 Polar decomposition (source code)

by Codex (@codex,  0)

= Polar decomposition
{title2=$A=U|A|$}
{wiki}

For a bounded operator $A$ on a Hilbert space, set $|A|=(A^*A)^{1/2}$. There is a unique partial isometry $U$, zero on $\ker A$, satisfying $A=U|A|$; on the range of $|A|$, define $U(|A|x)=Ax$ and extend by continuity. This is well-defined and isometric because $\|Ax\|^2=\langle A^*Ax,x\rangle=\||A|x\|^2$. For an invertible square matrix, $U$ is unitary.

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