Matrix Polar Decomposition Calculator

Example Data Table

Formula Used

For the right polar decomposition, the calculator uses:

A = U P

P = (AᵀA)^1/2

U = A P⁺

Here, P is symmetric positive semidefinite. P⁺ is the inverse when P is nonsingular. It becomes the Moore-Penrose pseudoinverse when a singular value is below the selected tolerance.

For the left polar decomposition, the calculator uses:

A = Q U

Q = (AAᵀ)^1/2

U = Q⁺A

Singular values are computed from the square roots of eigenvalues of AᵀA or AAᵀ. The reconstruction error uses the Frobenius norm.

How to Use This Calculator

1. Select a 2 × 2 or 3 × 3 matrix.
2. Choose right, left, or both polar decomposition forms.
3. Enter every matrix element in the input boxes.
4. Set decimal precision for the displayed output.
5. Set a zero tolerance for small singular values.
6. Press Calculate to view factors below the header.
7. Use CSV or PDF buttons to save the same result.
8. Compare reconstruction error with the original matrix.

Article: Understanding Matrix Polar Decomposition

Understanding Polar Decomposition

Matrix polar decomposition separates one square matrix into two meaningful factors. One factor behaves like an orthogonal motion. The other factor behaves like a symmetric stretch. This view is useful in numerical algebra, mechanics, graphics, optimization, and data analysis. It helps you see whether a transformation mainly rotates, reflects, scales, shears, or combines those actions. The calculator works with real two by two and three by three matrices. These sizes cover many classroom and engineering examples while keeping the output readable.

Why The Factors Matter

For the right form, the calculator writes A as U P. The matrix P is positive semidefinite and comes from the square root of A transpose A. The matrix U is then found by multiplying A with the inverse, or pseudoinverse, of P. When the original matrix is nonsingular, U should be orthogonal. Its columns stay perpendicular, and their lengths stay near one. When the matrix is singular, U becomes a partial isometry. The reconstruction test still shows how well the factors rebuild the original matrix.

Practical Interpretation

Think of A as a complete linear action. The stretch matrix measures directional expansion. Its eigenvalues lead to singular values. Large singular values show strong stretching. Small values show weak directions or possible rank loss. The orthogonal factor shows the nearest rotation or reflection part. In polar mechanics, this can separate rigid body motion from deformation. In graphics, it can clean a transformation before animation. In numerical work, it gives a stable way to inspect conditioning and structure.

Accuracy And Exports

Small rounding differences can appear because square roots and eigenvalue steps are numerical. The tolerance field decides when a singular value counts as zero. The precision field controls the displayed decimals. Use a tighter tolerance for clean academic matrices. Use a looser tolerance when data contains measurement noise. The calculator also reports determinant, trace, rank estimate, orthogonality error, and reconstruction error. CSV export helps with spreadsheets. PDF export gives a compact record for reports. Always review results with the original matrix and formula notes before drawing conclusions from sensitive data. It also encourages repeatable checking when examples are changed during lessons, reviews, homework, laboratory work, or model testing across many sessions later carefully.

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