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Matrix with exactly one 1 per row and column
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column
Permutation_matrix
Matrix with one nonzero entry in each row and column
mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly
Generalized permutation matrix
Generalized_permutation_matrix
Mathematical version of an order change
^{-1}\right)=+1.} The sign of a permutation is equal to the determinant of its permutation matrix (below). A permutation matrix is an n × n matrix that has exactly one
Permutation
Matrix in linear algebra
transpose. Specifically, the commutation matrix K(m,n) is the nm × mn permutation matrix which, for any m × n matrix A, transforms vec(A) into vec(AT): K(m
Commutation_matrix
Mathematical tool in group theory
example lets us create six permutation matrices (all elements 1 or 0, exactly one 1 in each row and column). The 6x6 matrix representing an element will
Cayley_table
Real square matrix whose columns and rows are orthogonal unit vectors
reflection matrix with θ = 90° generates a reflection about the line at 45° given by y = x and therefore exchanges x and y; it is a permutation matrix, with
Orthogonal_matrix
Type of matrix factorization
decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers
LU_decomposition
Matrix representing a Euclidean rotation
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [
Rotation_matrix
Type of square matrix
multiples of permutation matrices until we arrive at the zero matrix, at which point we will have constructed a convex combination of permutation matrices
Doubly_stochastic_matrix
of permutation matrices. One first represents G {\displaystyle G} as a permutation group and then maps each permutation to the corresponding matrix. Representing
Permutation_representation
Matrix of binary truth values
matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can
Logical_matrix
Norm on a vector space of matrices
{\displaystyle \phi (Px)=\phi (x)} for any permutation matrix P {\displaystyle P} . A norm is a unitarily invariant matrix norm if and only if it is a symmetric
Matrix_norm
Property in group theory
method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value
Parity_of_a_permutation
Square matrix used to represent a graph or network
are given. G1 and G2 are isomorphic if and only if there exists a permutation matrix P such that P A 1 P − 1 = A 2 . {\displaystyle PA_{1}P^{-1}=A_{2}
Adjacency_matrix
In mathematics, invariant of square matrices
an n × n {\displaystyle n\times n} matrix is an expression involving permutations and their signatures. A permutation of the set { 1 , 2 , … , n } {\displaystyle
Determinant
Linear algebra matrix
x^{n}-1)} . Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix P {\displaystyle P} : C = c 0 I
Circulant_matrix
Mathematical model
context. The permutation matrices are precisely the alternating sign matrices that don't contain −1. An example of an alternating sign matrix that is not
Alternating_sign_matrix
Tool for working with matrices
algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices. It was published by Garrett Birkhoff in 1946
Birkhoff_algorithm
Machine learning technique
n}} be permutation matrices; and D ∈ R m × n {\displaystyle D\in \mathbb {R} ^{m\times n}} an arbitrary matrix. The softmax function is permutation equivariant
Attention_(machine_learning)
Universal reversible logic gate, applied in quantum computing
bits are replaced by qubits. The truth table and permutation matrix are as follows (the permutation can be written (7,8) in cycle notation): An input-consuming
Toffoli_gate
Type of matrix factorization
and, although not always needed, a permutation matrix (P). It was developed by Prescott Durand Crout. The Crout matrix decomposition algorithm differs slightly
Crout_matrix_decomposition
Matrix equal to its transpose
{\displaystyle P} is a permutation matrix (arising from the need to pivot), L {\displaystyle L} a lower unit triangular matrix, and D {\displaystyle D}
Symmetric_matrix
(Mathematical) decomposition into a product
diagonal entries equal to one, an upper triangular matrix U, and a permutation matrix P; this is a matrix formulation of Gaussian elimination. By the fundamental
Factorization
sum of permutations Enumerations of specific permutation classes Factorial Falling factorial Permutation matrix Generalized permutation matrix Inversion
List_of_permutation_topics
within combinatorial matrix theory include: (0,1)-matrix, a matrix whose coefficients are all 0 or 1 Permutation matrix, a (0,1)-matrix with exactly one nonzero
Combinatorial_matrix_theory
Concepts from linear algebra
the cyclic permutation matrix A = [ 0 1 0 0 0 1 1 0 0 ] . {\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.} This matrix shifts the
Eigenvalues_and_eigenvectors
Equivalence under a change of basis (linear algebra)
similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and
Matrix_similarity
Sum of elements on the main diagonal
The trace of a Hermitian matrix is real, because the elements on the diagonal are real. The trace of a permutation matrix is the number of fixed points
Trace_(linear_algebra)
separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums. Separable permutations may be characterized
