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assimil anglais perfectionnement audio 1Q:

Square root of matrix being reconstructed out of orthogonal projection

Let $A$ be an $n\times n$ self-adjoint matrix and let $x\in \mathbb{R}^n$ be a column vector with a single nonzero element $a$. I wish to prove that the vector
$$y:=\sqrt{a}\, x$$
appears as a linear combination of a subset of the columns of $A^\frac12$.
Can anyone point me to a suitable reference?

A:

Theorem. Let $A$ be an $n \times n$ selfadjoint matrix, $
u \in {\mathbb C}^n$, $\|
u\| = 1$, $|a| = \frac{1}{2}$ and $y = \sqrt{a}\,x$. Then $Ay = \sqrt{a}Ay = \sqrt{a}A
u$.
Proof. We have
\begin{align}
\sqrt{a}Ay &= \sqrt{a}\langle y, Ay \rangle y \\
&= \sqrt{a}\langle y, A
u \rangle y\\
&= \sqrt{a}\langle y, A
u \rangle
u\\
&= \langle y, A
u \rangle \sqrt{a}
u\\
&= \langle x, Ax \rangle \sqrt{a}
u
\end{align}
where we have used the fact that the matrix norm of the positive definite matrix $AA^\ast = A$ is equal to the norm of the vector.

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