Paper 1, Section II, A

Let $A$ be a real, symmetric $n \times n$ matrix.

We say that $A$ is positive semi-definite if $\mathbf{x}^{T} A \mathbf{x} \geqslant 0$ for all $\mathbf{x} \in \mathbb{R}^{n}$. Prove that $A$ is positive semi-definite if and only if all the eigenvalues of $A$ are non-negative. [You may quote results from the course, provided that they are clearly stated.]

We say that $A$ has a principal square root $B$ if $A=B^{2}$ for some symmetric, positive semi-definite $n \times n$ matrix $B$. If such a $B$ exists we write $B=\sqrt{A}$. Show that if $A$ is positive semi-definite then $\sqrt{A}$ exists.

Let $M$ be a real, non-singular $n \times n$ matrix. Show that $M^{T} M$ is symmetric and positive semi-definite. Deduce that $\sqrt{M^{T} M}$ exists and is non-singular. By considering the matrix

$M\left(\sqrt{M^{T} M}\right)^{-1}$

or otherwise, show $M=R P$ for some orthogonal $n \times n$ matrix $R$ and a symmetric, positive semi-definite $n \times n$ matrix $P$.

Describe the transformation $R P$ geometrically in the case $n=3$.

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