The bookUnit 1 · Eigenvalues and Canonical Forms23 / 83

Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Why this is asked: 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.

Characteristic vs minimal polynomial — what each tells you

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See it move

What a matrix does to a vectorinteractive

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

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The trap here

“Same characteristic polynomial ⇒ similar” — false

0 matrix and [[0,1],[0,0]]

Both have char poly $x^{2}$ , different minimal polynomials (x vs $x^{2})$ .