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特征多项式

1. Introduction

Introduction

2. Characteristic polynomial of a 2x2

Which of the following is the characteristic polynomial of $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ ?

3. Eigenvalues of a triangular matrix

What are the eigenvalues of the upper triangular matrix

A = \begin{pmatrix} 1 & 7 & 2 & 4 \\ 0 & 1 & 3 & 11 \\ 0 & 0 & \pi & 101 \\ 0 & 0 & 0 & 0 \end{pmatrix}.

4. Vocabulary: characteristic polynomial

In one word: the polynomial $\det(A - \lambda I)$ is called the ______ polynomial of $A$ .

5. Eigenvalues of a triangular 2x2

Find the eigenvalues of $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ , entered in increasing order, separated by commas (for example: 1,2).

6. Eigenvalues of a 3x3

Find the eigenvalues of

A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix},

entered in increasing order, separated by commas (for example: 1,2,3).

7. Eigenvalues as roots

Which equation characterizes the eigenvalues of a square matrix $A$ ?

8. Roots of a characteristic polynomial

A $2\times 2$ matrix has characteristic polynomial $\lambda^2 - \lambda - 6$ . Enter its two eigenvalues in increasing order as $\lambda_1,\lambda_2$ .

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