The singular values of a matrix \(A\) are the square roots of the eigenvalues of the matrix \(A^T A\). They are a key component of Singular Value Decomposition (SVD).

The Process:

• 1. Form A^TA: First, calculate the transpose of the input matrix, \(A^T\). Then, multiply the transpose by the original matrix to get \(A^T A\). The result will always be a square, symmetric matrix.
• 2. Find Eigenvalues: Calculate the eigenvalues (\(\lambda_1, \lambda_2, ...\)) of the matrix \(A^T A\).
• 3. Calculate Singular Values: The singular values (\(\sigma_1, \sigma_2, ...\)) are the square roots of the corresponding eigenvalues.

\(\sigma_i = \sqrt{\lambda_i}\)