Linear Algebra and Its Applications · Problem 6.2.30

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

Without multiplying

A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 0 & 5 \end{bmatrix} \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}

, find (a) the determinant of

A

. (b) the eigenvalues of

A

. (c) the eigenvectors of

A

. (d) a reason why

A

is symmetric positive definite.