Polar Linear Algebra: Frameworks & Applications

• Polar Linear Algebra is a collection of frameworks that utilize polar grids, projective polarity, and matrix decompositions to unify concepts of spectral analysis, duality, and orthogonality.
• It employs techniques like circular convolution, Fourier diagonalization, and self-adjoint spectral constraints to enhance data analysis and learning architectures.
• Practical algorithms include FFT-based polar convolution and iterative polar decomposition methods, offering efficient solutions in both classical and quantum computational settings.

Polar Linear Algebra is used in the literature for several mathematically distinct but intersecting frameworks. In one usage, it is a linear–algebraic and spectral framework tailored to data living on polar grids, with a linear radial coordinate and a periodic angular coordinate (Guasti, 30 Mar 2026). In another, it is a purely linear-algebraic calculus of projective polarity on homogeneous coordinates, encoded by invertible $(n+2)\times(n+2)$ matrices (Nielsen et al., 5 Mar 2026). In matrix analysis, it centers on the polar decomposition $A=UH$, where $U$ has orthonormal columns and $H$ is Hermitian positive definite or positive semidefinite (Gawlik et al., 2016). The literature also uses the term for generalized polarization identities, tensor calculus in polar coordinates, cone-preserving linear maps, and matroidal polar invariants. This suggests a common emphasis on polar geometry, duality, orthogonality, or polarization as the organizing linear-algebraic structure.

1. Polar grids, convolution, and spectral operator algebras

In “Foundations of Polar Linear Algebra,” the basic object is a discrete polar grid

$r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$

together with “polar matrices” $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$. The radial dimension is treated as a stack of independent channels, while the angular dimension is treated as periodic and governed by circular convolution and the discrete Fourier transform. The central bilinear operation is the “polar product”

$$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$

with identity $E[r,\theta]=\delta_{\theta 0}$. The associated algebra is distributive, associative, scalar compatible, and commutative; this commutativity implies simultaneous diagonalization by the angular Fourier basis. The polar transpose and polar adjoint are defined by

$$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$

so that, in the real-valued case, self-adjointness reduces to evenness in $\theta$ (Guasti, 30 Mar 2026).

The spectral structure is explicit rather than incidental. For fixed radius $A=UH$0, the angular DFT diagonalizes the circulant convolution operator associated with a kernel $A=UH$1, and the polar convolution theorem takes the form

$A=UH$2

Each pair $A=UH$3 is therefore an independent spectral coordinate. Discrete rotations (“rotors”)

$A=UH$4

share the same Fourier eigenbasis, with eigenvalues $A=UH$5. The framework thus decomposes into orthogonal, non-interacting eigenmodes indexed by $A=UH$6, unless cross-mode couplings are added explicitly.

A central theoretical theme is the use of self-adjoint-inspired spectral constraints. For a real-valued polar matrix, $A=UH$7 implies that each associated circulant operator has a real spectrum. In learning architectures, this motivates constraining spectral multipliers $A=UH$8 to be real-valued, using real symmetric channel-mixing matrices, and designing angular gating filters with real Fourier coefficients. The stated effect is to remove arbitrary phase rotations across modes, bias operators toward energy-preserving or energy-selective behavior, yield more stable training dynamics, and facilitate interpretation because spectral responses become real scalings of modes.

The paper develops both “purely polar” and “fully spectral” neural operator models. The stated MNIST pipeline consists of a Cartesian-to-polar transform $A=UH$9, radial equalization $U$0, channel lifting by a $U$1 convolution, complex-valued features, a sequence of PolarFNO-like blocks, and a classifier head. The baseline polar FNO, for $U$2 channels, $U$3 blocks, and $U$4, reaches ~95.0% validation accuracy and ~94.9% test accuracy after 2 epochs. A fully spectral architecture, with a single FFT at input and spectral features consumed directly by the classifier head, reaches ~94.6% test accuracy on MNIST. A self-adjoint-inspired spectral architecture converges very fast, with $U$5 validation accuracy after 1 epoch, and after 16 epochs reaches 96.47% validation accuracy and 96.50% test accuracy with negligible generalization gap (Guasti, 30 Mar 2026).

