Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 9 down vote favorite
4

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is singular value just another name for eigenvalue?

share|improve this question
1  
They agree in finite dimensions, but not necessarily for infinite-dimensional operators. I've heard the term "singular value" applied to any value for which $(A-\lambda I)^{-1}$ either does not exist or is not continuous, while eigenvalues refer only to those values for which $(A-\lambda I)^{-1}$ does not exist. – Alex Becker Apr 3 '12 at 3:30
The singular value is a nonnegative scalar of a square or rectangular matrix while an eigenvalue is a scalar (any scalar) of a square matrix. – Hassan Muhammad Apr 3 '12 at 3:37
^Note that I was addressing square matrices specifically, or in the infinite-dimensional case, endomorphisms. – Alex Becker Apr 3 '12 at 3:56
My guess is that the question is about the singular value decomposition for matrices of finite-dimensional operators. – sydeulissie Apr 3 '12 at 4:02
They are not the same thing at all, and has nothing to do with dimension. They only agree in the special case where the matrix is symmetric. This agreement also extends (in a sense) for infinite dimensional compact operators. – Nick Alger Sep 30 '12 at 2:46
show 2 more comments

3 Answers

up vote 7 down vote

The singular values of a $M\times N$ matrix $X$ are the square roots of the eigenvalues of the $N\times N$ matrix $X^*X$ (where $*$ stands for the transpose-conjugate matrix if it has complex coefficients, or the transpose if it has real coefficients).

Thus, if $X$ is $N\times N$ real symmetric matrix with non-negative eigenvalues, then eigenvalues and singular values coincide, but it is not generally the case !

share|improve this answer
Correction: "positive semidefinite", not "with non-negative coefficients". For example, $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has an eigenvalue $-1$, which is not a singular value. – user31373 Sep 30 '12 at 4:49
Typo corrected, thx. – Student Oct 1 '12 at 23:33
up vote 3 down vote

is singular value just another name for eigenvalue?

No, singular values & eigenvalues are different.

What is the difference between Singular Value and Eigenvalue?

There are many possible answers to this question. Since I don't know what you're trying to prove, I'd recommend carefully comparing definitions between the two: eigendecomposition, singular value decomposition

[EDIT: You might find the first several chapters of the book "Numerical Linear Algebra" by Trefethen and Bau more useful than the Wikipedia article. They're available here.]

Two important points:

  • Notice in particular that the SVD is defined for any matrix, while the eigendecomposition is defined only for square matrices (and more specifically, normal matrices).

  • Notice that singular values are always real, while eigenvalues need not be real.

share|improve this answer
up vote 0 down vote

Also Very Good pdf by Matlab.....

http://www.mathworks.com/moler/eigs.pdf

share|improve this answer
4  
While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. – user53153 Jan 23 at 4:33

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.