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Relation between eigenvalue and singular value

July 11, 2016

For any square matrix A, we have

-\sigma_{n-k+1}(A) \le \lambda_k\left(\frac{A+A^T}{2}\right) \le \sigma_k(A)

where 0\le\sigma_n\le\dots\le\sigma_1 are the singular values, and \lambda_n\le\dots\le \lambda_1 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 \lambda_1\left(\frac{-A-A^T}{2}\right)=-\lambda_{n}\left(\frac{A+A^T}{2}\right) and \lambda_n\left(\frac{-A-A^T}{2}\right)=-\lambda_{1}\left(\frac{A+A^T}{2}\right).

Second, applying the right half of the inequality gives


Matlab mod

June 29, 2016

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


where n=4

Graphical meaning of power of adjacency matrix

June 6, 2016


Set default initial page view in adobe reader

May 23, 2016

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

There are three options provided by Adobe Reader.

  1. File->properties->Initial view
  2. Edit->preferences->Page display->Default layout and zoom
  3. 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


Summary: Useful Tikz commands

May 19, 2016
%% basics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% define a variable

% 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
-angle 90 % looks like ->
% path and text 
\path (1) edge[bend left=7,->,sloped] node[near start]{\contour{white}{\tiny -1}  }(2);

Read more…

Skew-symmetric in the complex case

March 8, 2016

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

For a real vector x, we have x^TSx=0. But for a complex vector x, we do not have x^*Sx=0. Instead, we have

  • x^*Sx is imaginary since (x^*Sx)^*=x^*S^*x=-x^*Sx (if the conjugate of a complex number has the opposite sign, then the complex number is imaginary)
  • x^*Hx is real since (x^*Hx)^*=x^*Hx (if the conjugate of a complex number is itself, then it is real)





Monotonicity of binomial coefficients

March 1, 2016

Here are two basic identities about the monotonicity of binormial coefficients