vector spaces.
B.If $W_{1}$ and $W_{2}$ are isomorphic vector spaces, then $T_{1}$ and $T_{2}$ have the same minimal polynomial.✓ Correct
C.If $W_{1} = W_{2} = V$ , then $T_{1}$ and $T_{2}$ are similar.
D.If $W_{1}$ and $W_{2}$ are isomorphic, then $T_{1}$ and $T_{2}$ have the same characteristic polynomial.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾

Let N be a 4 × 4 complex matrix with

N^{3} = 0

and

N^{2} \ne 0

. Which of the following are true?

A.Every eigenvalue of N is 0.✓ Correct
B.The Jordan form of N consists of one block of size 3 and one of size 1.✓ Correct
C.rank N = 3.
D.N is not diagonalisable.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

4. Rational canonical form

Lec-37 The Cyclic Decomposition Theorem I

NPTEL · Linear Algebra

Lec-38 The Cyclic Decomposition Theorem II. The Rational Form

NPTEL · Linear Algebra

Invariant factors and the rational canonical form — works over any field.

Rational canonical form and invariant factors free

In plain words

This form exists over every field and needs no roots at all. The payoff is that two matrices similar over a bigger field were already similar over the smaller one.

The form

Every operator on a finite-dimensional V over any field F is similar to a direct sum of companion matrices $C(f_{1})$ ⊕ ⋯ ⊕ C(f_k) with

$f_{1}$ | $f_{2}$ | ⋯ | f_k (the invariant factors).

f_k = the minimal polynomial; $- \prod{}f_{i} =$ the characteristic polynomial;
the list is unique, so two matrices are similar over F ⇔ they have the same invariant factors.

The companion matrix of $x^{n} +$ $a_{n-1}x^{n-1}$ $+ \cdots + a_{0}$ has that polynomial as both its characteristic and minimal polynomial.

Why it matters

Works over $\mathbb{Q}$ or 𝔽_p where the characteristic polynomial may not split — Jordan form does not.
Similarity is field-independent: two matrices over $\mathbb{Q}$ that are similar over $\mathbb{C}$ are already similar over $\mathbb{Q}$ , because the invariant factors are computed by row reduction over $\mathbb{Q} ($ Smith normal form of xI − A).

Elementary divisors vs invariant factors

Elementary divisors are the prime-power factors; grouping the largest of each prime gives f_k, the next largest give $f_{k-1}$ , and so on. Over an algebraically closed field the elementary divisors are $(x - \lambda)^m$ — exactly the Jordan blocks.

Example. $\chi = (x-1)^{2}(x-2)^{2}, m = (x-1)(x-2)$ : invariant factors are (x−1)(x−2) and (x−1)(x−2), so the matrix is diag(1,1,2,2) — diagonalisable.

Key takeaways

Invariant factors divide successively; last = minimal, product = characteristic.
Rational form exists over every field; Jordan needs the polynomial to split.
Similarity over an extension field implies similarity over the base field.

Rational canonical form, and when to prefer itinteractive

This visual is part of Trap Analysis

The form that exists over every field — and the divisibility chain that makes it unique.

Exam focus: Rational canonical form works over any field — use it when the characteristic polynomial does not split. Invariant factors divide one another; the last one is the minimal polynomial.

Practice

Part Cshow ▾

Let A be an n×n matrix over a field F. Which of the following are true?

A.The rational canonical form of A is a direct sum of companion matrices of its invariant factors.✓ Correct
B.The largest invariant factor of A equals its minimal polynomial.✓ Correct
C.The product of all the invariant factors of A equals its characteristic polynomial.✓ Correct
D.Two matrices with the same characteristic and minimal polynomials must be similar.

Solution

(1)–(3) are the defining facts about invariant factors and rational canonical form.

(4) is false: the characteristic and minimal polynomials are only two of the invariants that determine similarity class; when an eigenvalue has multiplicity ≥ 3 there can be two non-similar matrices sharing both, differing in how the remaining Jordan blocks split — e.g. diag $(J_{2}(0), J_{2}(0))$ and diag $(J_{2}(0), J_{1}(0), J_{1}(0))$ inside a 4×4 nilpotent both have characteristic polynomial $x^{4}$ and minimal polynomial $x^{2}$ , but different Jordan (hence rational canonical) forms.

