相似

Introduction

Similar Matrices

Geometry of Similar Matrices

Eigenvalues of Similar Matrices

Suppose the matrices $A$ and $B$ are similar, $A = CBC^{-1}$ with $C$ invertible. Which statement is true?

Which of the following matrices is similar to $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$?

In one word: two square matrices $A$ and $B$ such that $A = CBC^{-1}$ for some invertible matrix $C$ are called ______ .

The matrices $A$ and $B$ are similar, and the characteristic polynomial of $A$ is $\lambda^2 - 3\lambda + 2$. Enter the eigenvalues of $B$ in increasing order, separated by commas (for example: 1,2).

Let $J = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ and let $M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ (the reversal matrix, with $M^{-1} = M$). Compute $M^{-1}JM$ and enter the entry in the first row, second column (a number).

The matrices $A$ and $B$ are similar, meaning $A = CBC^{-1}$ for an invertible matrix $C$. Which statement must hold?

A matrix $A$ is similar to the diagonal matrix $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$. Enter $\det(A)$.