The computational claims are equally structural. Naive polar convolution costs $U$6, whereas FFT-based implementation costs $U$7. In spectral coordinates, a polar operator acts by

$U$8

which exposes three levels of natural parallelization: radial parallelism, angular-frequency parallelism, and channel parallelism. The work describes this as a mathematically clean route to spectral tensor parallelism, orthogonal to data parallelism, pipeline parallelism, and expert parallelism.

2. Projective polarity and convex duality

A different usage of Polar Linear Algebra appears in “Quadratic polarity and polar Fenchel-Young divergences from the canonical Legendre polarity.” Here the point of departure is projective geometry: a polarity is an involutive correspondence between points and hyperplanes, realized in homogeneous coordinates by a bilinear form

$U$9

For a convex set $H$0, the induced polarity is

$H$1

and the polar hyperplane of a point $H$2 is

$H$3

For closed convex $H$4, polarity is thus realized linearly in homogeneous coordinates, and for nondegenerate $H$5 it is an involution on closed convex sets. The paper describes this as a purely linear-algebraic calculus of polarities on $H$6 matrices (Nielsen et al., 5 Mar 2026).

The canonical case is the Legendre polarity, defined by the matrix

$H$7

If $H$8 is closed and proper convex, then the boundary of the polarity of the graph of $H$9 recovers the graph of the convex conjugate: $r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$0 This reformulates Legendre–Fenchel conjugation as a projective polarity statement. The paper’s main structural result is that every quadratic polarity can be represented either as a deformation of the output of the canonical Legendre polarity or as the Legendre polarity of a deformed primal convex body. If $r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$1, then

$r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$2

if $r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$3, then

$r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$4

The matrices $r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$5 and $r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$6 satisfy

$r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$7

This framework also yields a polarity-based account of information-geometric divergences. For the canonical Legendre polarity, the polar Fenchel–Young divergence is

$r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$8

When $r=0,\dots,N_r-1,\qquad \theta=0,\dots,N_\theta-1,$9, with $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$0 and $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$1, one obtains

$A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$2

so polar Fenchel–Young divergences reduce to classical Fenchel–Young, and hence Bregman, divergences. A normalized version,

$A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$3

with $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$4, gives a polarity interpretation of total Bregman divergences. In this strand of the subject, Polar Linear Algebra names a unifying matrix formalism for projective geometry, convex duality, and information geometry.

3. Polar decomposition of matrices

In matrix analysis, polar linear algebra is organized around the polar decomposition

$A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$5

for $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$6 with $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$7, where $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$8 and $A[r,\theta]\in\mathbb{R}^{N_r\times N_\theta}$9 is Hermitian positive definite when $$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$0 has full column rank. The map $$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$1 takes values in the Stiefel manifold and is the orthogonal projection, in Frobenius norm, onto

$$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$2

The Fréchet derivative of this map is a central object. “Computing the Fréchet Derivative of the Polar Decomposition” shows that if $$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$3 and

$$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$4

then

$$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$5

so the off-diagonal block of a matrix sign function yields the derivative of the polar factor. The same paper derives the Lyapunov equation

$$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$6

with $$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$7, and develops coupled Newton and Newton–Schulz iterations that compute both $$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$8 and $$(A\otimes B)[r,\theta]=\sum_{k=0}^{N_\theta-1} A[r,k]\;B[r,(\theta-k)\bmod N_\theta],$$9 for square and rectangular matrices (Gawlik et al., 2016).

The geometric interpretation of polar decomposition has also been developed directly. “Making matrices better: Geometry and topology of polar and singular value decomposition” treats $E[r,\theta]=\delta_{\theta 0}$0 as a sphere after radial projection and describes the orthogonal group as a submanifold, singular matrices as stratified varieties, and the polar factor as the nearest orthogonal neighbor. In this language, the orthogonal factor in $E[r,\theta]=\delta_{\theta 0}$1 is the Frobenius-norm projection onto $E[r,\theta]=\delta_{\theta 0}$2, while the singular value decomposition gives the nearest lower-rank neighbor by truncating singular values (DeTurck et al., 2017).

