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Tool in linear algebra and matrix analysis
The Schur complement is a key tool in the fields of linear algebra, the theory of matrices, numerical analysis, and statistics. It is defined for a block
Schur_complement
In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domain decomposition
Schur_complement_method
German mathematician (1875–1941)
Schur: Schur algebra Schur complement Schur index Schur indicator Schur multiplier Schur orthogonality relations Schur polynomial Schur product Schur
Issai_Schur
conjecture Schur complement method Schur complement Schur-convex function Schur decomposition Schur functor Schur index Schur's inequality Schur's lemma (from
List of things named after Issai Schur
List_of_things_named_after_Issai_Schur
Matrix defined using smaller matrices called blocks
are conformable with them for partitioning. Furthermore, A and the Schur complement of A in P: P/A = D − CA−1B must be invertible. Equivalently, by permuting
Block_matrix
Theorem in group theory
G} has a complement in G {\displaystyle G} . Moreover if either N {\displaystyle N} or G / N {\displaystyle G/N} is solvable then the Schur–Zassenhaus
Schur–Zassenhaus_theorem
Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix
{In} (H_{11})+\mathrm {In} (H/H_{11})} where H/H11 is the Schur complement of H11 in H: H / H 11 = H 22 − H 12 ∗ H 11 − 1 H 12 . {\displaystyle
Haynsworth inertia additivity formula
Haynsworth_inertia_additivity_formula
Homomorphisms between simple modules over the same ring are isomorphisms or zero
In mathematics, Schur's lemma is an elementary but useful statement in representation theory of groups and algebras. In the group case it says that if
Schur's_lemma
Topics referred to by the same term
(sometimes called an antonym) Complement (group theory) Complementary subspaces Orthogonal complement Schur complement Complement (complexity), relating to
Complement
Theorem in linear algebra
matrix. The result is named after Issai Schur (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in Journal für die reine und angewandte
Schur_product_theorem
American mathematician
worked in linear algebra and matrix theory. She gave the name to Schur complements and is the namesake of the Haynsworth inertia additivity formula.
Emilie_Virginia_Haynsworth
differences, the discretization of the Poincaré–Steklov operator is the Schur complement obtained by eliminating all degrees of freedom inside the domain. Note
Poincaré–Steklov_operator
In mathematics, invariant of square matrices
using either the Leibniz formula or a factorization involving the Schur complement, is det ( A 0 C D ) = det ( A ) det ( D ) = det ( A B 0 D ) . {\displaystyle
Determinant
Statistical measure in mathematical model
-j}=X_{j}^{T}X_{-j},r_{-j,j}=X_{-j}^{T}X_{j},r_{-j,-j}=X_{-j}^{T}X_{-j}} . By using Schur complement, the element in the first row and first column in r − 1 {\displaystyle
Variance_inflation_factor
Property of a mathematical matrix
proved with stronger conditions on the blocks, for instance, using the Schur complement. A general quadratic form f ( x ) {\displaystyle f(\mathbf {x} )} on
Definite_matrix
Theorem of matrix ranks
differential equations as the capacitance matrix. Sherman–Morrison formula Schur complement Matrix determinant lemma, formula for a rank-k update to a determinant
Woodbury_matrix_identity
Square root of the determinant of a skew-symmetric square matrix
and Boyd, Edinburgh, fourth edition, 1939. Zhang, Fuzhen, ed. The Schur complement and its applications. Vol. 4. Springer Science & Business Media, 2006
Pfaffian
where S := B ∗ A − 1 B {\displaystyle S:=B^{*}A^{-1}B} denotes the Schur complement. Since S {\displaystyle S} is symmetric positive-definite, we can apply
Uzawa_iteration
Matrix factorisation in mathematics
In linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary
Schur_decomposition
Mathematical circuit analysis technique
equivalent network with one less node. The equivalence follows from the Schur complement identity applied to the Kirchhoff matrix of the network. The equivalent
