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Back to Symmetric matrices and variance

Exercise 1

Avg difficulty 2.5

For each of the following matrices, find the eigenvalues and a basis for each eigenspace. Determine whether the matrix is diagonalizable and, if so, find a diagonalization. Determine whether the matrix is orthogonally diagonalizable and, if so, find an orthogonal diagonalization.

  1. [51−13]\begin{bmatrix} 5 & 1 \\ -1 & 3 \\ \end{bmatrix}

  2. [0110]\begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}

  3. [1002−20014]\begin{bmatrix} 1 & 0 & 0 \\ 2 & -2 & 0 \\ 0 & 1 & 4 \\ \end{bmatrix}

  4. [25−45−75−452]\begin{bmatrix} 2 & 5 & -4\\ 5 & -7 & 5 \\ -4 & 5 & 2 \\ \end{bmatrix}

Austin, Understanding Linear Algebra, §7.1 Symmetric matrices and variance, Exercise 1

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.