Show that this orthogonal diagonalization is a singular value decomposition.

In summary, the problem is to prove that if a positive definite symmetric matrix A is of size nxn, then it can be orthogonally diagonalized as A = PDP', where P' = transpose(P), and this is equivalent to a singular value decomposition of the form A = UEV', where E is an nxn matrix containing the eigenvalues of A. The key concept to use in the proof is the fact that A has an orthonormal set of n eigenvectors, which are the same as those of A^T.
  • #1
s_j_sawyer
21
0

Homework Statement



Prove that if A is an nxn positive definite symmetric matrix, then an orthogonal diagonalization A = PDP' is a singular value decomposition. (where P' = transpose(P))2. The attempt at a solution.

I really don't know how to start this problem off. I know that the singular value decomposition is of the form A = UEV' where E will be an nxn matrix containing the singular values of A, and in this case the eigenvalues of A as well. But that's about it. Any help would be greatly appreciated!
 
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  • #2
is A a real matrix? if A is symmetric, how are the eigenvectors of A related to that of A^T
 
  • #3
lanedance said:
is A a real matrix? if A is symmetric, how are the eigenvectors of A related to that of A^T

Well, A has an orthonormal set of n eigenvectors, which would therefore be the same as A^T, but I don't know how to use this in the proof.
 

FAQ: Show that this orthogonal diagonalization is a singular value decomposition.

1. What is an orthogonal diagonalization?

An orthogonal diagonalization is a process in linear algebra where a matrix is transformed into a diagonal matrix by using an orthogonal matrix.

2. What is a singular value decomposition?

A singular value decomposition is a factorization of a matrix into three components: a diagonal matrix of singular values, an orthogonal matrix of left singular vectors, and an orthogonal matrix of right singular vectors.

3. How is a singular value decomposition related to orthogonal diagonalization?

A singular value decomposition is essentially an extension of orthogonal diagonalization to non-square matrices. It involves finding the eigenvalues and eigenvectors of the matrix and then using them to construct the three components of the decomposition.

4. Why is a singular value decomposition useful?

A singular value decomposition is useful because it allows us to easily manipulate and analyze non-square matrices. It also has applications in various fields such as image processing, data compression, and machine learning.

5. How do you show that a given orthogonal diagonalization is a singular value decomposition?

To show that a given orthogonal diagonalization is a singular value decomposition, we need to verify that the three components (singular values, left singular vectors, and right singular vectors) satisfy the properties of a singular value decomposition. These properties include the fact that the singular values are non-negative, the left and right singular vectors are orthogonal, and their product results in the original matrix.

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