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Exercise 11.7.1

Avg difficulty 3.0

For each of the following symmetric matrices, find the eigenvalues, an orthonormal basis for each eigenspace, and then orthogonally diagonalize the matrix.

(a)A=[−1222],(b)A=[3113],(c)A=[110121011],(d)A=[212122225].(a)\quad A=\left[\begin{array}{rr}-1&2\\ 2&2\\ \end{array}\right],\quad(b)\quad A=\left[\begin{array}{rr}3&1\\ 1&3\\ \end{array}\right],\quad(c)\quad A=\left[\begin{array}{rrr}1&1&0\\ 1&2&1\\ 0&1&1\\ \end{array}\right],\quad(d)\quad A=\left[\begin{array}{rrr}2&1&2\\ 1&2&2\\ 2&2&5\\ \end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Diagonalization of real symmetric matrices, Exercise 11.7.1

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.