What are the singular values of a matrix multiplied by the identity matrix?

[azdang]

Homework Statement

Let A be a real mxn matrix, m>=n, with singular values $$\sigma$$_j.Show that the singular values of ($$\stackrel{I_{n}}{A}$$) are equal to $$\sqrt{1+\sigma_j^2}$$.

Homework Equations

The Attempt at a Solution

I know that an SVD for A is A = U($$\stackrel{\Sigma}{0}$$)v^T and so, the singular values of A are $$\sigma_j$$. I have no idea how to break this down. I assume I want to look at an SVD for ($$\stackrel{I_n}{A}$$), but I don't know how to figure out that the singular values would be $$\sqrt{1+\sigma_j^2}$$. Does anyone have any ideas? Thanks so much.

 

[azdang]

Also, sorry, I'm having a hard time figuring out how to have it typeset correctly to show you guys what's going on.

 

[HallsofIvy]

Don't use the html tags ^{and _{inside LaTex. Use _ for subscripts and ^ for superscripts.}}

 

[azdang]

Oh wow, thank you so much. It looks great.