How this instrument works
Polar decomposition writes a matrix A as a product U × P, echoing how a complex number splits into a direction on the unit circle times a magnitude: z = e^(iθ) × r. Here U carries the direction — an orthogonal matrix representing a pure rotation or reflection, nothing more — while P carries the magnitude, a symmetric, positive-definite matrix that does all of the actual stretching or squashing, always along a set of mutually perpendicular axes. Multiply U back into P and you recover A exactly, with each factor doing one job and one job only.
P's own trace and determinant fall directly out of A's two singular values, σ₁ and σ₂, without ever assembling it entry by entry. Its trace is simply their sum, σ₁ + σ₂, since the stretch factor shares those same amounts along its own principal axes; its determinant is their product, σ₁ × σ₂, which always equals |det(A)|, the absolute value of A's own determinant. For [[1,1],[0,1]], the singular values are the golden ratio, φ ≈ 1.618, and its reciprocal, 1⁄φ ≈ 0.618 — add them and you get √5, the number this calculator reports as trace(P).
This sheet shares its underlying arithmetic with two close relatives on this site. Singular Values reports σ₁ and σ₂ directly, the exact pair this page adds and multiplies to build trace(P) and det(P). QR Decomposition also splits a matrix into two pieces, but a genuinely different pair: an orthogonal factor Q alongside an upper-triangular factor R, rather than an orthogonal factor alongside a symmetric one. Both approaches separate 'which way' a matrix points from 'how far' it stretches, yet they arrive there through distinct factorizations that only sometimes agree on their numbers.
One limit worth stating plainly: this page hands back only two summary numbers, trace(P) and det(P), not the sixteen individual entries that make up the full U and P matrices. Those entries need the eigenvectors of AᵀA as well as the eigenvalues, a step beyond what a trace-and-determinant formula alone can supply, so treat this sheet as a fast check on P's overall size rather than a full reconstruction of the decomposition.
- Enter the matrix A's four entries into a (row 1, col 1), b (row 1, col 2), c (row 2, col 1), and d (row 2, col 2).
- Read Trace of P for σ₁ + σ₂, the combined stretch along P's two principal axes.
- Read Determinant of P and check it against |a×d − b×c| computed by hand — the two should match exactly.
- Try a=1, b=0, c=0, d=1 (the identity matrix) to see the baseline case: no rotation and no stretch at all.
- Compare the result against this site's Singular Values page, which reports σ₁ and σ₂ separately rather than combined.
Worked example — [[1,1],[0,1]] and [[3,0],[4,5]]
For A = [[1,1],[0,1]], the singular values work out to φ ≈ 1.618033988749895 and 1⁄φ ≈ 0.618033988749895. Trace(P) is their sum, √5 ≈ 2.23606797749979, and det(P) is their product, φ × (1⁄φ) = 1, which matches |det(A)| = |1×1 − 1×0| = 1 exactly, just as the formula requires.
A less tidy case shows the same rule holding. For A = [[3,0],[4,5]], det(A) = 3×5 − 0×4 = 15, so det(P) must equal 15 — and it does. The singular values here are √45 ≈ 6.7082 and √5 ≈ 2.2361, not √50 as a quick mental shortcut might suggest, and their sum gives trace(P) = 4√5 ≈ 8.94427190999916, the figure this calculator returns.
Questions
What does polar decomposition split a matrix into?
Two factors, U and P, multiplied together to rebuild the original matrix A exactly. U is orthogonal — a pure rotation or reflection that changes direction but never length. P is symmetric and positive-definite — a pure stretch along a set of perpendicular axes that changes length but never direction. Splitting the two apart separates 'which way' a transformation points from 'how far' it stretches.
Why is P always symmetric and positive-definite?
Because P is built from AᵀA, a product that is always symmetric by construction, and its eigenvalues — the squares of A's singular values — are always positive whenever A itself is invertible. A symmetric matrix with strictly positive eigenvalues is exactly the definition of positive-definite, which is why P never contains a reflection or a rotation of its own.
How does this page find trace(P) and det(P) without building the full matrix?
Both quantities depend only on A's singular values, σ₁ and σ₂, which this page derives directly from A's four entries. Trace(P) = σ₁ + σ₂ and det(P) = σ₁ × σ₂ = |det(A)|, so neither figure requires ever assembling P's individual entries or finding AᵀA's eigenvectors.
Does this calculator return the full U and P matrices?
No — it reports only trace(P) and det(P), two summary numbers describing P's overall size. Recovering every entry of U and P calls for AᵀA's eigenvectors as well as its eigenvalues, a longer calculation this page does not attempt, so treat the two figures here as a quick check rather than a full decomposition.
How is this different from the Singular Values page on this site?
Singular Values reports σ₁ and σ₂ as two separate numbers, the matrix's largest and smallest stretch factors. This page takes that same pair and combines them: adding them for trace(P), multiplying them for det(P). The underlying arithmetic is shared; only what gets reported differs.
How is this different from QR decomposition?
QR decomposition splits a matrix into an orthogonal factor Q and an upper-triangular factor R, while polar decomposition splits it into an orthogonal factor U and a symmetric, positive-definite factor P. Both separate direction from size, but the second factor is triangular in one case and symmetric in the other, so their numbers generally differ even for the same starting matrix.
What happens with the identity matrix?
Feeding in a=1, b=0, c=0, d=1 gives trace(P) = 2 and det(P) = 1, since both singular values equal 1. The identity matrix already has no rotation and no stretch built in, so U and P are each the identity too — the plainest possible baseline for comparing any other matrix against.