# Understanding the Vector Space of Polynomials of a Certain Type of Matrix [closed]

Let $$A$$ be a $$4 \times 4$$ complex diagonal matrix with exactly three distint entries on its diagonal.

(1) What is the dimension of the vector space of polynomials of $$A$$?

(2) What is the dimension of the vector space of $$4 \times 4$$ complex matrices that commute with $$A$$?

(3) If $$B$$ is a $$4 \times 4$$ complex diagonal matrix with exactly three distinct entries on its diagonal, is it similar to a polynomial of $$A$$?

I am looking for explainations more so than the actual answers.

## closed as off-topic by Brahadeesh, KReiser, Alexander Gruber♦Nov 30 '18 at 3:08

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• What are your thoughts on the problem? What have you tried? – Omnomnomnom Nov 29 '18 at 20:38
• Are you familiar with Jordan canonical form and with minimal polynomials? – Omnomnomnom Nov 29 '18 at 20:38
• @Omnomnomnom I have some familarity with Jordan canonical form. – LinearGuy Nov 29 '18 at 20:40

## 1 Answer

In order to make writing things easier, I will specifically consider the case where the repeated eigenvalue comes first. That is, we have $$A = \pmatrix{\lambda_1 \\ & \lambda_1 \\ && \lambda_2 \\ &&& \lambda_3} = \pmatrix{\lambda_1 I_{2}\\ & \lambda_2 \\ && \lambda_3}$$ where $$I_2$$ denotes a size $$2$$ identity matrix.

Hint for 1: Note that for any polynomial $$p$$, $$p(A) = \pmatrix{p(\lambda_1)I_{2} \\ & p(\lambda_2) \\ && p(\lambda_3)}$$

Hint for 2: Verify that any block matrix of the form $$B = \pmatrix{B_1\\ & b_2 \\ && b_3}$$ will commute with $$A$$, where $$B_1$$ can be any $$2 \times 2$$ matrix and the $$b_i$$ are scalars. Note too that these are the only matrices that commute with $$A$$.

Hint for 3: Every polynomial of $$A$$ is diagonal (and therefore diagonalizable). However, the matrix $$\pmatrix{\lambda_1&1 \\ & \lambda_1 \\ && \lambda_2 \\ &&& \lambda_3}$$ is neither diagonal nor diagonalizable.