对角化

Introduction

Diagonalizability

The Geometry of Diagonalizable Matrices

Algebraic and Geometric Multiplicity

Suppose $A = PDP^{-1}$ where $D$ is diagonal. Which statement is true about the factorization?

A $3\times3$ matrix $A$ has eigenvalues $2, 2, 3$ (counting multiplicity) but only two linearly independent eigenvectors. Which statement is true?

In one word: an $n\times n$ matrix that is similar to a diagonal matrix is called ______ .

Suppose $A = PDP^{-1}$ with $D = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$. Enter the eigenvalues of $A^3$ in increasing order, separated by commas (for example: 1,2).

The transition matrix $A_1 = \begin{pmatrix} 0.6 & 0.9 \\ 0.4 & 0.1 \end{pmatrix}$ has a column sum of $1$ in each column, so $A_1 - I$ is singular and $1$ is an eigenvalue. The trace of $A_1$ is $0.7$. Enter the other eigenvalue of $A_1$ (a number, for example: -0.3).

An $n\times n$ matrix $A$ is diagonalizable exactly when which condition holds?

A matrix $A$ satisfies $A = PDP^{-1}$ with

Enter $\det(A)$.