Star-mesh_transform
Power engineering technique
matrix entries to the reduced matrix entries. Star-mesh transform Schur complement Power-flow study Caliskan, Sina Yamac; Tabuada, Paulo (2014). "Towards
Kron_reduction
Generalization of the one-dimensional normal distribution to higher dimensions
matrix Σ ¯ {\displaystyle {\overline {\boldsymbol {\Sigma }}}} is the Schur complement of Σ22 in Σ. That is, the equation above is equivalent to inverting
Multivariate normal distribution
Multivariate_normal_distribution
_{21}(t)} . Since the Schur complement is positive definite for the real t {\displaystyle t} away from the poles and the Schur complement is a rational polynomial
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
Iterative method in conformal mapping
networks (PINN) Fourier neural operator (FNO) Domain decomposition Schur complement Fictitious domain Schwarz alternating additive abstract additive Neumann–Dirichlet
Schwarz_alternating_method
Probability distribution
}_{22}-{\mathbf {A} }_{21}{\mathbf {A} }_{11}^{-1}{\mathbf {A} }_{12}} is the Schur complement of A 11 {\displaystyle {\mathbf {A} _{11}}} in A {\displaystyle {\mathbf
Inverse-Wishart_distribution
Measure of covariance of components of a random vector
K Y | X {\displaystyle \operatorname {K} _{\mathbf {Y|X} }} is the Schur complement of K X X {\displaystyle \operatorname {K} _{\mathbf {XX} }} in Σ {\displaystyle
Covariance_matrix
matrix D − C A − 1 B {\displaystyle D-CA^{-1}B\,} is known as the Schur complement of A relative to [ A B C D ] . {\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}
Berezinian
Dimensionality reduction method
the rest constrained. Model order reduction Finite element method Schur complement - arising in the field of linear algebra when performing a block Gaussian
Guyan_reduction
Generalization of beta distribution
is p 2 × p 2 {\displaystyle p_{2}\times p_{2}} , then defining the Schur complement U 22 ⋅ 1 {\displaystyle U_{22\cdot 1}} as U 22 − U 21 U 11 − 1 U 12
Matrix variate beta distribution
Matrix_variate_beta_distribution
decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Matrix equivalence
Outline_of_linear_algebra
iteration for the iterative solution of the equations arising from the Schur complement method. This continuous iteration can be discretized by the finite
Neumann–Neumann_methods
Swiss mathematician (1909–1978)
"The Schur Complement and Its Applications", in Zhang, Fuzhen (ed.), Historical Introduction: Issai Schur and the Early Development of the Schur Complement
Eduard_Stiefel
{A} :=\mathbf {D} -\mathbf {C} \mathbf {A} ^{-1}\mathbf {B} } is the Schur complement of A. (A must be square, so that it can be inverted. Furthermore, A
Methods_of_matrix_inversion
Multivariable generalization of the Student's t-distribution
_{11|2}=\Sigma _{11}-\Sigma _{12}\Sigma _{22}^{-1}\Sigma _{21}} is the Schur complement of Σ 22 in Σ {\displaystyle \Sigma _{22}{\text{ in }}\Sigma } . d
Multivariate_t-distribution
Vector satisfying some of the criteria of an eigenvector
decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula Reducing subspace Relations
Generalized_eigenvector
Type of matrix factorization
}{\begin{pmatrix}0&0\\0&Q^{\frac {*}{2}}\end{pmatrix}},} where the Schur complement of A {\displaystyle {\begin{matrix}A\end{matrix}}} in the block matrix
Block_LU_decomposition
Calculating the determinant of this matrix in two ways: direct and via Schur complement formula essentially gives Capelli's identity and its generalization
Manin_matrix
Topics referred to by the same term
triangularizable, see Schur decomposition Schur test for boundedness of integral operators Schur's theorem Schur's property Schur complement This disambiguation
Schur's lemma (disambiguation)
Schur's_lemma_(disambiguation)
Method In numerical analysis
application of BDDC can be omitted. J.-M. Cros, A preconditioner for the Schur complement domain decomposition method, in Domain Decomposition Methods in Science
BDDC
) = {\displaystyle (D-CA^{-1}B)=} a much smaller matrix called the Schur complement of A {\displaystyle A} . This is the FKF method that may make it computationally
