Notes - Dissertation, The polar decomposition

Created: December 01, 2025 | Updated: December 01, 2025 | About these notes | View in context | Study these flashcards

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@State the polar decomposition theorem for a matrix $A \in \mathbb C^{m \times n}$.

There exists a matrix $U \in \mathbb C^{m \times n}$ and a unique Hermitian positive semi-definite matrix $H \in \mathbb C^{n \times n}$ such that

\[A = UH,\quad U^\ast U = I _ n\]

and if $A$ is nonsingular, then $H$ is positive definite and $U$ is uniquely determined.

The polar decomposition of a matrix $A \in \mathbb R^{n \times n}$ is given by

\[A = UP\]

where $U$ is orthogonal and $P$ is positive semi-definite symmetric.

Can you give a geometric interpretation of such a decomposition?

It is a rotation or a reflection $U$, followed by a scaling along a set of $n$ orthogonal axes.

The polar decomposition theorem for a matrix $A \in \mathbb C^{m \times n}$ says there exists a matrix $U \in \mathbb C^{m \times n}$ and a unique Hermitian positive semi-definite matrix $H \in \mathbb C^{n \times n}$ such that

\[A = UH,\quad U^\ast U = I _ n\]

and if $A$ is nonsingular, then $H$ is positive definite and $U$ is uniquely determined.

@State a theorem describing a best approximation property for the unitary factor $U$, and describe a useful special case.

Let $A, B \in \mathbb C^{m \times n}$ and let $B^\ast A \in \mathbb C^{n \times n}$ have the polar decomposition

\[B^\ast A = UH\]

then for any unitary $Z \in \mathbb C^{n \times n}$, we have

\[\vert \vert A - BU \vert \vert _ F \le \vert \vert A - BZ \vert \vert _ F \le \vert \vert A + BU \vert \vert _ F\]

when $m = n$ and $B = I$, this simplifies to

\[\vert \vert A - U \vert \vert _ F \le \vert \vert A - Z \vert \vert _ F \le \vert \vert A + U \vert \vert _ F\]

Hence the closest unitary matrix (as measured in the Frobenius norm) to $A$ is given by the unitary factor of the polar decomposition, and the furthest is $-U$.

Suppose we have the SVD of a matrix $A \in \mathbb C^{m \times n}$ given by

\[A = P \Sigma Q^\ast\]

where $P \in \mathbb C^{m \times m}$ and $Q \in \mathbb C^{n \times n}$ are unitary and $\Sigma = \text{diag}(\sigma _ 1, \ldots, \sigma _ n)$ where $\sigma _ 1 \ge \cdots \ge \sigma _ n \ge 0$, how may we compute the polar decomposition $A = UH$?

Partition $P = [\underbrace{P _ 1} _ {n} \mid \underbrace{P _ 2} _ {m -n}]$ where $P^\ast _ 1 = P _ 1 = I _ n$. Then $A$ has polar decomposition $A = UH$ where

\[\begin{aligned} U &= P _ 1 Q^\ast \\ H &= Q \Sigma Q^\ast \end{aligned}\]

Suppose we have the polar decomposition $A = UH \in \mathbb C^{n \times n}$. How may we compute the SVD of $A$?

Find a spectral decomposition of $H = Q \Sigma Q^\ast$, then

\[A = (UQ)\Sigma Q^\ast\]

Suppose we have the polar decomposition

\[A = UH \in \mathbb C^{n \times n}\]

How may we compute $H$ in terms of a matrix square root?
\[H = \sqrt{A^\ast A}\]

Suppose we have the polar decomposition

\[A = UH \in \mathbb C^{n \times n}\]

Can you state a relationship between the eigenvalues and singular values of $H$ and $A$?
\[\lambda(H) = \sigma(H) = \sigma(A)\]

Suppose we have the polar decomposition

\[A = UH \in \mathbb C^{n \times n}\]

Can you state a relationship between the condition numbers of $H$ and $A$?
\[\kappa _ 2(H) = \kappa _ 2(A)\]

Suppose we have the polar decomposition

\[A = UH \in \mathbb C^{n \times n}\]

Can you give a necessary and sufficient condition for $A$ to be normal?
\[UH = HU\]

The polar decomposition theorem for a matrix $A \in \mathbb C^{m \times n}$ says there exists a matrix $U \in \mathbb C^{m \times n}$ and a unique Hermitian positive semi-definite matrix $H \in \mathbb C^{n \times n}$ such that

\[A = UH,\quad U^\ast U = I _ n\]

and if $A$ is nonsingular, then $H$ is positive definite and $U$ is uniquely determined.

@State a theorem describing a best approximation property for the Hermitian polar factor $H$.
  • $\frac 1 2 (A + H)$ is a best Hermitian positive semi-definite approximation to $A$ in the $2$-norm
  • For any Hermitian positive semi-definite $X \in \mathbb C^{n \times n}$, we have $ \vert \vert A - H \vert \vert _ 2 \le 2 \vert \vert A - X \vert \vert _ 2$.

The polar decomposition theorem for a matrix $A \in \mathbb C^{m \times n}$ says there exists a matrix $U \in \mathbb C^{m \times n}$ and a unique Hermitian positive semi-definite matrix $H \in \mathbb C^{n \times n}$ such that

\[A = UH,\quad U^\ast U = I _ n\]

and if $A$ is nonsingular, then $H$ is positive definite and $U$ is uniquely determined.

@State a theorem giving perturbation bounds for these polar factors.

Suppose:

  • $\varepsilon = \vert \vert \Delta A \vert \vert _ F / \vert \vert A \vert \vert _ F$
  • $\kappa _ F(A) \epsilon < 1$

Then $A + \Delta A$ has the polar decomposition

\[A + \Delta A = (U + \Delta U)(H + \Delta H)\]

where

\[\begin{aligned} &\frac{ \vert \vert \Delta H \vert \vert _ 2}{ \vert \vert H \vert \vert _ 2} \le \sqrt 2 \varepsilon + O(\varepsilon^2) \\ &\frac{ \vert \vert \Delta H \vert \vert _ 2}{ \vert \vert H \vert \vert _ 2} \le (1 + \sqrt 2) \kappa _ F(A)\varepsilon + O(\varepsilon^2) \end{aligned}\]

Other notes

  • Can be extended to rectangular matrices
  • Taking the determinant of the decomposition gives the ordinary polar form of the determinant
  • The $P$ factor is always unique even if $A$ is singular, denoted $P = (A^\ast A)^{1/2}$. If $A$ is nonsingular then uniqueness is implied just by inverting
  • If the SVD of $A$ is $A = W \Sigma V^\ast$, then:
    • $P = V \Sigma V^\ast$
    • $U = WV^\ast$


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