Define the singular values of a matrix.

The singular values of a matrix are the square roots of the eigenvalues of the matrix's transpose times the matrix itself.

Singular values are a way of measuring the "stretching" or "compression" of a linear transformation represented by a matrix. They are always non-negative and can be thought of as the lengths of the semi-axes of an ellipsoid that is the image of the unit sphere under the transformation.

To find the singular values of a matrix A, we first compute the matrix A^T A. This is a symmetric matrix, so it has real eigenvalues and orthogonal eigenvectors. We can find these eigenvalues and eigenvectors using standard techniques. Let λ_1, λ_2, ..., λ_n be the eigenvalues of A^T A, listed in decreasing order. Then the singular values of A are the square roots of these eigenvalues, i.e. σ_1 = sqrt(λ_1), σ_2 = sqrt(λ_2), ..., σ_n = sqrt(λ_n).

The singular values of a matrix have many important applications in linear algebra and beyond. For example, they can be used to compute the rank of a matrix, to solve linear systems of equations, to perform principal component analysis, and to compress data. They also have connections to other areas of mathematics, such as functional analysis and geometry.