Some square matrices are diagonalizable or even orthogonally diagonalizable. An important fact about diagonalization is that the resulting diagonal matrix contains the eigenvalues of the original matrix on the main diagonal. However, not all matrices are diagonalizable. In such a case, the singular value decomposition (SVD) still exists. If is an matrix, its singular values are the square roots of the eigenvalues of the matrix .
The term singular value relates to the distance of the given matrix to a singular matrix. The idea behind SVD is that every matrix can be decomposed into a product , where and are orthogonal matrices and and .
This Demonstration shows the singular values of certain linear transformations in , including rotation, dilation, and the sheer transformation of factor . The yellow square (with blue arrows) is the original region and the black region (with red arrows) is the transformed region. The singular values of the standard matrix affiliated with the transformation can be found when the transformed grid is orthogonal.
Choose a transformation and rotation of the grid until it appears to be orthogonal; the length of the red arrows approaches the singular values of the standard matrix.