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{\sc\large This Section is a Draft, Subject to Changes}\\
{\sc\large Needs Numerical Examples}
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The polar decomposition of a matrix writes any matrix as the product of a unitary matrix (Definition
UM)and a positive semi-definite matrix (Definition
PSM). It takes its name from a special way to write complex numbers. If you've had a basic course in complex analysis, the next paragraph will help explain the name. If the next paragraph makes no sense to you, there's no harm in skipping it.
Any complex number $z\in\complexes$ can be written as $z=re^{i\theta}$ where $r$ is a positive number (computed as a square root of a function of the real amd imaginary parts of $z$) and $\theta$ is an angle of rotation that converts $1$ to the complex number $e^{i\theta}=\cos(\theta)+i\sin(\theta)$. The polar form of a square matrix is a product of a positive semi-definite matrix that is a square root of a function of the matrix together with a unitary matrix, which can be viewed as achieving a rotation (Theorem
UMPIP).
OK, enough preliminaries. We have all the tools in place to jump straight to our main theorem.
Theorem PDM Polar Decomposition of a Matrix
Suppose that $A$ is a square matrix. Then there is a unitary matrix $U$ such that $A=\sr{\left(A\adjoint{A}\right)}U$.
Proof
