Push-through identity

For a complex $m$-by-$n$ matrix $X$ and a real number $\lambda >0$, the push-through identity isThe 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,Associativity gives $(X^{\ast }X+\lambda I_n)X^{\ast }=X^{\ast }(X{X}^{\ast }+\lambda I_m)$. 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.