Generalized Right Projection: Theory & Applications

• Generalized right projection is a framework defining various right‐biased projection mechanisms across multiple disciplines including algebra, geometry, and engineering.
• It encompasses specific applications in flag varieties, LMI control theory, singular eigenvalue problems, Banach space projections, *-ring structures, and right‐handed computer graphics.
• Its study reveals structural asymmetry that ensures method consistency, enabling rational resolutions and preserving key properties in projected models.

Generalized right projection is a polysemous technical term used across several research areas to denote a right-sided extension of an ordinary projection. In algebraic geometry it refers to the projection $\pi:G/B\to G/P$ and its projected Richardson strata; in linear matrix inequality theory it refers to elimination via projections onto right and left nullspaces; in singular generalized eigenvalue problems it denotes Petrov–Galerkin reduction onto a right subspace of dimension equal to the normal rank; in Banach-space and cone-duality settings it denotes asymmetric or complementary generalized projections; in $*$-rings it denotes a projection controlling right annihilators of a power; and in computer graphics it denotes a right-handed matrix interpolating between perspective and orthographic projection (Knutson et al., 2010, Meijer et al., 2023, Hochstenbach et al., 2022, Khan et al., 2022, Németh et al., 29 Apr 2025, Khairnar et al., 28 Aug 2025, MacIntosh, 2022).

1. Terminological scope

The phrase appears in several mathematically unrelated literatures, but each use preserves a right-sided asymmetry. In the flag-variety setting, the right-sidedness comes from projection to right cosets $W/W_P$ and from the quotient map $G/B\to G/P$. In control and convex LMI analysis, it comes from projection onto the nullspace of the right multiplier $R$. In singular pencil methods, it comes from selecting a right deflating subspace $V$. In Banach-space and cone-duality work, it comes from fixing the datum in the right argument of an asymmetric functional or from passing to the complementary map $I-P$. In $*$-ring theory, it comes from right annihilators. In graphics, it comes from right-handed camera coordinates (Knutson et al., 2010, Meijer et al., 2023, Hochstenbach et al., 2022, Németh et al., 29 Apr 2025, Khairnar et al., 28 Aug 2025, MacIntosh, 2022).

Domain	Projected object	Meaning of “right”
Flag varieties	$G/B\to G/P$, $W\to W/W_P$	Right cosets, right representatives
LMI/control	$*$0	Right nullspace of $*$1
Singular pencils	$*$2	Right subspace $*$3
Banach/cone projection	$*$4, $*$5	Right argument or right complement
$*$6-rings	$*$7	Right annihilator structure
Graphics	$*$8	Right-handed camera space

A plausible implication is that “generalized right projection” is best understood as a family of right-biased projection mechanisms rather than a single invariant definition.

2. Flag varieties, Richardson strata, and $*$9-Bruhat order

In the geometric and combinatorial literature, the generalized right projection is the pair of maps

$W/W_P$0

where $W/W_P$1 is a connected reductive group over an algebraically closed field, $W/W_P$2 and $W/W_P$3 are opposite Borels, $W/W_P$4 is parabolic, $W/W_P$5 is the Weyl group, and $W/W_P$6 is the set of minimal right coset representatives. For $W/W_P$7, the Schubert, opposite Schubert, and Richardson varieties in $W/W_P$8 are

$W/W_P$9

For $G/B\to G/P$0, the projected Richardson variety is

$G/B\to G/P$1

with open stratum $G/B\to G/P$2 (Knutson et al., 2010).

A central point is that $G/B\to G/P$3 need not be a Richardson variety on $G/B\to G/P$4. The paper gives an $G/B\to G/P$5 counterexample: projecting the big open Richardson cell in $G/B\to G/P$6 to $G/B\to G/P$7 yields $G/B\to G/P$8 minus two points, whereas certain Richardson varieties in $G/B\to G/P$9 give complements of three lines. The projected image is therefore typically a projected Richardson variety rather than a Richardson variety in the target.

The geometry nonetheless retains many of the standard properties of Richardson varieties. Projected Richardson varieties $R$0 are normal and Cohen–Macaulay, admit a rational resolution, and therefore have rational singularities in characteristic $R$1. Under the standard Frobenius splitting on $R$2 induced from the canonical splitting on $R$3, they are exactly the compatibly split subvarieties. If $R$4 and $R$5 is a Richardson model mapping onto $R$6, then

$R$7

and for any ample line bundle $R$8 on $R$9,

$V$0

The stratification is controlled by the $V$1-Bruhat order. One writes $V$2 when $V$3 in Bruhat order and $V$4, and defines $V$5 as the transitive closure. The equivalence relation on $V$6-Bruhat intervals produces the poset $V$7, and each class has a unique representative with top element in $V$8. If $V$9, then

$I-P$0

the open stratum $I-P$1 is smooth, every point of $I-P$2 lies in a unique open projected Richardson stratum, and the closure relation is

$I-P$3

Unions of projected Richardson varieties intersect reducedly.

