Free Polar Decomposition Calculator

Introduction to Matrix Polar Decomposition

The Matrix Polar Decomposition is a factorization technique that expresses any square or rectangular matrix as the product of a unitary matrix (the polar factor) and a positive semidefinite matrix. This decomposition parallels the polar form of a complex number, where the unitary matrix corresponds to the angle (rotation) and the positive semidefinite matrix corresponds to the magnitude (scaling). The Polar Decomposition Calculator automates this factorization, enabling quick computation of both the unitary and positive semidefinite components for matrices of various sizes.

Mathematical Definition

For a given matrix $A$, the right polar decomposition is written as:

$$A = U P$$

where $U$ is a unitary matrix ($U^{*}U = I$) and $P$ is a positive semidefinite Hermitian matrix. An alternative left polar decomposition also exists:

$$A = P' U'$$

with $U'$ unitary and $P'$ positive semidefinite. The two forms are related through the singular value decomposition (SVD) of $A$.

Computing the Polar Factor via SVD

The most reliable approach to obtain the polar decomposition uses the SVD. Let

$$A = W \Sigma V^{*}$$

where $W$ and $V$ are unitary matrices and $\Sigma$ is a diagonal matrix containing the singular values $\sigma_i$. Then the unitary polar factor and the positive semidefinite component are:

$$U = W V^{*}, \qquad P = V \Sigma V^{*}$$

For the left decomposition, $U' = W V^{*}$ (same as $U$) and $P' = W \Sigma W^{*}$. The polar factor $U$ represents the nearest unitary matrix to $A$ in the Frobenius norm, a property exploited in alignment problems.

Key Properties of the Decomposition

• Uniqueness: If $A$ is invertible, the polar decomposition is unique.
• Rectangular Matrices: Polar decomposition can be extended to non-square matrices; the spectral norm of the difference between $A$ and its polar factor is minimal.
• Polar Factor as Rotation: In 3D graphics and robotics, the unitary part represents rotation (or reflection if $U$ has determinant $-1$), while the positive semidefinite part encodes stretching and shearing.

Applications of Matrix Polar Decomposition

Field	Typical Use
Robotics	Extracting rotation from a transformation matrix for inverse kinematics
Quantum Mechanics	Decomposing operators into unitary evolution and contraction
Computer Graphics	Strain analysis and shape interpolation
Numerical Linear Algebra	Computing the matrix square root and solving orthogonal Procrustes problems
Aerospace Engineering	Attitude estimation from noisy measurements

Using the Polar Decomposition Calculator

This free online tool accepts a square or rectangular matrix (real or complex) and returns:

• The unitary polar factor $U$
• The positive semidefinite factor $P$ (and optionally $P'$ for the left form)
• The singular values of the input matrix

The computation is performed via a numerically stable SVD algorithm, ensuring accurate results even for ill-conditioned matrices. Simply enter the matrix entries, choose the decomposition side (right or left), and click compute.

The Matrix Polar Decomposition is a versatile mathematical tool, and this calculator makes it instantly usable for education, research, and engineering. Whether you need the polar factor for a rotation estimation problem or you want to study the structure of a matrix, the calculator provides both the numerical result and a step‑by‑step explanation.