Last updated: July 15, 2026
Polar Decomposition Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The Polar Decomposition Calculator factors an invertible 2×2 real matrix into A = U·P, where U is orthogonal and P is the symmetric positive semi-definite square root of AᵀA. It also reports the determinant, a rotation-angle summary, and whether the orthogonal factor includes a reflection.
Polar decomposition rewrites a 2×2 matrix as an orthogonal part times a symmetric stretch part, so you can separate rotation or reflection from scaling behavior.
Key Takeaways
- Polar decomposition rewrites an invertible 2×2 matrix as A = U·P.
- U captures rotation or reflection, while P captures symmetric stretch.
- The calculator derives P from √(AᵀA) and then recovers U from AP⁻¹.
- A symmetric positive-definite matrix should return U equal to the identity.
- Determinant, angle, and reflection status together explain the geometry.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
A = U·P, where U is orthogonal and P = √(AᵀA) is symmetric positive semi-definite
Where:
- A=Input matrix
- U=Orthogonal factor
- P=Symmetric positive semi-definite factor
- \det(A)=Determinant of the input matrix
- \theta=Rotation angle extracted from U(degrees)
Worked Examples
Lower-triangular matrix with positive determinant
This verified example shows a non-symmetric matrix splitting into a rotation-type U and a symmetric stretch matrix P.
- 1Compute det(A) = 3×5 - 0×4 = 15, so the matrix is invertible and orientation preserving.
- 2Form AᵀA and use the closed-form square-root formula to obtain P = [[4.472136, 2.236068], [2.236068, 4.472136]].
- 3Multiply A by P⁻¹ to get U = [[0.894427, -0.447214], [0.447214, 0.894427]].
- 4Read the orthogonal factor as a rotation of 26.565051° with no reflection.
Symmetric positive-definite matrix
A symmetric input is a strong sanity check because the orthogonal factor should collapse to the identity.
- 1Compute det(A) = 2×2 - 1×1 = 3, so the matrix is invertible.
- 2Because A is already symmetric positive definite, √(AᵀA) reproduces the original matrix.
- 3The resulting factors are U = [[1, 0], [0, 1]] and P = [[2, 1], [1, 2]].
- 4The angle is 0° and the decomposition confirms there is no reflection.
Reflection case with negative determinant
This example confirms that the orthogonal factor can include a reflection when the original matrix reverses orientation.
- 1Compute det(A) = 1×4 - 2×3 = -2, so orientation is reversed.
- 2The symmetric factor is P = [[2.057983, 2.40098], [2.40098, 3.772969]].
- 3Right-multiplying by P⁻¹ gives U = [[-0.514496, 0.857493], [0.857493, 0.514496]].
- 4The angle summary is 120.963757° and the reflection flag is yes because det(A) is negative.
Introduction
The Polar Decomposition Calculator turns a 2×2 real matrix into a geometric story. Instead of treating four entries as an opaque block of arithmetic, it separates the transformation into an orthogonal factor U and a symmetric positive semi-definite factor P using the exact identity A = U·P with P = √(AᵀA). That makes it much easier to distinguish rigid motion from directional stretching, verify whether a matrix preserves orientation, and connect classroom formulas to practical linear-algebra workflows. The calculator is especially useful when you want a numerically stable benchmark for hand calculations, migration tests, or quick interpretation of small matrix transforms.
What polar decomposition means for a 2×2 matrix
Polar decomposition rewrites an invertible real matrix as the product A = U·P, where U preserves lengths and angles up to a possible reflection, and P stores the symmetric stretching part. For a 2×2 matrix this is especially intuitive because every linear transformation in the plane can be understood as a combination of turning, flipping, and stretching. The calculator separates those roles cleanly. If the determinant is positive, the orthogonal factor behaves like a pure rotation. If the determinant is negative, the orthogonal factor includes a reflection. The positive semi-definite factor then explains how the matrix changes scale along principal directions. This viewpoint is useful in linear algebra, numerical analysis, computer graphics, and mechanics because it isolates geometry from magnitude. Instead of reading four entries as unrelated numbers, you get a structured interpretation of the map. That makes it easier to check whether a matrix mostly rotates, mostly stretches, or mixes both effects in a way that still has a clean geometric meaning.
A splits into an orthogonal factor and a stretch factor.
U captures rotation or reflection behavior in the plane.
P is symmetric and positive semi-definite by construction.
The decomposition turns raw entries into geometric meaning.
How the formula is built from AᵀA
The most important structural idea is that P comes from the square root of AᵀA. Because AᵀA is symmetric and positive definite whenever A is invertible, it has a well-behaved positive square root. For a 2×2 matrix, the calculator uses a closed-form route instead of an iterative algorithm. It first forms the entries M11, M12, M21, and M22 of AᵀA, then uses the trace and determinant of that symmetric matrix to compute a denominator that produces P directly. After P is known, the inverse of P is multiplied on the right to recover U = AP⁻¹. This order matters because it ensures the stretch factor stays symmetric while the orthogonal factor stays geometry preserving. The same logic appears in more advanced matrix analysis courses, but this calculator makes it accessible with explicit outputs. Each displayed matrix entry comes from the exact algebraic recipe, so the result is reproducible and easy to compare against hand calculations or lecture notes.
