In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... For the square matrix section, see square matrix. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. ...

where r is the absolute value of z (a positive real number), and e^iθ is the complex sign of z. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

Matrix polar decomposition

The polar decomposition of complex matrix A is a matrix decomposition of the form In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ...

where U is a unitary matrix and P is a positive-definite Hermitian matrix. This decomposition exists and is unique as long as A is non-singular. Note that In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

gives the corresponding polar decomposition of the determinant of A, since detP = r = | detA | and detU = e^iθ. In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

The matrix P is given by

where A* denotes the conjugate transpose of A. This expression is meaningful since a positive-definite Hermitian matrix has a unique positive square root. The matrix U is then given by In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or...

U = AP ^{− 1}.

One can check that P is positive-definite and that U is unitary.

One can also decompose A in the form

Here U is the same as before and P′ is given by

The matrix A is normal if and only if P′ = P. A complex square matrix A is a normal matrix iff where A* is the conjugate transpose of A (if A is a real matrix, this is the same as the transpose of A). ...

The map from the general linear group GL(n,C) to the unitary group U(n) defined by mapping A onto its unitary piece U gives rise to a homotopy equivalence since the space of positive-definite matrices is contractible. In fact U(n) is the maximal compact subgroup of GL(n,C). In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i. ... In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. ...

Hilbert space

The polar decomposition of any bounded linear operator A between complex Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative self-adjoint operator. In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In functional analysis a partial isometry is a linear map W between Hilbert spaces H, K such that there is a closed vector subspace H1 of H such that W restricted to H1 is an isometric map and W restricted to the orthogonal complement of H1 is zero. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

Viewing complex numbers as the bounded linear mappings acting by multiplication on the complex numbers, this factorizes any bounded linear mapping z : C → C uniquely as a product of the non-negative self-adjoint operator r and the unitary operator e^iθ.

This generalizes as follows: if A is a bounded linear operator then there is a unique factorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator and the initial space of U is the closure of the range of P. In fact the only possibility for P is to take

since

(the second equality here follows from the fact that U ^* U is the projection onto the closure of the range of P). We then have that

Observe

so there is an isometry U defined uniquely on the closure of the range of (A* A)^1/2 with the required properties.

Unbounded operators

If A is a closed, densely defined unbounded operator between complex Hilbert spaces then it still has a (unique) polar decomposition In functional analysis (a branch of mathematics), a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in...

where |A| is a (possibly unbounded) non-negative self adjoint operator with the same domain as A, and U is a partial isometry vanishing on the orthogonal complement of the range R(A).

The main step in the proof of this is to show that A^*A is a self adjoint operator; once this is done, the remainder of the argument is similar to the case of bounded operators.

See also

• Singular value decomposition