Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

In one sentence

The characteristic polynomial tells you which eigenvalues there are, the minimal polynomial tells you how badly the matrix fails to be diagonal, and almost every question here turns on the second rather than the first.

Why the exam asks it

Because the characteristic polynomial is what everyone computes and the minimal polynomial is what actually decides things. Cayley–Hamilton gets misremembered as saying the two are equal, and the options are built on the gap between "$A$ satisfies its characteristic polynomial" and "the characteristic polynomial is the smallest thing $A$ satisfies".

The idea

$\lambda$ is an eigenvalue of $A$ if $Av = \lambda v$ for some $v \ne 0$, equivalently if $\det(A - \lambda I) = 0$.

The characteristic polynomial $\chi_A(x) = \det(xI - A)$ has degree exactly $n$. Its roots are the eigenvalues, and:

• $\operatorname{tr}(A)$ is the sum of the eigenvalues with multiplicity;
• $\det(A)$ is their product.

Both follow from comparing coefficients, and both are quick ways to eliminate options.

Two multiplicities

For an eigenvalue $\lambda$:

• algebraic multiplicity: its multiplicity as a root of $\chi_A$;
• geometric multiplicity: $\dim\ker(A - \lambda I)$, the number of independent eigenvectors.

Always $1 \le \text{geometric} \le \text{algebraic}$. The inequality can be strict, and where it is strict the matrix is not diagonalisable. For $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ the eigenvalue $1$ has algebraic multiplicity $2$ and geometric multiplicity $1$, since $A - I$ has rank $1$.

Cayley–Hamilton

$\chi_A(A) = 0$: every matrix satisfies its own characteristic polynomial.

For $2\times2$ this reads $A^2 - \operatorname{tr}(A)A + \det(A)I = 0$, which is worth knowing outright because it turns any polynomial in a $2\times2$ matrix into a linear one. It also gives inverses cheaply: from $A^2 - 5A - 2I = 0$ we get $A^{-1} = \tfrac12(A - 5I)$ whenever $\det A \ne 0$.

What Cayley–Hamilton does not say is that $\chi_A$ is the smallest polynomial killing $A$. The identity matrix satisfies $x - 1$, of degree $1$, while its characteristic polynomial has degree $n$.

The minimal polynomial

$m_A$ is the monic polynomial of least degree with $m_A(A) = 0$. It divides every polynomial that annihilates $A$ — in particular, by Cayley–Hamilton, it divides $\chi_A$.

The two share all their roots: every eigenvalue is a root of $m_A$, and every root of $m_A$ is an eigenvalue. They differ only in multiplicities, and that difference is the entire content of the topic:

$$A \text{ is diagonalisable} \iff m_A \text{ is a product of \emph{distinct} linear factors.}$$

So over $\mathbb{C}$, where everything splits, diagonalisability is precisely the absence of repeated factors in $m_A$.

Computing $m_A$ without theory. For a single eigenvalue $\lambda$, the multiplicity of $(x - \lambda)$ in $m_A$ is the smallest $k$ with $\ker(A-\lambda I)^k$ stable — equivalently the smallest $k$ with $(A - \lambda I)^k = 0$ on the generalised eigenspace. In practice: compute $A - \lambda I$, square it, cube it, and stop when the rank stops dropping.

Standard consequences

• $A$ is invertible $\iff$ $0$ is not an eigenvalue.
• $A$ is nilpotent $\iff$ $\chi_A(x) = x^n$ $\iff$ all eigenvalues are $0$.
• Eigenvalues of $A^k$ are $\lambda^k$; of $A^{-1}$ are $1/\lambda$; of $A + cI$ are $\lambda + c$.
• $A$ and $A^T$ have the same characteristic polynomial, hence the same eigenvalues.
• Eigenvectors for distinct eigenvalues are independent.

Where intuition breaks

"Cayley–Hamilton says the minimal polynomial is the characteristic polynomial." It says $\chi_A(A) = 0$, so $m_A \mid \chi_A$. The identity has $m_A = x - 1$ and $\chi_A = (x-1)^n$. This is the single most common misreading in the topic.

"A real matrix has real eigenvalues." $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ has characteristic polynomial $x^2 + 1$. Real entries constrain the polynomial to have real coefficients, so non-real roots come in conjugate pairs — they do not fail to exist. Real symmetric is the hypothesis that forces real eigenvalues.

"Geometric multiplicity equals algebraic multiplicity." Only for diagonalisable matrices; that equality is diagonalisability, eigenvalue by eigenvalue.

"Eigenvalues of $A + B$ are sums of eigenvalues of $A$ and $B$." Badly false unless they commute and are simultaneously triangularisable. Take $A = \begin{pmatrix}0&1\\0&0\end{pmatrix}$ and $B = \begin{pmatrix}0&0\\1&0\end{pmatrix}$: both have all eigenvalues $0$, and $A + B$ has eigenvalues $\pm1$. The same warning applies to products.

"The characteristic polynomial determines the matrix up to similarity." It does not, and it is the reason canonical forms exist at all.

"$m_A = \chi_A$, so $A$ is diagonalisable." Backwards. $m_A = \chi_A$ means the matrix is as far from having repeated structure as possible in the block sense — it is cyclic, one Jordan block per eigenvalue. For $\begin{pmatrix}1&1\\0&1\end{pmatrix}$, $m_A = \chi_A = (x-1)^2$ and it is not diagonalisable.

"$A^n = 0$ for some $n$, so $A = 0$." Nilpotent matrices are not zero. All eigenvalues zero does not make the matrix zero — only if it is also diagonalisable.

The exam's angle

1. Compute the characteristic polynomial, then ask what the minimal polynomial could be. Its candidates are the divisors of $\chi_A$ with the same roots, and there are usually only two or three.
2. Use trace and determinant as a checksum. They pin the sum and product of the eigenvalues in seconds and often eliminate half the options.
3. For multiplicity questions, compute $\operatorname{rank}(A - \lambda I)$. Geometric multiplicity is $n$ minus that rank, and it is usually faster than finding eigenvectors.
4. For $2\times2$, write down Cayley–Hamilton immediately. It reduces every polynomial in $A$ to $\alpha A + \beta I$ and answers most computational questions directly.
5. Watch for options about $A + B$ or $AB$. Eigenvalues are not additive or multiplicative across matrices, and this is set often.

The night before

• $\chi_A$ has degree $n$; $\operatorname{tr}A = \sum\lambda_i$, $\det A = \prod \lambda_i$.
• Geometric $\le$ algebraic, always. Equality for every eigenvalue $\iff$ diagonalisable.
• Cayley–Hamilton: $\chi_A(A) = 0$, so $m_A \mid \chi_A$ — not $m_A = \chi_A$.
• $2\times2$: $A^2 - (\operatorname{tr}A)A + (\det A)I = 0$.
• $\chi_A$ and $m_A$ have the same roots, different multiplicities.
• Diagonalisable $\iff$ $m_A$ is a product of distinct linear factors.
• $m_A = \chi_A$ means cyclic, one block per eigenvalue — not diagonalisable.
• Real matrices can have non-real eigenvalues; rotation by $90°$ has $x^2+1$.
• Eigenvalues of $A+B$ are not sums: $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ and its transpose.