Recent work has also recast polar decomposition as a nonconvex optimization problem on the orthogonal group. “A geodesic convexity-like structure for the polar decomposition of a square matrix” studies

$E[r,\theta]=\delta_{\theta 0}$3

which is equivalent to the square orthogonal Procrustes problem and whose minimizer is the orthogonal polar factor of $E[r,\theta]=\delta_{\theta 0}$4. The paper proves a geodesic weak quasi-convexity and weak quasi-strong-convexity structure, shows that $E[r,\theta]=\delta_{\theta 0}$5 is $E[r,\theta]=\delta_{\theta 0}$6-smooth with $E[r,\theta]=\delta_{\theta 0}$7, and establishes that gradient descent in the orthogonal group computes the polar factor with linear convergence rate if the matrix is invertible and with an algebraic one if the matrix is singular (Alimisis et al., 2024).

On the algorithmic side, large-scale polar factorization has been mapped efficiently to accelerator hardware. “Large Scale Distributed Linear Algebra With Tensor Processing Units” uses the rectangular polar decomposition

$E[r,\theta]=\delta_{\theta 0}$8

with $E[r,\theta]=\delta_{\theta 0}$9 and $$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$0 Hermitian positive semidefinite, and implements a preconditioned Newton–Schulz iteration

$$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$1

together with a preconditioning polynomial. The complete algorithm uses about 25 iterations, corresponding to roughly 50 large matrix multiplications, and on a full TPUv3 pod with 2048 cores the estimated polar decomposition time for a dense square matrix with $$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$2 is about 20 minutes (Lewis et al., 2021).

A quantum version of the same object appears in “Fast algorithm for quantum polar decomposition, pretty-good measurements, and the Procrustes problem.” For

$$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$3

the canonical polar isometry is

$$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$4

Using the quantum singular value transform, the paper gives a block-encoding based algorithm with query complexity $$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$5 for the singular-vector transformation, and applies it to pretty-good measurements and the quantum Procrustes problem, obtaining polynomial advantages in size and condition number and an exponential speedup in precision over density-matrix-exponentiation-based approaches (Quek et al., 2021).

4. Scalar products, polarization, and indefinite metrics

A separate but related strand studies how quadratic data determine bilinear or sesquilinear structure. “Polarization Identities” generalizes the classical real and complex polarization identities by replacing the scalar field with a finite-dimensional real associative unital $$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$6-algebra $$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$7. If $$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$8 is an $$A^{T_p}[r,\theta]=A[r,(-\theta)\bmod N_\theta],\qquad A^{\dagger_p}[r,\theta]=\overline{A[r,(-\theta)\bmod N_\theta]},$$9-valued Hermitian form and $\theta$0, then for a polarizing subgroup $\theta$1 one has

$\theta$2

where $\theta$3. In the complex case this recovers

$\theta$4

The paper classifies polarizable algebras as finite direct sums

$\theta$5

with standard conjugate-transpose involution, and proves a generalized Jordan–von Neumann theorem: a quadrance $\theta$6 arises from a Hermitian form if and only if the parallelogram identity holds (Bender et al., 2022).

Indefinite and weighted scalar products lead to new forms of polar decomposition. “A New Polar Decomposition in a Scalar Product Space” considers nonsingular matrices $\theta$7 and $\theta$8, the $\theta$9-adjoint

$A=UH$00

and factorizations

$A=UH$01

The new right and left decompositions exist and are unique for any nonsingular matrices $A=UH$02 and $A=UH$03 when $A=UH$04 satisfies

$A=UH$05

and $A=UH$06 or $A=UH$07 is nonsingular. The factor $A=UH$08 is selfadjoint with respect to suitably defined scalar products and has eigenvalues only in the open right half-plane. The paper emphasizes that its assumptions are in some respects weaker and in some respects stronger than those of previous generalized polar decompositions (Sui et al., 2016).

