# Singular value decomposition

• Apr 16th 2011, 08:28 PM
alexmahone
Singular value decomposition
Suppose $\displaystyle A$ is a 2 by 2 symmetric matrix with unit eigenvectors $\displaystyle u_1$ and $\displaystyle u_2$. If its eigenvalues are 3 and -2, what are the matrices $\displaystyle U$, $\displaystyle \sum$, $\displaystyle V'$ in its SVD?

($\displaystyle V'$ is the transpose of $\displaystyle V$.)
• Apr 16th 2011, 11:27 PM
FernandoRevilla
Hint

The eigenvalues of A^hA are 9 and 4 so, the singular values of A are 3 and 2 .
• Apr 16th 2011, 11:53 PM
alexmahone
Quote:

Originally Posted by FernandoRevilla
Hint

The eigenvalues of A^hA are 9 and 4 so, the singular values of A are 3 and 2 .

Thanks. So I get:

http://quicklatex.com/cache3/ql_4d8e...4a997cf_l3.png

Is that right?
• Apr 17th 2011, 01:56 AM
FernandoRevilla
Quote:

Originally Posted by alexmahone
Is that right?

Using the standard method, the factorization is

[ u_1, -u_2 ]^t A [ u_1, u_2 ] = diag [ 3, 2 ]
• Apr 17th 2011, 02:52 AM
alexmahone
Quote:

Originally Posted by FernandoRevilla
Using the standard method, the factorization is

[ u_1, -u_2 ]^t A [ u_1, u_2 ] = diag [ 3, 2 ]

Are you saying that my answer in post #3 is wrong? (Sorry for being so slow; I'm self studying Linear Algebra from Strang and he isn't very clear most of the time.)
• Apr 17th 2011, 04:02 AM
FernandoRevilla
Quote:

Originally Posted by alexmahone
Are you saying that my answer in post #3 is wrong? (Sorry for being so slow; I'm self studying Linear Algebra from Strang and he isn't very clear most of the time.)

Your answer is right. I also provided the matrix U , that is

V = [ u_1, - u_2] , U = [ u_1 , u_2 ]