Start from AᵀA because it is symmetric by definition.
Use trace and determinant to build the square-root formula.
Recover U only after P has been computed and inverted.
The method avoids iterative approximations for 2×2 inputs.
Preparing the input matrix correctly
Reliable results begin with a clear matrix entry order. The calculator expects a11 and a12 in the first row, followed by a21 and a22 in the second row. That sounds obvious, but many mistakes come from transposed entry order or from copying values out of a system matrix without matching the intended row and column positions. The second validation step is invertibility. Polar decomposition for singular matrices can still be discussed theoretically, but this migration intentionally follows the exact invertible 2×2 algorithm listed in the specification. That is why the determinant must stay away from zero by at least the stated numerical tolerance. Finite numeric input matters as well, because Infinity or NaN would contaminate AᵀA and every downstream quantity. Before calculating, it helps to estimate the determinant and think about the transformation qualitatively. A matrix with large off-diagonal entries often mixes axes strongly, while a symmetric matrix often leads to an identity-like U. Those quick checks help you recognize whether the returned factorization makes geometric sense.
Enter row 1 before row 2 to preserve matrix order.
Check the determinant before trusting any decomposition.
Only finite numbers are valid for this exact algorithm.
A quick geometric estimate makes output review faster.
How to interpret the orthogonal factor U
The orthogonal matrix U is the geometry-preserving part of the factorization. In the plane, an orthogonal matrix has columns of unit length that are mutually perpendicular, so it does not create shear or arbitrary stretch. Instead, it rotates vectors, reflects them, or combines both. That is why the calculator reports the four entries of U together with a rotation angle and an isReflection flag. When the determinant of the original matrix is positive, det(U) is +1 and the map acts like a pure rotation. When the determinant is negative, det(U) is -1 and the factor includes a reflection. Reading the matrix entries alongside the angle helps you connect numerical and geometric interpretations. For example, a U close to the identity means the original matrix was already aligned with the coordinate axes or was itself symmetric positive definite. A U with large off-diagonal entries indicates a substantial change in orientation. In applications, this factor is often the part you keep when you want a clean rigid motion without the stretch that P contributes.
U preserves lengths because it is orthogonal.
A positive determinant implies a rotation-type U.
A negative determinant signals reflection behavior in U.
The angle output summarizes orientation in one scalar.
How to interpret the symmetric factor P
The matrix P contains the pure stretching information. Because it is symmetric positive semi-definite, it can be diagonalized by an orthogonal change of basis, which means its action can be understood as stretching along perpendicular principal directions. In practice, that makes P the part of A that changes size without adding extra turning beyond what U already handles. Symmetry matters here because it guarantees matching off-diagonal entries and a stable geometric interpretation. Positive semi-definiteness matters because it keeps the stretch from reversing orientation on its own. When the original matrix is itself symmetric positive definite, U becomes the identity and P reproduces the original matrix exactly, which is a useful sanity check and one of the examples included in this calculator. Comparing the P entries with the original matrix can also reveal how much of the transformation came from shear-like mixing versus directional scaling. If P is close to a scalar multiple of the identity, then the main behavior of A is nearly a uniform scale followed by the orthogonal action encoded in U.
P stores stretch without adding extra rotation.
Symmetry means P12 and P21 should match numerically.
Positive semi-definiteness keeps the stretch well behaved.
Symmetric input matrices often make U equal the identity.
Why determinant, angle, and reflection all matter
The decomposition becomes much more informative when you read the supporting scalar outputs with the matrices. The determinant of A tells you the signed area scaling of the transformation, so it answers whether orientation is preserved and how strongly the plane is expanded or contracted overall. The rotation angle is extracted from atan2(U21, U11), which is the cleanest way to summarize the orthogonal factor when you want a single orientation measure. The isReflection flag then removes ambiguity by telling you whether U belongs to the rotation family or the reflection family. Those three outputs complement one another. A small angle does not mean the matrix is gentle if P contains large stretching, and a large determinant does not automatically imply rotation. Together, the outputs give a compact diagnosis of the transformation. In engineering and graphics settings, that compact diagnosis is often more useful than staring at raw matrix entries, because it tells you how orientation, area scaling, and symmetric stretch interact inside the same linear map.
The determinant measures signed area scaling.
atan2 provides a stable angle summary for U.
Reflection status distinguishes det(U)=+1 from det(U)=-1.
Scalars and matrix entries should be read together.
