How this calculator works
Polar decomposition is the matrix analogue of writing a complex number as a rotation times a stretch. It factors a square matrix as Q P, where Q is orthogonal and P is symmetric positive semidefinite. This calculator obtains the factors from the singular value decomposition A = U Sigma V transpose, taking Q = U V transpose and P = V Sigma V transpose. The factor Q is a rotation or reflection, and P shows how each direction is stretched.
Formula
A = Q P with Q^T Q = I and P = P^T positive semidefinite. From A = U Sigma V^T, Q = U V^T and P = V Sigma V^T.
Worked example
The diagonal matrix with rows 2, 0 and 0, 3 is already in polar form with Q equal to the identity and P equal to the matrix itself. A quarter-turn rotation with rows 0, -1 and 1, 0 has Q equal to the matrix and P equal to the identity, because the rotation does not stretch anything.
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Frequently asked questions
Is the polar decomposition unique?
For an invertible matrix it is unique. For a singular matrix, the orthogonal factor is not unique in directions that collapse to zero, although the positive semidefinite factor remains unique.
What does the factor P represent?
It is the symmetric positive semidefinite square root of A transpose A. Its eigenvalues are the singular values of A and show the stretch along orthogonal directions.
Can Q be a reflection?
Yes. Q is orthogonal, so its determinant is plus or minus one. Whenever the determinant of A is negative, Q includes a reflection.