Fast_Kalman_filter
cosets of H. The Schur–Zassenhaus theorem guarantees the existence of complements of normal Hall subgroups of finite groups. Complements generalize both
Complement_(group_theory)
American mathematician
year at Harvard and MIT (1970-1971), he wrote “Manifestations of the Schur Complement’’, one of his most cited papers. In 1974, he started working on “The
Richard_W._Cottle
Subfield of convex optimization
^{T}x\\c^{T}x&d^{T}x\end{array}}\right]} We can use the theory of Schur Complements to see that D ⪰ 0 {\displaystyle D\succeq 0} (Boyd and Vandenberghe
Semidefinite_programming
Method for numerical differential equations
networks (PINN) Fourier neural operator (FNO) Domain decomposition Schur complement Fictitious domain Schwarz alternating additive abstract additive Neumann–Dirichlet
Gradient discretisation method
Gradient_discretisation_method
Ukrainian mathematician
Nekrashevych, Volodymyr.Self-similar groups, operator algebras and Schur complement.J. Mod. Dyn. 1 (2007), no. 3, 323–370. Editorial Statement, Algebra
Rostislav_Grigorchuk
Simplifying model for electrical grids
method that eliminates internal nodes from the network by applying the Schur complement to the admittance matrix. Ward reduction, the extended version of Kron
Power_system_reduction
in the interiors of the subdomain, thus reducing the problem to the Schur complement on the subdomain interface. Since the BDD preconditioner involves the
Balancing domain decomposition method
Balancing_domain_decomposition_method
English scientist and engineer
standard text in the field and is described as a "classic" in The Schur Complement and Its Applications by Fuzhen Zhang. Bishop calls it a "masterpiece"
Arthur_Roderick_Collar
Virginia Haynsworth (1916–1985), American linear algebraist known for Schur complements and Haynsworth inertia additivity formula Olive Hazlett (1890–1974)
List_of_women_in_mathematics
maps tangential electric field onto the equivalent electric current Schur complement method — early and basic method on subdomains that do not overlap Schwarz
List of numerical analysis topics
List_of_numerical_analysis_topics
U_{22(i)}} is p 2 × p 2 {\displaystyle p_{2}\times p_{2}} . Writing the Schur complement U 22 ⋅ 1 ( i ) = U 21 ( i ) U 11 ( i ) − 1 U 12 ( i ) {\displaystyle
Matrix variate Dirichlet distribution
Matrix_variate_Dirichlet_distribution
Approximation method
inverse can be computed by using recursion to compute inverses and Schur complements of diagonal blocks and combining both using the matrix-matrix multiplication
Hierarchical_matrix
is created by inverting A 11 {\displaystyle A_{11}} , putting the Schur complement A / A 11 {\displaystyle A/A_{11}} in place of A 22 {\displaystyle A_{22}}
Partial_inverse_of_a_matrix
algebras) Frobenius's theorem (group theory) Frobenius conjecture Frobenius–Schur indicator Perron–Frobenius theorem Quadratic Frobenius test Rouché–Frobenius
List of things named after Ferdinand Georg Frobenius
List_of_things_named_after_Ferdinand_Georg_Frobenius
Concept in mathematics
is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem. In particular a finite group
Frobenius_group
Area of mathematics
Alternating polynomials Symmetric polynomials Schur functor Robinson–Schensted correspondence Schur–Weyl duality Jucys–Murphy element Garnir relations
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
On the other hand, if A is a π′-group, then by the Schur–Zassenhaus theorem A has a complement in H, which is a Hall π-subgroup of G. Any finite group
Hall_subgroup
Set of lectures by Sigmund Freud
describing it privately as "coarse work, intended for the multitude". Max Schur, who became Freud's personal physician, was present at the original 1915
Introduction to Psychoanalysis
Introduction_to_Psychoanalysis
Result about when a matrix can be diagonalized
decomposed is Hermitian, the spectral decomposition is a special case of the Schur decomposition (see the proof in case of normal matrices below). The spectral
Spectral_theorem
