Question

Suppose $Q$ is an orthogonal matrix. (a) Prove that $K=2 \mathrm{I}-Q-Q^T$ is a positive semi-definite matrix. (b) Under what conditions is $K>0$ ?

   Suppose $Q$ is an orthogonal matrix. (a) Prove that $K=2 \mathrm{I}-Q-Q^T$ is a positive semi-definite matrix. (b) Under what conditions is $K>0$ ?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 8, Problem 46

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Step 1

An orthogonal matrix $Q$ satisfies $Q^T Q = Q Q^T = I$, where $I$ is the identity matrix and $Q^T$ is the transpose of $Q$. This implies that $Q^T = Q^{-1}$. Show more…

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Suppose $Q$ is an orthogonal matrix. (a) Prove that $K=2 \mathrm{I}-Q-Q^T$ is a positive semi-definite matrix. (b) Under what conditions is $K>0$ ?

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