What is the Polar Decomposition?

A polar decomposition of $A\in\mathbb{C}^{m \times n}$ with $m \ge n$ is a factorization $A = UH$ , where $U\in\mathbb{C}^{m \times n}$ has orthonormal columns and $H\in\mathbb{C}^{n \times n}$ is Hermitian positive semidefinite. This decomposition is a generalization of the polar representation $z = r \mathrm{e}^{\mathrm{i}\theta}$ of a complex number, where $H$ corresponds to $r\ge 0$ and $U$ to $\mathrm{e}^{\mathrm{i}\theta}$ . When $A$ is real, $H$ is symmetric positive semidefinite. When $m = n$ , $U$ is a square unitary matrix (orthogonal for real $A$ ).

We have $A^*A = H^*U^*UH = H^2$ , so $H = (A^*\!A)^{1/2}$ , which is the unique positive semidefinite square root of $A^*A$ . When $A$ has full rank, $H$ is nonsingular and $U = AH^{-1}$ is unique, and in this case $U$ can be expressed as

$U = \displaystyle\frac{2}{\pi} A \displaystyle\int_{0}^{\infty} (t^2I+A^*\!A)^{-1}\, \mathrm{d}t.$

An example of a polar decomposition is

$A = \left[\begin{array}{@{}rr} 4 & 0\\ -5 & -3\\ 2 & 6 \end{array}\right] = \sqrt{2}\left[\begin{array}{@{}rr} \frac{1}{2} & -\frac{1}{6}\\[\smallskipamount] -\frac{1}{2} & -\frac{1}{6}\\[\smallskipamount] 0 & \frac{2}{3} \end{array}\right] \cdot \sqrt{2}\left[\begin{array}{@{\,}rr@{}} \frac{9}{2} & \frac{3}{2}\\[\smallskipamount] \frac{3}{2} & \frac{9}{2} \end{array}\right] \equiv UH.$

For an example with a rank-deficient matrix consider

$A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix},$

for which $A^*A = \mathrm{diag}(0,1,1)$ and so $H = \mathrm{diag}(0,1,1)$ . The equation $A = UH$ then implies that

$U = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ \theta & 0 & 0 \end{bmatrix}, \quad |\theta| = 1,$

so $U$ is not unique.

The polar factor $U$ has the important property that it is a closest matrix with orthonormal columns to $A$ in any unitarily invariant norm. Hence the polar decomposition provides an optimal way to orthogonalize a matrix. This method of orthogonalization is used in various applications, including in quantum chemistry, where it is called Löwdin orthogonalization. Most often, though, orthogonalization is done through QR factorization, trading optimality for a faster computation.

An important application of the polar decomposition is to the orthogonal Procrustes problem¹

$\min \bigl\{\, \|A-BW\|_F: W \in \mathbb{C}^{n\times n},\; W^*W = I \,\bigr\},$

where $A,B\in\mathbb{C}^{m\times n}$ and the norm is the Frobenius norm $\|A\|_F^2 = \sum_{i,j} |a_{ij}|^2$ . This problem, which arises in factor analysis and in multidimensional scaling, asks how closely a unitary transformation of $B$ can reproduce $A$ . Any solution is a unitary polar factor of $B^*\!A$ , and there is a unique solution if $B^*\!A$ is nonsingular. Another application of the polar decomposition is in 3D graphics transformations. Here, the matrices are $3\times 3$ and the polar decomposition can be computed by exploiting a relationship with quaternions.

For a square nonsingular matrix $A$ , the unitary polar factor $U$ can be computed by a Newton iteration:

$X_{k+1} = \displaystyle\frac{1}{2} (X_k + X_k^{-*}), \quad X_0 = A.$

The iterates $X_k$ converge quadratically to $U$ . This is just one of many iterations for computing $U$ and much work has been done on the efficient implementation of these iterations.

If $A = P \Sigma Q^*$ is a singular value decomposition (SVD), where $P\in\mathbb{C}^{m\times n}$ has orthonormal columns, $Q\in\mathbb{C}^{n\times n}$ is unitary, and $\Sigma$ is square and diagonal with nonnegative diagonal elements, then

$A = PQ^* \cdot Q \Sigma Q^* \equiv UH,$

where $U$ has orthonormal columns and $H$ is Hermitian positive semidefinite. So a polar decomposition can be constructed from an SVD. The converse is true: if $A = UH$ is a polar decomposition and $H = Q\Sigma Q^*$ is a spectral decomposition ( $Q$ unitary, $D$ diagonal) then $A = (UQ)\Sigma Q^* \equiv P \Sigma Q^*$ is an SVD. This latter relation is the basis of a method for computing the SVD that first computes the polar decomposition by a matrix iteration then computes the eigensystem of $H$ , and which is extremely fast on distributed-memory manycore computers.

The nonuniqueness of the polar decomposition for rank deficient $A$ , and the lack of a satisfactory definition of a polar decomposition for $m < n$ , are overcome in the canonical polar decomposition, defined for any $m$ and $n$ . Here, $A = UH$ with $U$ a partial isometry, $H$ is Hermitian positive semidefinite, and $U^*U = HH^+$ . The superscript “ $+$ ” denotes the Moore–Penrose pseudoinverse and a partial isometry can be characterized as a matrix $U$ for which $U^+ = U^*$ .

Generalizations of the (canonical) polar decomposition have been investigated in which the properties of $U$ and $H$ are defined with respect to a general, possibly indefinite, scalar product.

References

This is a minimal set of references, which contain further useful references within.

Nicholas J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008. (Chapter 8.)
Nicholas Higham, Christian Mehl, and Françoise Tisseur, The canonical generalized polar decomposition, SIAM J. Matrix Anal. Appl. 31(4), 2163–2180, 2010.
Nicholas J. Higham and Vanni Noferini, An algorithm to compute the polar decomposition of a $3 \times 3$ matrix, Numer. Algorithms 73, 349–369, 2016.
Yuji Nakatsukasa and Nicholas J. Higham, Stable and efficient spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD, SIAM J. Sci. Comput. 35(3), A1325–A1349, 2013.
Dalal Sukkari, Hatem Ltaief, Aniello Esposito, and David Keyes, A QDWH-based SVD software framework on distributed-memory manycore systems, ACM Trans. Math. Software 45, 18:1–18:21, 2019.

Footnotes:

Procrustes: an ancient Greek robber who tied his victims to an iron bed, stretching their legs if too short for it, and lopping them if too long.

2 thoughts on “What is the Polar Decomposition?”

I’m very much enjoying this “What is” series.

One question: At the beginning of this post you write: “This decomposition is a generalization of the polar representation z = r*e^(i*theta) of a complex number”. Can this analogy be taken further using the matrix exponential, setting U=e^V for some other matrix V?

Yes – there are standard results on writing an orthogonal or unitary matrix as exp(S) for a skew-symmetric or skew-Hermitian matrix.

What is the Polar Decomposition?

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