# Singular values of block lower triangular matrix

Given a fat matrix $$A \in \mathbb{R}^{m \times n}$$ (where $$m) with singular values $$\sigma_1, \dots, \sigma_m$$, can I express the singular values $$\tilde{\sigma}_1, \dots, \tilde{\sigma}_{m+n}$$ of the following block lower triangular matrix

$$T = \begin{pmatrix} I & 0 \\ A & I \end{pmatrix}$$

in terms of the singular values of $$A$$? Or can I give some upper bounds on the singular values of $$T$$?

Kind regards and thanks in advance!

EDIT:

I think I got a solution myself: on a standard textbook on matrix analysis I found the property $$\sigma_i(C) - \sigma_i(B) \leq \Vert C - B \Vert$$ for all $$i$$ with $$A,B \in \mathbb{R}^{m \times n}$$.

Now, defining $$C = T$$ and $$B = \begin{pmatrix} 0 & 0 \\ A & 0 \end{pmatrix}$$ we have $$C - B = I$$ and $$\sigma_{max}(B) = \sigma_{max}(A)$$. Since $$I$$ has norm 1, then, we have

$$\sigma_{max}(T) = \sigma_{max}(C) \leq 1 + \sigma_{max}(A).$$

Am I forgetting anything? It looks too beautiful to be true.

• I think I got a solution myself: on a standard textbook on matrix analysis I found the property $\sigma_i(C) - \sigma_i(B) \leq \Vert C - B \Vert$ for all $i$ with $A,B \in \mathbb{R}^{m \times n}$. Now, defining $C = T$ and $$B = \begin{pmatrix} 0 & 0 \\ A & 0 \end{pmatrix}$$ we have $C - B = I$ and $\sigma_{max}(B) = \sigma_{max}(A)$. Since $I$ has norm 1, then, we have $$\sigma_{max}(T) = \sigma_{max}(C) \leq 1 + \sigma_{max}(A).$$ Am I forgetting anything? It looks too beautiful to be true.
– Trb2
Commented Nov 28, 2020 at 0:11
• Feel free to edit that in your question. Commented Nov 28, 2020 at 0:12
• @Argyll what should I edit?
– Trb2
Commented Nov 28, 2020 at 0:13
• Eigenvalues of $T T^\top$. Commented Nov 28, 2020 at 22:32
• @Trb2 I obtained the same (or close). I am not very happy with it either. Please consider editing your question again and including the work in your comment. Commented Dec 5, 2020 at 10:52

The upper bound is trivial since

$$\sigma_{\max}(B) = \|B\|_2 = \|I + E \|_2 \le 1 + \|E\|_2 = 1 + \|A\|_2 = 1 + \sigma_{\max}(A).$$