复特征值

Introduction

Matrices with Complex Eigenvalues

Rotation-Scaling Matrices

Geometry of $2\times 2$ Matrices with a Complex Eigenvalue

Block Diagonalization

A real $2\times2$ matrix $A$ has $2+3i$ as an eigenvalue. Which of the following must be true?

Let $R$ be the rotation by $90^\circ$: $R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Which statement is true?

In one word: the numbers $a+bi$ and $a-bi$ are called ______ of each other.

The matrix $A = \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$ has eigenvalues $1 \pm bi$. Enter the number $b$ (the positive imaginary part, a number).

Find the eigenvalues of

entered with the positive-imaginary eigenvalue first, separated by a comma (for example: 2i,-2i).

A real $2\times 2$ matrix $A$ has $2 + 3i$ as an eigenvalue. Which statement is correct?

The matrix $A = \begin{pmatrix} 0 & -3 \\ 3 & 0 \end{pmatrix}$ has eigenvalues $\lambda = \pm\,bi$ for some positive number $b$. Enter $b$.