## Relation between eigenvalue and singular value

For any square matrix A, we have

where are the singular values, and are the eigenvalues.

**Proof of the right half of the inequality:**

See Topics in matrix analysis: page 151, Corollary 3.1.5

**Proof of the left half of the inequality:**

First, we have

To verify, it is easy to see and .

Second, applying the right half of the inequality gives

## Matlab mod

if you would like to achieve something like this:

1 2 3 4->1 2 3 4

-1 0->2 4

5 6->1 2

you should use

**mod(i-1,n)+1**

where n=4

## Graphical meaning of power of adjacency matrix

## Set default initial page view in adobe reader

It’s not that easy, but I finally fixed it.

There are three options provided by Adobe Reader.

- File->properties->Initial view
- Edit->preferences->Page display->Default layout and zoom
- Edit->preferences->Accessibility->Override page display

Many people recommend the second way, but it does not work for me.

The first option works, but only for one single PDF.

The third one works for all PDF for me!!! I found it here

http://superuser.com/questions/349971/how-do-i-set-default-view-preferences-in-adobe-acrobat

## Summary: Useful Tikz commands

%% basics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % define a variable \def\length{3} % coordinate \coordinate (A) at (0,0); % relative coordinate: example \draw ($(A)+(0,1)$) -- ($(A)+(1,0)$); % arrow and text \draw[->,thin] (A)--(A1) node[above] {$x$}; % the text x is beside the node A1 % arrow types -> ->> -latex -stealth -angle 90 % looks like -> % path and text \path (1) edge[bend left=7,->,sloped] node[near start]{\contour{white}{\tiny -1} }(2);

## Skew-symmetric in the complex case

Suppose S is a real skew-symmetric matrix; H is a real symmetric matrix.

For a real vector x, we have . But for a complex vector x, we do not have . Instead, we have

- is imaginary since (if the conjugate of a complex number has the opposite sign, then the complex number is imaginary)
- is real since (if the conjugate of a complex number is itself, then it is real)

## Monotonicity of binomial coefficients

Here are two basic identities about the monotonicity of binormial coefficients

- for any

In addition,

For proof, see the post in stackexchange by Mario

- for any

For proof, see the post in stackexchange