How this calculator works
Every matrix has a singular value decomposition, and the singular values are the diagonal entries of its middle factor. They are always non-negative and are listed from largest to smallest. A matrix with m rows and n columns has the smaller of m and n singular values, counting multiplicity. For a square matrix, the singular values are the absolute values of the eigenvalues only when the matrix is symmetric and positive semidefinite; in general they are the square roots of the eigenvalues of A transpose A.
Formula
A = U Sigma V^T, and the singular values are the diagonal entries sigma_i of Sigma. Also sigma_i = sqrt(eigenvalue_i(A^T A)).
Worked example
The diagonal matrix with rows 3, 0 and 0, 4 has singular values 4 and 3. The matrix with rows 1, 2 and 3, 4 has singular values 5.464985704 and 0.3659661906, which are the square roots of 15 plus and minus the square root of 221.
Advertisement
Frequently asked questions
Why are they always non-negative?
They are lengths of semi-axes, so they cannot be negative. A negative eigenvalue of A transpose A is impossible because that matrix is positive semidefinite.
How many singular values are there?
The count equals the smaller of the row and column counts. A 2 by 3 matrix therefore has two singular values, and a 3 by 3 matrix has three.
What is the largest singular value used for?
It is the 2-norm of the matrix, the maximum stretch on a unit vector. The ratio of largest to smallest non-zero singular value gives the 2-norm condition number.