Positive Semidefinite Matrix Square Root

For a symmetric positive semidefinite (PSD) matrix A, there exists a unique symmetric PSD matrix X such that X² = A. This is the principal square root of A.

Experiment

Open the Matrix Square Root module to interact with this concept:

1. Set A11 to 5, A12 to 4, and A22 to 5.
2. Observe that the displayed matrix X entries are indeed 2, 1, 1, 2, matching our worked example.
3. Check the eigenvalues of A (1, 9) and the eigenvalues of X (1, 3).
4. The maximum absolute entry of X² − A is zero up to floating-point rounding. It is calculated from unrounded entries, not reconstructed from the displayed six decimals.
5. Change A11 to 1, A12 to 0, and A22 to -1. The system correctly identifies this as "Outside PSD domain" because it has a negative eigenvalue (-1).
6. Set A11, A12, and A22 all to 0. The output is exactly the zero matrix with a 0 residual, demonstrating correct handling of the edge case.