The same right projection governs a combinatorial theory. For any Bruhat interval $I-P$4, the projected simplicial complex $I-P$5 on the vertex set $I-P$6 is shellable and pure of dimension $I-P$7; its maximal faces are in bijection with the saturated $I-P$8-Bruhat chains from $I-P$9 to $*$0. In the minuscule case, the Gröbner degeneration of each projected Richardson variety is the Stanley–Reisner scheme of this shellable ball: $*$1 The stratification also appears in total positivity, Poisson geometry, and quantum geometry, where it aligns respectively with totally nonnegative cells, symplectic leaves of standard Poisson structures, and strata associated to $*$2-prime ideals in quantum partial flag varieties.

3. Elimination lemmas, right nullspaces, and non-strict LMIs

In LMI theory, the generalized right projection is the right-sided formulation of the non-strict projection lemma. The classical strict projection lemma states that for arbitrary complex matrices $*$3, $*$4, and Hermitian $*$5, there exists $*$6 such that

$*$7

if and only if

$*$8

where $*$9 and $G/B\to G/P$0 are annihilators of $G/B\to G/P$1 and $G/B\to G/P$2. In $G/B\to G/P$3 notation with $G/B\to G/P$4, $G/B\to G/P$5, $G/B\to G/P$6, the right/left form is

$G/B\to G/P$7

if and only if

$G/B\to G/P$8

or, equivalently,

$G/B\to G/P$9

with $W\to W/W_P$0 and $W\to W/W_P$1 (Meijer et al., 2023).

The non-strict generalization adds a coupling condition. There exists $W\to W/W_P$2 such that

$W\to W/W_P$3

if and only if

$W\to W/W_P$4

together with

$W\to W/W_P$5

In the $W\to W/W_P$6 form, the existence of $W\to W/W_P$7 such that

$W\to W/W_P$8

is equivalent to

$W\to W/W_P$9

and

$*$00

This coupling condition is the decisive distinction between strict and non-strict projection. It is not an LMI and cannot be reduced to the two kernel inequalities alone. The necessity is illustrated by the example

$*$01

for which $*$02, but no $*$03 exists making $*$04. Helmersson’s earlier non-strict lemma, which assumes $*$05, appears as a special case in which the coupling condition holds automatically.

The theorem is used to eliminate bilinear terms in several control-theoretic settings. For discrete-time marginal stability, the existence of $*$06 with $*$07 is reformulated as the existence of $*$08 and $*$09 such that

$*$10

The same framework yields robust polytopic and switching extensions, a matrix $*$11-lemma that allows $*$12 to be singular, and a matrix dilation result: there exists $*$13 with

$*$14

if and only if

$*$15

4. Singular generalized eigenvalue problems and projected regular pencils

For singular generalized eigenvalue problems, generalized right projection is a reduction method for the singular pencil $*$16. If the normal rank is

$*$17

the method chooses a right subspace $*$18 whose columns span a deflating subspace that excludes the right singular structure, together with a matching left projection $*$19, and forms the projected pencil

$*$20

This Petrov–Galerkin projection is designed so that the reduced pencil is regular and retains the finite eigenvalues of the regular part of the original singular pencil (Hochstenbach et al., 2022).

The algorithmic realization uses random orthonormal projectors $*$21 and $*$22, with $*$23, and studies

$*$24

The main theorem states that there is a generic set of left projectors $*$25 and right projectors $*$26 such that $*$27 is regular and strictly equivalent to

$*$28

where $*$29 is the regular part of $*$30 and $*$31 is a regular pencil whose eigenvalues are all simple and distinct from those of $*$32.

This construction does not merely reduce dimension; it supplies tests that distinguish true eigenvalues from random eigenvalues introduced by the singular structure. If $*$33 is an eigenvalue of $*$34 with right eigenvector $*$35 and left eigenvector $*$36, then $*$37 is an eigenvalue of the original pencil if and only if

$*$38

with the obvious modification using $*$39 for $*$40. The paper also introduces

$*$41

as a weak condition-number proxy for classifying finite versus infinite eigenvalues.

The projection method is presented as the most attractive version for generic singular pencils because of its efficiency. Projecting costs up to $*$42 flops, the reduced GEP has size $*$43 and costs $*$44, and the genericity of random $*$45 ensures regularity with probability $*$46 in floating-point practice. The paper contrasts this with an augmented-pencil approach, which can be favorable when linear systems with the augmented pencil can be solved efficiently.

