Earlier I asked a question concerning the spectrum of a certain augmented system $$M=\left[\begin{array}{cc}I & A\\ A^T & 0\end{array}\right],$$ where $A$ is an invertible real $n\times n$ matrix, and where the singular values of A are in the interval $[1,\kappa_2(A)]$. $\kappa_2(A)$ denotes the (2-norm) condition number of A. The spectrum was found to be lying inside $$\mathcal I=\left[\frac{1-\sqrt{1+4\kappa(A)^2}}2,\ \frac{1-\sqrt{5}}2\right]\cup \left[\frac{1+\sqrt{5}}2,\ \frac{1+\sqrt{1+4\kappa(A)^2}}2\right].$$ Does this generalize further if we add the normal equations as well? That is, can we still use the block determinant formula trick to find an interval for the spectrum of the following: $$\widetilde M=\left[\begin{array}{cc}I+AA^T & A\\ A^T & A^TA\end{array}\right]$$
1 Answers
You can get expressions for the eigenvalues of $M$ and $\tilde{M}$ in terms of the singular values of $A$. Let $A=USV^T$ be the SVD of $A$. Then $$ M\sim \begin{bmatrix}I & S \\ S & 0\end{bmatrix} \quad\text{and}\quad \tilde{M}\sim\begin{bmatrix}I+S^2 & S \\ S & S^2\end{bmatrix}, $$ where $X\sim Y$ means "$X$ is similar to $Y$" (in this case orthogonally). In particular, $$ M=\begin{bmatrix}U&0\\0&V\end{bmatrix}\begin{bmatrix}I & S \\ S & 0\end{bmatrix}\begin{bmatrix}U&0\\0&V\end{bmatrix}^T \quad\text{and}\quad \tilde{M}=\begin{bmatrix}U&0\\0&V\end{bmatrix}\begin{bmatrix}I+S^2 & S \\ S & S^2\end{bmatrix}\begin{bmatrix}U&0\\0&V\end{bmatrix}^T, $$ The eigenvalues of $M$ and $\tilde{M}$ are thus simply the eigenvalues of the $2\times 2$ matrices $$ \begin{bmatrix} 1 & \sigma_i \\ \sigma_i & 0\end{bmatrix} \quad\text{and}\quad \begin{bmatrix}1 + \sigma_i^2 & \sigma_i \\ \sigma_i & \sigma_i^2\end{bmatrix}, \quad i=1,\ldots,n, $$ respectively, where $\{\sigma_i\}_{i=1}^n$ are the singular values of $A$.
Therefore, $$ \lambda(M)=\bigcup_{i=1}^n\left\{\frac{1}{2}\left(1\pm\sqrt{1+4\sigma_i^2}\right)\right\} \quad\text{and}\quad \lambda(\tilde{M})=\bigcup_{i=1}^n\left\{\frac{1}{2}\left(1\pm\sqrt{1+4\sigma_i^2}\right)+\sigma_i^2\right\}. $$ The desired bounds you can hence obtain by examining the behaviour of the (four) functions $$ f_{M,\pm}(\sigma)=\frac{1}{2}\left(1\pm\sqrt{1+4\sigma^2}\right) \quad\text{and}\quad f_{\tilde{M},\pm}(\sigma)=\frac{1}{2}\left(1\pm\sqrt{1+4\sigma^2}\right)+\sigma^2 $$ for $\sigma\in[1,\kappa(A)]$. Both $f_{\tilde{M},+}$ and $f_{\tilde{M},-}$ seem to be monotonically increasing on the considered interval, so my tip for the spectrum bounds for the second matrix would be $$ \lambda(\tilde{M})\subset\left[\frac{3-\sqrt{5}}{2},\frac{2\kappa^2(A)+\sqrt{1+4\kappa^2(A)}}{2}\right]. $$
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