Separable_permutation
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Method of random selection
odd/even permutation property of the ghost leg. An odd number of legs represents an odd permutation, and an even number of legs gives an even permutation. It
Ghost_leg
Combinatorial optimization problem
Beckmann in the following form. Given square matrices D and T, find the permutation matrix X that minimizes the double-dot product of T with A = X D X ⊺ {\displaystyle
Quadratic_assignment_problem
Pair of positions in a sequence where two elements are out of sorted order
i {\displaystyle i} . The permutation matrix of the inverse is the transpose, therefore v {\displaystyle v} of a permutation is r {\displaystyle r} of
Inversion (discrete mathematics)
Inversion_(discrete_mathematics)
denoted A and B is a doubly stochastic matrix D such that DA = BD. If the doubly stochastic matrix is a permutation matrix, then it constitutes a graph isomorphism
Fractional_graph_isomorphism
Determinant of a product of rectangular matrices
the permutation matrix for π, ( R g ) S , [ m ] {\displaystyle (R_{g})_{S,[m]}} is the permutation matrix for σ, and LfRg is the permutation matrix for
Cauchy–Binet_formula
symmetric permutation of the arrowhead matrix, P T A P {\displaystyle P^{T}AP} , where P is a permutation matrix, is a (permuted) arrowhead matrix. Real symmetric
Arrowhead_matrix
Theorem in linear algebra
&*\\0&0&0&\cdots &B_{h}\end{smallmatrix}}\right)} where P is a permutation matrix and each Bi is a square matrix that is either irreducible or zero. Now if A is non-negative
Perron–Frobenius_theorem
Matrix whose only nonzero entries lie on the lower-left-to-upper-right diagonal
elementary product from an anti-diagonal matrix will have a different sign depending on whether the permutation related to it is odd or even: More precisely
Anti-diagonal_matrix
Measure of covariance of components of a random vector
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the
Covariance_matrix
Group of symmetries of an n-dimensional hypercube
n {\displaystyle \mathbb {R} ^{n}} , the matrix of such a transformation must be a signed permutation matrix: it must have exactly one nonzero entry in
Hyperoctahedral_group
Mathematical term
However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SLn, we can take one of the nonzero
Bruhat_decomposition
Matrix decomposition method
Formally, if A is an n × n positive semidefinite matrix of rank r, then there is at least one permutation matrix P such that P A PT has a unique decomposition
Cholesky_decomposition
Algorithm for modelling sequential data
PM_{\text{causal}}P^{-1}} , where P {\displaystyle P} is a random permutation matrix. An encoder consists of an embedding layer, followed by multiple encoder
Transformer_(deep_learning)
Antisymmetric permutation object acting on tensors
epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It
Levi-Civita_symbol
Array of numbers
In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and
Matrix_(mathematics)
Representation of a matrix as a product
triangular, and P is a permutation matrix. Existence: An LUP decomposition exists for any square matrix A. When P is an identity matrix, the LUP decomposition
Matrix_decomposition
Polynomial-time algorithm for the assignment problem
matrix C to minimize the trace of a matrix, min P Tr ( P C ) , {\displaystyle \min _{P}\operatorname {Tr} (PC)\;,} where P is a permutation matrix.
Hungarian_algorithm
Matching which covers every node of the graph
adjacent to exactly one edge in M. The adjacency matrix of a perfect matching is a symmetric permutation matrix. A perfect matching is also called a 1-factor;
Perfect_matching
Similarity measure between two invertible matrices
scale and permutation matrix, i.e. the product of a diagonal matrix and a permutation matrix. The Amari distance is invariant to permutation and scaling
Amari_distance
Specialized notation for multivariable calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various
Matrix_calculus
Mathematical operation on matrices
a zero matrix, and k is a scalar. Non-commutative: In general, A ⊗ B and B ⊗ A are different matrices. However, A ⊗ B and B ⊗ A are permutation equivalent
Kronecker_product
Probability distribution
V^{T}\mathbf {X} V} , moreover Φ {\displaystyle \mathbf {\Phi } } can be a permutation matrix which exchanges diagonal elements. It follows that the diagonal elements
Inverse-Wishart_distribution
Polynomial of the elements of a matrix
determinant of A in that the signatures of the permutations are not taken into account. The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes
Permanent_(mathematics)
Special kind of square matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal
Triangular_matrix
Method of shuffling a deck of cards
{\begin{matrix}4\\5\\6\\3\\7\\2\\8\\9\\1\\10\end{matrix}}} A theorem called "the ultimate Gilbreath principle" states that, for a permutation π {\displaystyle
Gilbreath_shuffle
Matrix representation of a graph
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a
Laplacian_matrix
Concept in mathematics
sense that taking A to be a permutation matrix, the pair (P,Q) will be the pair of standard tableaux associated to the permutation under the Robinson–Schensted
