# Symmetric Matrices and orthognal

• Apr 29th 2005, 12:58 PM
akscola
Symmetric Matrices and orthogonal
Hi,
I want to show that if A(nxn) is symmetric then it is orthogonally equivalent to a symmetric matrix wit equal elements in the diagonal.For example for 2x2 if A=[a b;b c] there exist a Q O(2) (Q is orthogonal matrix)such thet :

Q 'AQ=[d e;e d]

Something which I have found till now :

1-In case 2x2 if orthogonal matrix is Q= [cos(teta) -sin(teta);sin(teta) cos(teta)] and A=[a b;b c] then for teta=.5*atan(c-a)/2b Q'AQ=[ f h;h f].

2-Since A is symmetric then it can be written as A=V'YV such that Y=V'AV and Y is a diagonal mat with eigen value on its diagonal and then same story for Y which is easier to find a Q matrix and then define a function f same as above and ....

3-It is possible that we may not find a general form for nxn.We may define f:R(nxn)----->R f(x)=SUM(yii-yjj) , Y=X'AX and note that f is a continuous function.And then we have to prove that f has a global minima Q* in O(n) and for that f(Q*)=0
• Apr 29th 2005, 07:04 PM
hpe
Quote:

Originally Posted by akscola
Hi,
I want to show that if A(nxn) is symmetric then it is orthognally equvalent to a symmetric matrix wit equal elements in the diagonal.

That sounds all OK. The issue is now whether for a given diagonal matrix D you can find an orthogonal matrix X such that B = XDX' is symmetric (that's automatic) and constant along the diagonal. For this purpose, you might try to make a tridiagonal matrix B. It has the same number of parameters as an arbitrary diagonal matrix.