4th episode of the 7th season of Parks and Recreation
by series co-creator Michael Schur and directed by Beth McCarthy-Miller. In an interview with Entertainment Weekly, Schur revealed that the story for the
Leslie_and_Ron
Subgroup of a root system's isometry group
ISBN 978-0-273-08517-1, Zbl 0483.57002 Howlett, Robert B. (1988), "On the Schur Multipliers of Coxeter Groups", J. London Math. Soc., 2, 38 (2): 263–276
Weyl_group
Representations of finite groups, particularly on vector spaces
theory of Richard Brauer was developed. Character theory Real representation Schur orthogonality relations McKay conjecture Burnside ring (Serre 1977, p. 47)
Representation theory of finite groups
Representation_theory_of_finite_groups
Theorem in functional analysis
⋯ + ξ k ( A ) {\textstyle \xi _{1}(A)+\dots +\xi _{k}(A)} is concave. (Schur-Horn inequality) ξ 1 ( A ) + ⋯ + ξ k ( A ) ≤ a i 1 , i 1 + ⋯ + a i k , i
Min-max_theorem
Concerns the decomposition of representations of a finite group into irreducible pieces
{\displaystyle G} -invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the
Maschke's_theorem
Sporadic simple group
{Co} _{3}} is maximal in C o 0 {\displaystyle \mathrm {Co} _{0}} . The Schur multiplier and the outer automorphism group are both trivial. Co3 acts on
Conway_group_Co3
History of a branch of mathematics
Sylow. This period saw Hans Zassenhaus's famous Schur-Zassenhaus theorem on the existence of complements to Hall's generalization of Sylow subgroups, as
History_of_group_theory
Commutative group (mathematics)
Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier (Berlin, Heidelberg: Springer, 2017), p. 206. "Abel Prize Awarded:
Abelian_group
Species of plant
squeezed kabosu is added. In Japanese cuisine it is traditionally used as a complement to grilled freshwater fish, but not saltwater fish.[citation needed] In
Persicaria_hydropiper
German mathematician (1849–1917)
Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical
Ferdinand_Georg_Frobenius
Branch of algebraic geometry
the same form as the first Jacobi-Trudi identity, expressing arbitrary Schur functions s a {\displaystyle s_{\mathbf {a} }} as determinants in terms
Schubert_calculus
2003 album by Blink-182
release date kept getting missed and pushed back to the point where Jordan Schur, then-president of Geffen, made calls asking, "What is the absolute last
Blink-182_(album)
Mathematical study of linear operators
{\displaystyle A} is normal if and only if it is unitarily diagonalizable: By the Schur decomposition, we have A = U T U ∗ {\displaystyle A=UTU^{\ast }} , where
Operator_theory
Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
1990 studio album by Public Enemy
itself tries to complement the album as a whole, fitting into the overall context." In his essay on hip hop aesthetics, Richard Schur interpreted such
Fear_of_a_Black_Planet
Number in {..., –2, –1, 0, 1, 2, ...}
practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between
Integer
Ethnoreligious group native to the Levant
(1260–1804)] (in Hebrew). בית הוצאה כתר, ירושלים; יד יצחק בן-צבי. p. 258. Schur, Nathan (1 July 1986). "The Samaritans, as Described in Christian Itineraries
Samaritans
Autoantibody that binds to contents of the cell nucleus
1080/08916930400022715. PMID 15804705. S2CID 1627719. Benito-Garcia E, Schur, PH, Lahita, R, American College of Rheumatology Ad Hoc Committee on Immunologic
Antinuclear_antibody
2014 Marvel Studios film
2013). "Why Are Rob Lowe And Rashida Jones Leaving 'Parks And Rec'? EP Mike Schur Says Decision Was Mutual". The Huffington Post. Archived from the original
Guardians of the Galaxy (film)
Guardians_of_the_Galaxy_(film)
Non-commutative group with 6 elements
swaps. This decomposition is also a consequence (particular case) of the Schur–Zassenhaus theorem. In terms of permutations the two group elements of G
Dihedral_group_of_order_6
Type of group in mathematics
(thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower." Thus the natural inclusion
Orthogonal_group
Mathematical proof at least partially generated by computer
