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HomeMatlabWhat's the Polar Decomposition? – Nick Higham

What’s the Polar Decomposition? – Nick Higham

A polar decomposition of $Ainmathbb{C}^{m times n}$ with $m ge n$ is a factorization $A = UH$, the place $Uinmathbb{C}^{m times n}$ has orthonormal columns and $Hinmathbb{C}^{n times n}$ is Hermitian constructive semidefinite. This decomposition is a generalization of the polar illustration $z = r mathrm{e}^{mathrm{i}theta}$ of a fancy quantity, the place $H$ corresponds to $rge 0$ and $U$ to $mathrm{e}^{mathrm{i}theta}$. When $A$ is actual, $H$ is symmetric constructive semidefinite. When $m = n$, $U$ is a sq. unitary matrix (orthogonal for actual $A$).

We have now $A^*A = H^*U^*UH = H^2$, so $H = (A^*!A)^{1/2}$, which is the distinctive constructive semidefinite sq. root of $A^*A$. When $A$ has full rank, $H$ is nonsingular and $U = AH^{-1}$ is exclusive, and on this case $U$ could be expressed as

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

An instance 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 instance with a rank-deficient matrix contemplate

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

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 $U$ isn’t distinctive.

The polar issue $U$ has the necessary property that it’s a closest matrix with orthonormal columns to $A$ in any unitarily invariant norm. Therefore the polar decomposition gives an optimum solution to orthogonalize a matrix. This technique of orthogonalization is utilized in varied purposes, together with in quantum chemistry, the place it’s known as Löwdin orthogonalization. Most frequently, although, orthogonalization is finished by way of QR factorization, buying and selling optimality for a sooner computation.

An necessary utility of the polar decomposition is to the orthogonal Procrustes downside

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

the place $A,Binmathbb{C}^{mtimes n}$ and the norm is the Frobenius norm $|A|_F^2 = sum_{i,j} |a_{ij}|^2$. This downside, which arises in issue evaluation and in multidimensional scaling, asks how carefully a unitary transformation of $B$ can reproduce $A$. Any answer is a unitary polar issue of $B^*!A$, and there’s a distinctive answer if $B^*!A$ is nonsingular. One other utility of the polar decomposition is in 3D graphics transformations. Right here, the matrices are $3times 3$ and the polar decomposition could be computed by exploiting a relationship with quaternions.

For a sq. nonsingular matrix $A$, the unitary polar issue $U$ could be computed by a Newton iteration:

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

The iterates $X_k$ converge quadratically to $U$. This is only one of many iterations for computing $U$ and far work has been executed on the environment friendly implementation of those iterations.

If $A = P Sigma Q^*$ is a singular worth decomposition (SVD), the place $Pinmathbb{C}^{mtimes n}$ has orthonormal columns, $Qinmathbb{C}^{ntimes n}$ is unitary, and $Sigma$ is sq. and diagonal with nonnegative diagonal components, then

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

the place $U$ has orthonormal columns and $H$ is Hermitian constructive semidefinite. So a polar decomposition could be constructed from an SVD. The converse is true: if $A = UH$ is a polar decomposition and $H = QSigma Q^*$ 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 premise of a way for computing the SVD that first computes the polar decomposition by a matrix iteration then computes the eigensystem of $H$, and which is extraordinarily quick on distributed-memory manycore computer systems.

The nonuniqueness of the polar decomposition for rank poor $A$, and the dearth of a passable definition of a polar decomposition for $m < n$, are overcome within the canonical polar decomposition, outlined for any $m$ and $n$. Right here, $A = UH$ with $U$ a partial isometry, $H$ is Hermitian constructive semidefinite, and $U^*U = HH^+$. The superscript “$+$” denotes the Moore–Penrose pseudoinverse and a partial isometry could be characterised as a matrix $U$ for which $U^+ = U^*$.

Generalizations of the (canonical) polar decomposition have been investigated during which the properties of $U$ and $H$ are outlined with respect to a normal, probably indefinite, scalar product.

References

It is a minimal set of references, which comprise additional helpful references inside.

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