Singular Values Calculator

For any given real matrix $A$ of dimensions $m \times n$, the singular values are the square roots of the eigenvalues of $A^T A$ (or $A A^T$). The process begins by calculating the product of the matrix and its transpose. Next, we find the characteristic polynomial by determining the determinant of $(A^T A - \lambda I) = 0$ to extract the eigenvalues (\(\lambda\)). Finally, the singular values (\(\sigma\)) are derived by taking the square root of each eigenvalue: \(\sigma_i = \sqrt{\lambda_i}\), conventionally sorted in descending order.

Let us calculate the singular values for a simple 2x2 matrix $A$ where row 1 is [3, 0] and row 2 is [0, 4]. First, we compute the transpose product $A^T A$. Since $A$ is a diagonal matrix, $A^T A$ yields a matrix with diagonal elements $3^2 = 9$ and $4^2 = 16$, while off-diagonal elements are zero. Next, we find the eigenvalues of this resulting matrix, which are simply 16 and 9. Taking the square root of each eigenvalue gives our singular values: \(\sigma_1 = \sqrt{16} = 4\) and \(\sigma_2 = \sqrt{9} = 3\). Thus, the singular values of matrix $A$ are 4 and 3.

Always verify your matrix dimensions before entry; mismatched row and column configurations will lead to calculation errors. When dealing with large datasets, remember that singular values dictate the rank and condition number of a matrix, which helps identify near-singular or ill-conditioned systems. Finally, note that singular values are always real and non-negative, even if the original matrix contains complex numbers.