Polar Decomposition

• categories: Linear algebra, Factorization

Definition
The polar decomposition of a Matrix $A\in {\mathbb{C}}^{m\times n}$ (or ${\mathbb{R}}^{m\times n}$) expresses $A$ as the product of two matrices:
$A=UP,$
where:

1. $U\in {\mathbb{C}}^{m\times n}$ (or ${\mathbb{R}}^{m\times n}$) is a unitary matrix (or Orthogonal matrix if $A$ is real), satisfying $U^{\ast }U=I$ (or $U^{T}U=I$).
2. $P\in {\mathbb{C}}^{n\times n}$ (or ${\mathbb{R}}^{n\times n}$) is a Positive Semi-Definite Matrix and Hermitian (or Symmetric Matrix if $A$ is real), satisfying $P=P^{\ast }$ and $P\ge 0$.

If $A$ is square and invertible, $U$ is unitary and $P$ is Positive Definite Matrix.

Intuition
The polar decomposition separates a matrix $A$ into two parts:

1. $U$ represents the “rotation” or “orthogonal transformation” component.
2. $P$ represents the “stretching” or “scaling” component.

For square matrices, this is analogous to decomposing a complex number $z=r{e}^{i\theta }$ into a unit-magnitude rotation $e^{i\theta }$ and a positive scaling $r$.

Key Properties

1. Existence and Uniqueness:

   • The polar decomposition always exists for any matrix $A$.
   • $P$ is uniquely determined. $U$ is unique if $A$ is full rank.

2. Computation of $P$:
   $P=\sqrt{A^{\ast }A},$
   where $\sqrt{A^{\ast }A}$ is the unique positive semi-definite square root of $A^{\ast }A$.

3. Computation of $U$:
   $U=A{P}^{-1},$
   if $P$ is invertible. If $A$ is not full rank, $U$ can be determined using a projection.

4. Norm Preservation:
   The unitary (or orthogonal) matrix $U$ preserves the norm of vectors: $\Vert Uv\Vert =\Vert v\Vert$ for any vector $v$.

5. Special Case (Square Matrices):
   If $A$ is invertible, $U$ is unitary, and $P$ is positive definite.

Applications

• Numerical Linear Algebra: Polar decomposition is used in algorithms requiring matrix decompositions with orthogonal and positive components.
• Computer Graphics: Used to decompose affine transformations into rotation and scaling parts.
• Continuum Mechanics: Describes deformation gradients in materials, separating pure rotation from stretch.
• Signal processing: Polar decomposition helps in solving least squares problems and analyzing signal transformations.