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Exercise 11.8.2

Avg difficulty 3.0

Calculate the eigenvalues of each symmetric matrix, then determine for each matrix whether it is positive definite, positive semidefinite, or neither.

A=[2225],B=[4−6−69],C=[21011−10−12],D=[3−42−440204].A=\left[\begin{array}{rr}2&2\\ 2&5\\ \end{array}\right],\quad B=\left[\begin{array}{rr}4&-6\\ -6&9\\ \end{array}\right],\quad C=\left[\begin{array}{rrr}2&1&0\\ 1&1&-1\\ 0&-1&2\\ \end{array}\right],\quad D=\left[\begin{array}{rrr}3&-4&2\\ -4&4&0\\ 2&0&4\\ \end{array}\right].

is positive definite, positive semidefinite, or neither.

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Positive semidefinite and positive definite matrices, Exercise 11.8.2

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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.