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Section8.1Eigenvalues and Eigenvectors

SubsectionExercises

Compute the eigenvalues and eigenvectors of the following matrices. Eigenvalues should match exactly with the answers given below, but eigenvectors are equivalent up to scalar multiplication, so there is no single correct eigenvector for a given eigenvalue.

1

\(\begin{bmatrix} 8 \amp -6 \\ 3 \amp -1 \end{bmatrix}\)

Answer
2

\(\begin{bmatrix} 7 \amp -6 \\ 12 \amp -10 \end{bmatrix}\)

Answer
3

\(\begin{bmatrix} 2 \amp 0 \amp 0 \\ 2 \amp -2 \amp -1 \\ -2 \amp 6 \amp 3 \end{bmatrix}\)

Answer
4

\(\begin{bmatrix} 4 \amp -3 \amp 1 \\ 2 \amp -1 \amp 1 \\ 0 \amp 0 \amp 2 \end{bmatrix}\)

Answer

Find the complex conjugate eigenvalues and corresponding complex eigenvectors of the following matrices. Note that not only do eigenvalues come in complex conjugate pairs, eigenvectors will be complex conjugates of each other as well. Thus you only need to compute one eigenvector, the other eigenvector must be the complex conjugate.

5

\(\begin{bmatrix} 0 \amp 1 \\ -1 \amp 0 \end{bmatrix}\)

Answer
6

\(\begin{bmatrix} 0 \amp -3 \\ 12 \amp 0 \end{bmatrix}\)

Answer
7

Prove that if \(\lambda = 0\) is an eigenvalue of the matrix \(A\text{,}\) then \(A\) is not invertible.

Hint
8

The following idea is crucial to the Google Page Rank algorithm. The proof relies upon Mathematical Induction.

Let \(\lambda\) be an eigenvalue of the matrix \(A\) with corresponding eigenvector \(\vec{v}\text{,}\) show that for any \(n = 1, 2, 3, \ldots\text{,}\) \(\lambda^n\) is an eigenvector of \(A^n\) with eigenvector \(\vec{v}\text{.}\)

Hint