Polar Decomposition Calculator

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Polar Decomposition Calculator

Decompose a 2x2 matrix into a product of an orthogonal matrix and a symmetric positive semi-definite matrix.

Last updated: March 2026 | By ForgeCalc Engineering

Polar Decomposition Solver

Matrix factorization

Calculation Steps

1.Matrix A = [[2, 1], [1, 2]]
2.Compute A^T A = [[5, 4], [4, 5]]
3.Find P as the positive semi-definite square root of A^T A.
4.Find U by multiplying A by P^-1.
5.Result: A = U P
Orthogonal Matrix U
1
0
0
1
Symmetric Matrix P
2
1
1
2

What Polar Decomposition Means

Polar decomposition writes a matrix as a pure rotation or reflection multiplied by a symmetric stretching matrix. It helps separate direction from scaling.

How to Read Polar Decomposition

  1. Form A^T A from the original matrix.
  2. Take the positive semi-definite square root to get P.
  3. Compute U by multiplying A by P^-1.
  4. Check that A = U P.
A = U P

Worked Example

Example: A = [[2, 1], [1, 2]] can be decomposed into U and P.

A = U P

Frequently Asked Questions

What does U represent?

U represents the orthogonal part, which preserves length and angle.

What does P represent?

P represents the symmetric stretching part of the matrix.

Does this accept decimals?

Yes. Any finite real matrix entries are accepted.

Is the decomposition unique?

For many matrices, the polar decomposition is unique under the standard conditions.

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