Part Cshow ▾

Let A be a 3×3 real matrix with characteristic polynomial

(x^{2} + 1)(x - 3)

. Which of the following are true?

A.A has exactly one real eigenvalue, namely 3.✓ Correct
B.The minimal polynomial of A equals its characteristic polynomial, $(x^{2} + 1)(x - 3)$ .✓ Correct
C.The rational canonical form of A is the companion matrix of $x^{3} - 3x^{2} + x - 3$ .✓ Correct
D.A is diagonalisable over $\mathbb{R}$ .

Solution

$(1) x^{2} + 1$ contributes no real roots. (2) both irreducible factors appear to the first power in the characteristic polynomial, so the minimal polynomial — which must have the same irreducible factors, each to some power between 1 and its multiplicity there — is forced to equal it. (3) follows: A is non-derogatory, so its rational canonical form is the single companion block of its characteristic polynomial. (4) is false: $x^{2} + 1$ does not split into real linear factors, so the minimal polynomial does not split over $\mathbb{R}$ , and A is not diagonalisable there (it is diagonalisable over $\mathbb{C}$ , with eigenvalues 3, i, −i).

Part Cshow ▾

Let A, B be n×n matrices over

\mathbb{Q}

. Which of the following are true?

A.If A and B are similar over $\mathbb{C}$ , they are similar over $\mathbb{Q}$ ✓ Correct
B.A and B have the same rational canonical form iff they are similar✓ Correct
C.The last invariant factor is the minimal polynomial✓ Correct
D.The rational canonical form requires the characteristic polynomial to split

Solution

The invariant factors come from the Smith normal form of xI − A over $\mathbb{Q}[x]$ , so they are field-independent and no splitting is needed.

Part Cshow ▾

Let A be the companion matrix of

p(x) = x^{3} - 2x^{2} + x - 2 = (x - 2)(x^{2} + 1)

over

\mathbb{Q}

. Which of the following are true?

A.p(x) is both the characteristic and the minimal polynomial of A.✓ Correct
B.A has exactly one rational eigenvalue.✓ Correct
C.A is diagonalisable over $\mathbb{Q}$ .
D.A is diagonalisable over $\mathbb{C}$ .✓ Correct

Solution

(1) a companion matrix is always non-derogatory: its characteristic and minimal polynomials coincide, equal to p(x) itself.

(2) the roots are 2, i, −i; only $2 \in \mathbb{Q}$ .

(4) over $\mathbb{C}, p$ splits as (x−2)(x−i)(x+i) into three distinct linear factors, so A is diagonalisable there.

(3) is false precisely because $x^{2} + 1$ has no root in $\mathbb{Q}$ — the minimal polynomial doesn't split over $\mathbb{Q}$ , so no basis of $\mathbb{Q}^{3}$ can diagonalise A, even though A is perfectly diagonalisable once you allow complex entries.

Part Cshow ▾

Let A be a 4×4 matrix over

\mathbb{Q}

with minimal polynomial

(x - 1)(x - 2)^{2}

and characteristic polynomial

(x - 1)^{2}(x - 2)^{2}

. Which of the following are true?

A.The rational canonical form of A consists of a 1×1 block and a 3×3 block.✓ Correct
B.The eigenvalue 1 has geometric multiplicity 2, matching its algebraic multiplicity.✓ Correct
C.The eigenvalue 2 has geometric multiplicity 1, strictly less than its algebraic multiplicity 2.✓ Correct
D.A is diagonalisable.

Solution

The invariant factors multiply to the characteristic polynomial and the largest equals the minimal polynomial, so they are (x−1) and $(x-1)(x-2)^{2}$ , giving blocks of size 1 and 3 — (1). Geometric multiplicity of an eigenvalue equals the number of invariant factors it divides: eigenvalue 1 divides both invariant factors (multiplicity 2, matching its algebraic multiplicity 2), eigenvalue 2 divides only the second (multiplicity 1, against algebraic multiplicity 2) — (2), (3). (4) is false: since $(x-2)^{2}$ appears in the minimal polynomial with a repeated root, A cannot be diagonalisable.