Polar Decomposition Calculator

The polar decomposition factors a real matrix A into a product QP. For a nonsingular square matrix, Q is orthogonal and P is symmetric positive definite. An orthogonal matrix preserves lengths and angles, so it represents a rotation or reflection. The positive factor stretches space along perpendicular principal directions without adding rotation. This separation is the matrix counterpart of writing a nonzero complex number as a magnitude multiplied by a unit direction. This calculator handles real nonsingular two-by-two matrices. It first forms the symmetric Gram matrix A transpose times A. The positive square root of that matrix is P. For a two-by-two positive-definite matrix, the square root can be evaluated directly from its trace and determinant, avoiding an iterative approximation. The orthogonal factor then follows from Q equals A times the inverse of P. Displayed entries are rounded to six decimal places, while the internal calculations retain JavaScript double precision. The factors have useful identities that make the answer easy to check. Multiplying Q transpose by Q gives the identity matrix, apart from rounding error. Matrix P equals its own transpose and has positive eigenvalues. Finally, multiplying Q by P reconstructs the original matrix A. For the identity matrix, both factors are the identity. For a matrix that combines a quarter-turn with unequal axis scaling, Q isolates the quarter-turn while P records the two stretch factors. Polar decomposition appears in mechanics, graphics, numerical analysis, optimization, and statistics. In continuum mechanics it separates deformation into rigid rotation and pure stretch. In computer graphics it helps extract rotation from a transformation that also includes scale or shear. The orthogonal Procrustes problem uses the polar factor to find the closest orthogonal matrix, and numerical algorithms use polar iterations when preserving geometric structure matters. A singular matrix can also possess a polar decomposition, but its orthogonal factor may not be unique and calculating it requires extra conventions. This interactive calculator restricts input to nonsingular matrices so that P has an inverse and the displayed factorization is unambiguous. Very ill-conditioned matrices can amplify floating-point error because one stretch direction is close to zero. Use the result as a clear educational and computational aid, and verify high-stakes numerical work with software designed for arbitrary precision or larger matrix dimensions.