hdl:2445/192693. S2CID 28258285. Heule, Marijn J. H. (2017). "Schur Number Five". arXiv:1711.08076 [cs.LO]. "Schur Number Five". www.cs.utexas.edu. Retrieved 2021-10-06
Computer-assisted_proof
English mathematician
Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical
William_Burnside
Planned space for displaying plants and other forms of nature
Archived from the original on 29 September 2007. Retrieved 24 May 2007. Schur, Norman; Ehrlich, Eugene; Ehrlich, Richard (1987). British English from
Garden
Writing Lie algebra sets as matrices
etc. A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts: If V, W are irreducible g {\displaystyle {\mathfrak
Lie_algebra_representation
Matrix used in complex analysis
unit disk or a pair of holomorphic functions on the unit disk and its complement. The Grunsky inequalities express boundedness properties of these matrices
Grunsky_matrix
Vector space of infinite sequences
entry. The space ℓ1 has the Schur property: In ℓ1, any sequence that is weakly convergent is also strongly convergent (Schur 1921). However, since the weak
Sequence_space
Branch of mathematics that studies abstract algebraic structures
be reducible. The definition of an irreducible representation implies Schur's lemma: an equivariant map α : ( V , ψ ) → ( V ′ , ψ ′ ) {\displaystyle
Representation_theory
Particular projective representations of the orthogonal or special orthogonal groups
n = 2m + 1 is odd, the isomorphism B: S → S∗ is unique up to scale by Schur's lemma, since S is irreducible, and it defines a nondegenerate invariant
Spin_representation
Robertson–Seymour theorem (graph theory) Schnyder's theorem (graph theory) Schur's theorem (Ramsey theory) Schwenk's theorem (graph theory) Sensitivity theorem
List_of_theorems
in the proof of the Kazhdan–Margulis theorem. One can recover the Jordan–Schur theorem as a corollary to the existence of Zassenhaus neighbourhoods. Jorgensen's
Margulis_lemma
Mathematical concept
∖ K ) {\displaystyle \pi _{1}(S^{3}\setminus K)} of nontrivial knot complements fall into this category and therefore are not hyperbolic. This is also
Hyperbolic_group
{\displaystyle C} is in the center of the universal enveloping algebra, Schur's lemma tells us that C {\displaystyle C} acts as multiple c λ {\displaystyle
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
Discrete group of Möbius transformations
and usually denoted Λ ( Γ ) {\displaystyle \Lambda (\Gamma )} . The complement Ω ( Γ ) = S ∞ 2 − Λ ( Γ ) {\displaystyle \Omega (\Gamma )=S_{\infty }^{2}-\Lambda
Kleinian_group
In mathematics, element that equals its square
irreducible idempotent is an idempotent a for which aR is a simple module. By Schur's lemma, EndR(aR) = aRa is a division ring, and hence is a local ring, so
Idempotent_(ring_theory)
Human viral infection
Alpert E, Isselbacher KJ, Schur PH (July 1971). "The pathogenesis of arthritis associated with viral hepatitis. Complement-component studies". The New
Hepatitis_B
Subset of a group that forms a group itself
left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if p is the lowest prime dividing the order of a finite
Subgroup
Representation of a group or algebra that is a direct sum of simple representations
representation with the inner product given by the averaging argument, the Schur orthogonality relations say: the irreducible characters (characters of simple
Semisimple_representation
Theorem for proving more complex theorems
Nakayama's lemma Noether normalization lemma Poincaré's lemma Riesz's lemma Schur's lemma Schwarz's lemma Sperner's lemma Urysohn's lemma Vitali covering lemma
Lemma_(mathematics)
Algebraic structure with addition and multiplication
f\mapsto (f_{ij}).} Any ring homomorphism R → S induces Mn(R) → Mn(S). Schur's lemma says that if U is a simple right R-module, then EndR(U) is a division
Ring_(mathematics)
Concept in Hlibert spaces mathematics
_{i=1}^{n}a_{i}b_{i}\,.} Lieb–Thirring inequality – Inequality in mathematical physics Schur–Horn theorem – Characterizes the diagonal of a Hermitian matrix with given
Trace_inequality
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