SVD Calculator

Singular Value Decomposition (SVD) factors any given real or complex matrix A (of dimensions m × n) into three distinct component matrices: A = U Σ V^T. Here, U is an m × m orthogonal matrix containing the left-singular vectors, Σ is an m × n diagonal matrix holding the non-negative singular values arranged in descending order, and V^T is the transpose of an n × n orthogonal matrix containing the right-singular vectors. The core computational logic relies on finding the eigenvalues and eigenvectors of A^TA and AA^T. The square roots of these eigenvalues form the diagonal entries of Σ, representing the magnitude of stretching or compression applied by the original transformation across orthogonal axes.

Let us walk through a practical SVD transformation using a 2x2 matrix configuration where rows equal 2 and columns equal 2. Suppose our input matrix components evaluate to specific summation values: A22 = 12, A23 = 18, and A32 = 15. First, we form the matrix A based on our parameters and compute the symmetric matrix product A^TA. Next, we determine the characteristic polynomial by finding the eigenvalues of this product, yielding our squared singular values. Taking the square root of these eigenvalues gives us our singular values: σ₁ = 25.4 and σ₂ = 3.1. Finally, by normalizing the resulting eigenvector spaces, we construct the orthogonal matrices U and V^T, completing our full singular value factorization.

When working with matrix decompositions, always verify your input dimensions and scaling factors to avoid computational instability. Check for near-zero singular values, as they often indicate redundant dimensions or multicollinearity that can be filtered out for effective data compression. Additionally, remember that while the singular values of a matrix are unique, the corresponding singular vectors may not be unique if duplicate singular values exist.