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Exercise 11.8.1

Avg difficulty 3.0

Determine by direct calculation (i.e., without calculating the characteristic polynomial or the eigenvalues) which of the following matrices are positive definite, positive semidefinite, or neither.

A=[1111],B=[1−2−25],C=[2313],D=[211110101],E=[10002000−1].A=\left[\begin{array}{rr}1&1\\ 1&1\\ \end{array}\right],\quad B=\left[\begin{array}{rr}1&-2\\ -2&5\\ \end{array}\right],\quad C=\left[\begin{array}{rr}2&3\\ 1&3\\ \end{array}\right],\quad D=\left[\begin{array}{rrr}2&1&1\\ 1&1&0\\ 1&0&1\\ \end{array}\right],\quad E=\left[\begin{array}{rrr}1&0&0\\ 0&2&0\\ 0&0&-1\\ \end{array}\right].

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Positive semidefinite and positive definite matrices, Exercise 11.8.1

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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.