Reference cases that build intuition quickly
Benchmark matrices are useful because they show what the decomposition should look like before you calculate. A symmetric positive-definite matrix should return U equal to the identity, since the matrix is already a pure stretch in an orthogonal basis. A matrix with negative determinant should produce an orthogonal factor that includes a reflection, even if the angle still looks like an ordinary orientation value. Upper- or lower-triangular examples often highlight how off-diagonal entries feed both the rotation part and the symmetric part. Studying a few reference cases also helps you debug transcription mistakes. If a matrix you expected to be symmetric suddenly returns a nontrivial U, the first thing to check is whether an entry was copied into the wrong position. The table below uses the calculator’s verified examples to anchor those expectations. Reviewing these cases before an exam or before coding a larger matrix routine makes the structure more memorable than trying to memorize formulas alone.
Symmetric positive-definite inputs make U equal the identity.
Negative determinant cases force reflection behavior.
Triangular matrices still split into rigid and stretch parts.
Reference cases are the fastest sanity check for hand work.
| Matrix A | det(A) | Angle | Reflection |
|---|---|---|---|
| [[3, 0], [4, 5]] | 15 | 26.565051° | no |
| [[2, 1], [1, 2]] | 3 | 0° | no |
| [[1, 2], [3, 4]] | -2 | 120.963757° | yes |
| Identity I | 1 | 0° | no |
A dependable validation workflow after calculation
After the calculator returns U and P, a quick validation routine can confirm that the factorization is trustworthy. First, check whether the reported determinant matches the simple hand computation a11a22 - a12a21. Second, inspect P for symmetry by comparing P12 and P21; they should match after rounding to six decimals. Third, remember the defining reconstruction test: multiplying U by P should recover the original matrix up to rounding noise. Finally, think about the geometry. If the input matrix was clearly symmetric, U should be close to the identity. If the determinant was negative, the reflection flag should say yes. These checks are lightweight, but together they catch the most common mistakes: swapped entries, mistaken signs, or confusion about how the factorization is ordered. This workflow is especially valuable when you are using the calculator as a migration benchmark, because it lets you verify both numerical outputs and conceptual meaning without needing a separate algebra system for every example.
Recompute the determinant directly from the four entries.
Confirm that P stays symmetric after rounding.
Check that U·P reconstructs the original matrix.
Compare the geometry with what you expected beforehand.
| Checkpoint | What to test | Expected pattern |
|---|---|---|
| Invertibility | det(A) | Non-zero and finite |
| Orthogonality | columns of U | Unit and perpendicular |
| Symmetry | P12 vs P21 | Equal after rounding |
| Reconstruction | U·P | Matches A to 6 decimals |
Quick Reference Card
Polar Decomposition Quick Reference
Quick reference • Polar Decomposition Calculator
A = U·P, with P = √(AᵀA) and U = AP⁻¹Valid range: Use finite entries for an invertible 2×2 real matrix with |det(A)| > 1e-9.
Common Values
⚠ Watch Out
- •Singular matrices are rejected by this exact migration because P cannot be inverted safely.
- •Enter the matrix entries in the correct row and column order.
- •Read the angle together with the reflection flag, not by itself.
- •Rounded outputs may hide tiny floating-point noise, so reconstruct U·P for a strict check.
Pro Tips
- →Estimate the determinant before calculating to catch data-entry mistakes early.
- →If A is symmetric positive definite, expect U to be the identity matrix.
- →Use P12 and P21 as a fast symmetry check on the stretch factor.
- →Compare U with a standard rotation matrix to build geometric intuition.
FAQs
What does the polar decomposition separate?
It separates an invertible 2×2 real matrix into an orthogonal factor **U** and a symmetric positive semi-definite factor **P**. In geometric terms, that means you can read the rigid motion separately from the stretch.
Why must the determinant be non-zero?
This calculator follows the exact closed-form invertible algorithm. If the determinant is zero or numerically indistinguishable from zero, the matrix cannot be inverted cleanly in the step that recovers **U = AP⁻¹**.
What does the reflection output mean?
If **isReflection** is **yes**, then the orthogonal factor has determinant -1 and includes a reflection. If it is **no**, the orthogonal factor behaves like a pure rotation with determinant +1.
Why can a symmetric matrix return U = I?
If the matrix is already symmetric positive definite, then it already acts like a pure stretch in an orthogonal basis. In that case the polar decomposition keeps **P = A** and makes **U** the identity matrix.
Is the reported angle always enough to describe U?
The angle is a compact summary extracted from **atan2(U21, U11)**, but you should still read it together with **isReflection**. A reflected orthogonal matrix can share angle-like values with an ordinary rotation while representing a different geometry.
Why is P symmetric?
The construction starts from **AᵀA**, which is symmetric. Its positive square root remains symmetric, so the resulting factor **P** always has matching off-diagonal entries apart from rounding noise.
How can I verify the calculator result by hand?
Recompute the determinant, compare **P12** with **P21**, and multiply **U·P** to see whether it reconstructs the original matrix to six-decimal accuracy. Those checks validate both the algebra and the interpretation.