5. Banach spaces, Bregman-type asymmetry, and cone duality

In Banach-space analysis, generalized right projection denotes an asymmetric projection induced by the normalized duality mapping. Let $*$47 be a real Banach space with dual $*$48, duality mapping $*$49, and generating functional

$*$50

For a nonempty subset $*$51, the generalized metric projection is

$*$52

Because $*$53 is fixed in the second, or right, argument of the asymmetric functional, $*$54 is described as the right generalized projection induced by $*$55. In smooth settings with $*$56, this coincides with the left Bregman projection $*$57 (Khan et al., 2022).

The behavior depends strongly on geometry. Reflexivity guarantees nonemptiness of $*$58 for closed convex $*$59, and if the space is reflexive, strictly convex, and smooth, then $*$60 is single-valued. In Hilbert spaces, $*$61, $*$62, and $*$63. In general Banach spaces, $*$64 may be multivalued or empty, may fail to be nonexpansive or firmly nonexpansive, and can differ substantially from the metric projection.

For uniformly convex and uniformly smooth Banach spaces, Alber’s generalized metric projection

$*$65

has a precise variational characterization: $*$66 Its Gâteaux directional differentiability has been worked out explicitly for several sets. For a closed ball $*$67, $*$68 inside the ball and $*$69 outside; the directional derivative satisfies

$*$70

and

$*$71

where $*$72 is the directional derivative of the norm. For closed convex cones, the map is radially homogeneous, and identities such as $*$73 and $*$74 for $*$75 are established. For cylinders in $*$76, the exact values of $*$77 can be highly coupled across coordinates, and the paper explicitly states that even in a special case $*$78 is extremely complicated (Li, 2023).

A distinct but related cone-duality framework replaces norm-based asymmetry by a strongly quasiconvex generator $*$79. For a closed convex cone $*$80,

$*$81

The natural right projection is then the complementary retraction

$*$82

Its image is exactly the kernel of the left projection: $*$83 Under suitable conditions, notably quadratic norms $*$84 with $*$85 symmetric positive definite or, more generally, global bipolar norms, the kernel is a closed convex cone and $*$86 and $*$87 are mutually polar retractions on convex cones. In the Euclidean case, this recovers Moreau’s decomposition and the identity

$*$88

The kernel also admits a gradient characterization: $*$89 Counterexamples based on minimal wedges with noncoherent meridians show that convexity of the kernel can fail for smooth, strictly convex, symmetric norms without the global bipolar property (Németh et al., 29 Apr 2025).

6. $*$90-rings and right-handed graphics matrices

In $*$91-ring theory, the generalized right projection of an element $*$92 is an idempotent self-adjoint element encoding right annihilator information at some power of $*$93. A projection $*$94 is $*$95 if there exists $*$96 such that

$*$97

A $*$98-ring is generalized Rickart $*$99 if for every $W/W_P$00 there exists $W/W_P$01 and a projection $W/W_P$02 such that

$W/W_P$03

In that case every element possesses a generalized right projection, and for some $W/W_P$04,

$W/W_P$05

The generalized right projection reduces to the classical right projection $W/W_P$06 when one can take $W/W_P$07, and it is left-right symmetric through

$W/W_P$08

The paper further relates GRP/GLP to the parallelogram law, partial comparability of projections, and orthogonal decompositions of projection pairs (Khairnar et al., 28 Aug 2025).

In computer graphics, the phrase refers to a generalized right-handed projection matrix that unifies perspective and orthographic projection. In right-handed camera space, with camera looking toward $W/W_P$09, near plane $W/W_P$10, and far plane $W/W_P$11, a single parameter $W/W_P$12 controls the fourth row: $W/W_P$13 The generalized right-handed matrix $W/W_P$14 linearly blends the coefficients of the standard right-handed perspective and orthographic matrices. At $W/W_P$15 it reproduces the canonical perspective matrix exactly; at $W/W_P$16 it reproduces the canonical orthographic matrix exactly. For intermediate $W/W_P$17, the mapping after the perspective divide becomes linear-fractional in $W/W_P$18, yielding what the paper calls a partially-orthographic projection. Only the third row changes between the OpenGL-style $W/W_P$19 and D3D/Vulkan-style $W/W_P$20 conventions; the fourth row remains

$W/W_P$21

This is a literal right-handed formulation: the sign pattern is determined by the convention $W/W_P$22 in the perspective limit (MacIntosh, 2022).

Taken together, these usages show that generalized right projection is not a single theory but a recurring structural idea. What persists across the literatures is a deliberate asymmetry: projection relative to right cosets, right nullspaces, right singular subspaces, right arguments of asymmetric functionals, right annihilators, or right-handed coordinate systems.