進階矩陣代數：練習題

The matrix \( \mathbf{A} \) is defined as \( \mathbf{A} = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix} \). Find the set of all eigenvalues for \( \mathbf{A} \).

Given that \( \lambda = -2 \) is an eigenvalue of the matrix \( \mathbf{A} = \begin{pmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{pmatrix} \), find a normalized eigenvector corresponding to this eigenvalue.

Given that \( \mathbf{A} \) is a \( 3 \times 3 \) non-singular matrix such that \( \det(\mathbf{A}) = 4 \), determine the exact value of \( \det(2\mathbf{A}^{-1}) \).

The symmetric matrix \( \mathbf{M} \) is defined as \( \mathbf{M} = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \). Find an orthogonal matrix \( \mathbf{P} \) such that \( \mathbf{P}^{\text{T}}\mathbf{M}\mathbf{P} \) is a diagonal matrix.

Find the inverse of the matrix \( \mathbf{M} = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 3 & 0 & 1 \end{pmatrix} \).

Find the eigenvalues of the matrix \( \mathbf{C} = \begin{pmatrix} 5 & -2 \\ 6 & -2 \end{pmatrix} \) and determine a corresponding eigenvector for each eigenvalue.

A \( 3 \times 3 \) matrix \( \mathbf{A} \) is given by \( \mathbf{A} = \begin{pmatrix} 2 & 2 & -3 \\ 2 & 2 & 3 \\ -3 & 3 & 3 \end{pmatrix} \).
(a) Given that \( \mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} \) is an eigenvector of \( \mathbf{A} \), find its corresponding eigenvalue.
(b) Find the other two eigenvalues of \( \mathbf{A} \).