These two lines of work are distinct in scope but closely related in algebraic spirit. One reconstructs Hermitian forms from quadratic data by averaging over compact multiplicative groups; the other constructs polar-type factorizations in bilinear and sesquilinear settings where the relevant adjoint is not Euclidean.

5. Coordinate-adapted calculus and noncommutative phase space

Polar linear algebra also appears as a coordinate-adapted operator calculus. “Tensor calculus in polar coordinates using Jacobi polynomials” develops spectral bases on the unit disk for scalar, vector, and tensor partial differential equations. With

$A=UH$09

the radial basis automatically satisfies regularity conditions at $A=UH$10. Differentiation is represented by sparse raising and lowering operators: $A=UH$11

$A=UH$12

Together with multiplication operators $A=UH$13 and conversion operators $A=UH$14, these satisfy a Heisenberg algebra,

$A=UH$15

so that covariant derivatives, multiplication by azimuthally symmetric functions, and tensorial relations all become banded matrix operations on coefficient vectors. The paper’s stated innovation is to use a larger set of possible bases to achieve maximum bandedness of linear operations (Vasil et al., 2015).

A noncommutative version is developed in “Polar Coordinates and Noncommutative Phase Space.” The Weyl transform

$A=UH$16

maps $A=UH$17 unitarily onto Hilbert–Schmidt operators on $A=UH$18, with

$A=UH$19

The radial operator is

$A=UH$20

and the angular momentum operator satisfies

$A=UH$21

Using Hermite operators $A=UH$22, their reindexing $A=UH$23, and Laguerre–Gaussian functions, the paper realizes the polar transformation $A=UH$24 through explicit orthogonal function expansions. In this setting, polar coordinates become an operator basis in noncommutative phase space rather than a mere change of variables (Krueger, 2016).

6. Representation theory, cones, and matroidal polar invariants

Several further uses of Polar Linear Algebra emphasize global geometry and combinatorics. In representation theory, a polar representation of a compact connected Lie group $A=UH$25 is an orthogonal representation on $A=UH$26 admitting a linear subspace $A=UH$27 that intersects every orbit and does so orthogonally. With the polar group

$A=UH$28

where $A=UH$29 is the principal isotropy subgroup, one has

$A=UH$30

“A Note on Polar Representations” proves that a polar representation of a connected compact Lie group is determined up to linear isomorphism by a history and its dimension, where the history is the collection of isotropy subgroups that occur along a chamber in a section (Gozzi, 2017).

Cone-theoretic optics provides another variant. In “The Cone of Mueller Matrices,” Stokes vectors form the future light cone

$A=UH$31

and a Mueller matrix is a real $A=UH$32 matrix $A=UH$33 satisfying

$A=UH$34

The set

$A=UH$35

is the Mueller cone, a solid cone in the 16-dimensional vector space of real $A=UH$36 matrices. The paper studies its cone-theoretic properties and gives computational programs to decide whether a matrix is Mueller, approximate a matrix by a Mueller matrix, approximate a Mueller matrix by Mueller invertibles or by a Stokes-cone-primitive Mueller matrix, and implement an Eigenvalue Calibration Method (Takane et al., 2023).

At the intersection of projective duality and matroid theory, “Polar Degrees of Matroids” computes the polar degrees of reciprocal linear spaces. If $A=UH$37 is a linear subspace not contained in any coordinate hyperplane, $A=UH$38 is its coordinate-wise inverse, and $A=UH$39 is the associated matroid, then the polar degrees $A=UH$40 are governed by the reduced characteristic polynomial $A=UH$41: $A=UH$42 The proof compares the geometry of conormal varieties of reciprocal linear spaces with the combinatorial conormal fan $A=UH$43 inside the bipermutohedral fan. As a corollary, $A=UH$44 is dual defective if and only if $A=UH$45 is not connected, and if $A=UH$46 is connected then

$A=UH$47

twice the beta invariant of $A=UH$48 (Briand et al., 24 Jun 2026).

Taken together, these strands suggest that Polar Linear Algebra is not a single universal formalism but a family of precise frameworks in which polar geometry, polarity, polar decomposition, polarization, or cone structure controls spectral behavior, duality, orthogonality, or combinatorial invariants.