= Push-through identity
For a complex $m$-by-$n$ <matrix> $X$ and a real number $\lambda>0$, the push-through identity is
$$
X^*(XX^*+\lambda I_m)^{-1}=(X^*X+\lambda I_n)^{-1}X^*.
$$
The star is the <conjugate transpose>; for a real <matrix>, replace it by the <matrix transpose>. Both inverse <matrices> exist because the shifted products are <positive-definite matrices>: for example,
$$
v^*(X^*X+\lambda I_n)v=\|Xv\|^2+\lambda\|v\|^2>0\qquad(v\ne0).
$$
<Associativity> gives $(X^*X+\lambda I_n)X^*=X^*(XX^*+\lambda I_m)$. Multiplying by the respective inverse <matrices> on the left and right proves the identity. It permits a calculation involving an $m$-by-$m$ inverse to be rewritten using an $n$-by-$n$ inverse, or conversely, choosing the smaller dimension when convenient.
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