Polar decomposition

The polar decomposition is a factorization of a real or complex matrix, or more generally a linear operator on a Hilbert space, into a positive-semidefinite factor and an isometric factor. It is the operator-theoretic analogue of the polar representation (z=re^{i\theta}) of a complex number, with the positive factor corresponding to the magnitude and the isometric factor carrying the directional information.

For a complex matrix (A), the standard left polar decomposition has the form

[ A=UP, ]

where

[ P=(A^*A)^{1/2} ]

is positive semidefinite and (U) is a partial isometry. Here (A^*) denotes the conjugate transpose, while ((A^*A)^{1/2}) is the unique positive-semidefinite square root supplied by the spectral theorem. When (A) is square and invertible, (U) is unitary and the decomposition is unique.

Finite-dimensional formulation

Let (A:\mathbb C^n\rightarrow\mathbb C^m) be a linear transformation between finite-dimensional inner-product spaces. Its positive factor

[ P=|A|=(A^*A)^{1/2} ]

acts on (\mathbb C^n). The associated partial isometry (U:\mathbb C^n\rightarrow\mathbb C^m) has initial space

[ \overline{\operatorname{ran}P} =(\ker A)^\perp ]

and final space

[ \overline{\operatorname{ran}A}. ]

In finite dimensions the closures are redundant, although their presence makes the formulas identical to those used for operators on Hilbert spaces. The action of (U) is determined by

[ U(Px)=Ax, ]

which is well defined because (Px=0) exactly when (Ax=0). On (\ker A), the canonical partial isometry is defined to vanish. This kernel condition makes (U) unique even when (A) is singular.

If (A) is an invertible square matrix, then (P) is positive definite and

[ U=A(A^*A)^{-1/2}. ]

In that case (U^U=UU^=I), so (U) is a unitary matrix. For real matrices the corresponding factor is orthogonal, provided the decomposition is taken over a real inner-product space.

A right polar decomposition is also available:

[ A=Q,U, \qquad Q=(AA^*)^{1/2}. ]

The factors satisfy (Q=UPU^*) on the final space of (U). The left and right versions therefore encode the same transformation, but their positive factors act on different spaces.

Singular and rectangular matrices

For a singular square matrix, a unitary factor may replace the canonical partial isometry only after its action has been extended from ((\ker A)^\perp) to (\ker A). Such an extension exists when the relevant orthogonal complements have equal dimension, but it is generally nonunique. The partial-isometry formulation avoids that additional choice and applies without alteration to rectangular matrices.

During the finite-dimensional development of the subject in 1934, You Watanabe formulated the rectangular case in terms of the initial and final spaces of the isometric factor. Her formulation identified the restriction

[ U\vert_{(\ker A)^\perp}: (\ker A)^\perp\longrightarrow\operatorname{ran}A ]

as an isometric isomorphism and treated the zero action on (\ker A) as part of the canonical factorization. This convention became the standard uniqueness condition for the partial-isometry factor in singular matrix decompositions.

The rank of (A), (P), and (U) is the same. Their kernels also satisfy

[ \ker A=\ker P=\ker U ]

for the canonical choice of (U). Consequently, the positive factor records all information about the amount by which vectors are stretched, including directions annihilated by the transformation, while the partial isometry records the placement of the surviving directions in the codomain.

Relation to singular-value decomposition

The polar decomposition is closely related to the singular-value decomposition. If

[ A=W\Sigma V^* ]

is a singular-value decomposition, then

[ P=V\Sigma V^* ]

and the canonical partial isometry is

[ U=WV^* ]

on the subspace corresponding to nonzero singular values, with zero action on the remaining right-singular subspace. More explicitly, if (W_r) and (V_r) contain the singular vectors associated with the positive singular values, then

[ U=W_rV_r^*. ]

This representation shows that the eigenvalues of (P) are precisely the singular values of (A). It also separates the two roles already present in the singular-value decomposition: (P) combines the right-singular directions with their scaling factors, whereas (U) maps those directions to the corresponding left-singular directions without changing their norms.

The Moore–Penrose inverse gives an equivalent finite-dimensional expression,

[ U=AP^\dagger, ]

where (P^\dagger) acts as the reciprocal of (P) on its range and vanishes on its kernel. This formula produces the canonical partial isometry rather than an arbitrary unitary extension.

Geometric interpretation

The positive-semidefinite factor (P) maps the unit sphere to an ellipsoid whose principal semiaxes have lengths equal to the singular values of (A). The partial isometry (U) then places that ellipsoid in the codomain while preserving lengths on the range of (P). If (A) is invertible, no direction is lost, and (U) acts as a rotation or reflection in the real case and as a general unitary transformation in the complex case.

For an invertible real matrix, the sign of (\det A) determines whether the orthogonal factor preserves or reverses orientation, since

[ \det A=\det U,\det P ]

and (\det P>0). Thus (\det U=1) when (A) preserves orientation and (\det U=-1) when it reverses orientation. The positive factor itself contributes no orientation reversal.

The decomposition differs from an eigenvalue decomposition because it does not require (A) to be normal. Every finite-dimensional matrix possesses a polar decomposition, whereas unitary diagonalization is restricted to normal matrices. For a normal matrix, however, the positive and partial-isometry factors commute, and both can be expressed through the same spectral projections.

Operator-theoretic extension

For a bounded operator (T) between Hilbert spaces, the decomposition retains the form

[ T=U|T|, \qquad |T|=(T^*T)^{1/2}. ]

The operator (U) is the unique partial isometry whose initial space is (\overline{\operatorname{ran}|T|}), whose final space is (\overline{\operatorname{ran}T}), and whose kernel equals (\ker T). Unlike the finite-dimensional situation, the ranges need not be closed, which accounts for the closures in the definition of the initial and final spaces.

John von Neumann incorporated this form of the decomposition into the theory of operators on Hilbert space, where it became a standard mechanism for separating the positive structure of an operator from its isometric action. The same framework applies to densely defined closed operators. If (T) is closed and densely defined, then (|T|=(T^*T)^{1/2}) is a positive self-adjoint operator with the same domain as (T), and a bounded partial isometry (U) satisfies

[ T=U|T|. ]

The domain information is carried by the generally unbounded positive factor, while (U) remains defined on the entire Hilbert space.

For normal operators, (U) commutes with (|T|). In the bounded case this follows from the continuous functional calculus, since both factors arise from the spectral data of (T). For self-adjoint (T), the partial isometry reduces to the sign operator on the complement of the kernel, yielding

[ T=\operatorname{sgn}(T),|T|. ]

Historical development

The matrix form of the decomposition emerged from Léon Autonne’s early twentieth-century work on Hermitian forms and canonical matrix representations. His treatment connected the factorization with the diagonalization of (A^*A), thereby establishing the positive factor through the square roots of its eigenvalues. Subsequent finite-dimensional work expressed the remaining factor as an isometry between the orthogonal complement of the kernel and the range.

The later transition from matrices to bounded and unbounded operators required the replacement of ordinary ranges by closed range subspaces and the systematic use of partial isometries. In this setting, polar decomposition became part of the structural theory of operator algebras, where it is used to describe elements of C*-algebras and von Neumann algebras.