Exact Frobenius moments of wide Gaussian pseudoinverses

Write $G=\left[\begin{array}{c}g\\ H\end{array}\right]$ with first row $g$ and remaining $k-1$ rows $H$. If $k=1$, set $P=I_{k+p}$; then $G{G}^{\ast }=g{g}^{\ast }=gP{g}^{\ast }$ and the inverse-diagonal identity is immediate. If $k>1$, [F2] gives full row rank of $H$, so $P=I-H^{\ast }(H{H}^{\ast }{)}^{-1}H$ is the orthogonal projector onto $\ker H$, of dimension $(k+p)-(k-1)=p+1$. To obtain the inverse entry without an assumed block-inverse formula, solve $G{G}^{\ast }\left(\genfrac{}{}{0px}{}{x}{y}\right)=\left(\genfrac{}{}{0px}{}{1}{0}\right)$. The lower equation gives $y=-(H{H}^{\ast }{)}^{-1}H{g}^{\ast }x$, and substitution into the upper equation gives $(gP{g}^{\ast })x=1$. Full row rank of $G$ from [F2] makes $G{G}^{\ast }$ invertible and $gP{g}^{\ast }>0$, hence $((G{G}^{\ast }{)}^{-1}{)}_{11}=x=(gP{g}^{\ast }{)}^{-1}$.

For each fixed $H$, choose orthonormal coordinates in which $P$ is diagonal with $p+1$ ones. Independence of $g$ from $H$ and invariance [F7] show that $gP{g}^{\ast }$ is a sum of $p+1$ independent squared real Gaussians, or squared moduli of normalized complex Gaussians. Apply [F10] separately on the open half-lines via $x=\sqrt{u}$ and $x=-\sqrt{u}$ (absolute derivative $1\mathrm{/}(2\sqrt{u})$); the omitted point has zero measure. Adding the two densities gives the real square density $u^{-1\mathrm{/}2}{e}^{-u\mathrm{/}2}\mathrm{/}\sqrt{2\pi }=u^{-1\mathrm{/}2}{e}^{-u\mathrm{/}2}\mathrm{/}(\sqrt{2}\mathrm{\Gamma }(1\mathrm{/}2))$ by [F6]. For the complex square law, [F11] applied to the normalized density gives $1={\pi }^{-1}\sigma (S^1){\int }_0^{\mathrm{\infty }}{e}^{-r^2}rdr=\sigma (S^1)\mathrm{/}(2\pi )$, so $\sigma (S^1)=2\pi$. Applying the same formula with any nonnegative Borel function of $r^2$, and then [F10] with $u=r^2$, shows that the squared modulus has density $e^{-u}$ on $(0,\mathrm{\infty })$. The first real square is thus Gamma$(1\mathrm{/}2,2)$ and the complex square Gamma$(1,1)$, the latter normalized by [F12]. If independent Gamma variables have common scale $\theta$ and shapes $a,b$, [F5] and the diffeomorphism $(u,v)\mapsto (u,s=u+v)$ of the positive quadrant onto $0<u<s$, of determinant one in [F10], show that their sum has convolution density with normalization $1\mathrm{/}({\theta }^{a+b}\mathrm{\Gamma }(a)\mathrm{\Gamma }(b))$ and numerator $e^{-s\mathrm{/}\theta }{\int }_0^s{u}^{a-1}(s-u{)}^{b-1}du$. For each $s>0$, substituting $u=st$ by [F10] and using [F3] makes this $e^{-s\mathrm{/}\theta }{s}^{a+b-1}B(a,b)$, which is the Gamma density of shape $a+b$ and scale $\theta$. Iterating this finite convolution gives $gP{g}^{\ast }\sim \mathrm{\Gamma }((p+1)\mathrm{/}2,2)$ over $\mathbb{R}$ and $gP{g}^{\ast }\sim \mathrm{\Gamma }(p+1,1)$ over $\mathbb{C}$. Tonelli [F5] integrates first over $g$ and then over $H$, so the reciprocal law holds unconditionally for the first diagonal entry; row permutation invariance gives the same law for every index.

Integrating the Gamma density with the scaling $u=\theta t$ from [F10] yields $\mathbb{E}{U}^{-1}={\theta }^{-1}\mathrm{\Gamma }(a-1)\mathrm{/}\mathrm{\Gamma }(a)$ for $U\sim \mathrm{\Gamma }(a,\theta )$ and $a>1$. Using the Gamma recurrence [F12] gives $\mathrm{\Gamma }(a)=(a-1)\mathrm{\Gamma }(a-1)$. Thus step 2.1 gives $\mathbb{E}((G{G}^{\ast }{)}^{-1}{)}_{11}=1\mathrm{/}(p-1)$ over $\mathbb{R}$ and $1\mathrm{/}p$ over $\mathbb{C}$.

By step 2.1 all inverse diagonal entries have the same reciprocal-Gamma law and finite mean in the stated ranges, so $\mathbb{E}\mathrm{tr}(G{G}^{\ast }{)}^{-1}$ is $k$ times the value in step 3.1. Put $B=G^{\ast }(G{G}^{\ast }{)}^{-1}$. Then $GB=I_k$, $BG$ is self-adjoint and idempotent, so $GBG=G$, $BGB=B$, $(GB{)}^{\ast }=GB$, and $(BG{)}^{\ast }=BG$; these are the four Penrose equations [F8], hence uniqueness [F9] gives $B=G^{†}$. Also $B^{\ast }B=(G{G}^{\ast }{)}^{-1}$, and therefore $\mathrm{\parallel }{G}^{†}{\mathrm{\parallel }}_F^2=\mathrm{tr}(G{G}^{\ast }{)}^{-1}$ almost surely.