Robinson–Schensted–Knuth correspondence
Robinson–Schensted–Knuth_correspondence
Type of (mathematical) permutation with no fixed element
cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has
Cyclic_permutation
Concept in combinatorics
The statistics of random permutations, such as the cycle structure of a random permutation, are of fundamental importance in the analysis of algorithms
Random_permutation_statistics
Matrix decomposition
beginning of each new step—column pivoting— and thus introduces a permutation matrix P: A P = Q R ⟺ A = Q R P T {\displaystyle AP=QR\quad \iff \quad A=QRP^{\textsf
QR_decomposition
Exact statistical hypothesis test
A permutation test (also called re-randomization test or shuffle test) is an exact statistical hypothesis test. A permutation test involves two or more
Permutation_test
Matrix operation which flips a matrix over its diagonal
involves a complicated permutation of the data elements that is non-trivial to implement in-place. Therefore, efficient in-place matrix transposition has been
Transpose
Number line and triangular tiling's symmetry mathematical structure
{\displaystyle u:\mathbb {Z} \to \mathbb {Z} } is an affine permutation, the corresponding matrix has entry 1 at position ( i , u ( i ) ) {\displaystyle (i
Affine_symmetric_group
Mathematical function generalizing the determinant and permanent
multilinear in the rows and columns of the matrix; and the immanant is invariant under simultaneous permutations of the rows or columns by the same element
Immanant
sum of permutations are two operations to combine shorter permutations into longer ones. Given a permutation π of length m and the permutation σ of length
Skew and direct sums of permutations
Skew_and_direct_sums_of_permutations
Problem in linear algebra
naively expands the formula, summing over all permutations and within the sum multiplying out each matrix entry. This requires n! n arithmetic operations
Computing_the_permanent
Square matrix whose entries are 1 along the anti-diagonal and 0 elsewhere
(also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrices, where the 1 elements
Exchange_matrix
Topics referred to by the same term
group or ring Commutation matrix, a permutation matrix which is used for transforming the vectorized form of another matrix into the vectorized form of
Commute
Computer system for solving algebra problems
for free, through that institution. Group theory Magma includes permutation, matrix, finitely presented, soluble, abelian (finite or infinite), polycyclic
Magma (computer algebra system)
Magma_(computer_algebra_system)
Orthogonal matrix
Walsh matrix can be derived from the ordering of the Hadamard matrix by first applying the bit-reversal permutation and then the Gray-code permutation: W
Walsh_matrix
Quantum logic gate
+d|10\rangle } The action of the CNOT gate can be represented by the matrix (permutation matrix form): CNOT = [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] . {\displaystyle
Controlled_NOT_gate
Algorithm for shuffling a finite sequence
until no elements remain. The algorithm produces an unbiased permutation: every permutation is equally likely. The modern version of the algorithm takes
Fisher–Yates_shuffle
Asymmetric encryption algorithm developed by Robert McEliece
k} binary non-singular matrix S {\displaystyle S} . Alice selects a random n × n {\displaystyle n\times n} permutation matrix P {\displaystyle P} . Alice
McEliece_cryptosystem
Four finite groups derived from the Leech lattice
conjugates inside the monomial subgroup. Any matrix in this conjugacy class has trace 0. A permutation matrix of shape 2818 can be shown to be conjugate
Conway_group
Matrix with non-zero elements only in a diagonal band
representation of a matrix with minimal bandwidth by means of permutations of rows and columns is NP-hard. Diagonal matrix Graph bandwidth Random matrix Golub & Van
Band_matrix
Universal reversible logic gate, applied in quantum computing
Truth table Permutation matrix form Input Output C I1 I2 C O1 O2 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 1 1
Fredkin_gate
Measure of distinguishability between two quantum states
_{i}p_{i}(\log p_{i}-\log(\sum _{j}q_{j}P_{ij}))} if and only if (Pi j) is a permutation matrix, which implies ρ = σ, after a suitable labeling of the eigenvectors
Quantum_relative_entropy
Large language model developed by Google AI
&\ddots &\vdots \\0&0&0&\dots &0\end{bmatrix}}} permuted by a random permutation matrix to P M causal P − 1 {\displaystyle PM_{\text{causal}}P^{-1}} . The
XLNet
Matrix decomposition in mathematics
The permutation has to ensure that the highest powers of z {\displaystyle z} are decreasing. Denote P , D {\displaystyle P,D} the permutation matrix, and
Birkhoff_factorization
Algorithms for matrix decomposition
least if B is a non-negative monomial matrix. In this simple case it will just correspond to a scaling and a permutation. More control over the non-uniqueness
Non-negative matrix factorization
Non-negative_matrix_factorization
transformation made of the reverse permutation matrix. In three dimensions, the dihedral group D ( 3 ) {\displaystyle D(3)} over a matrix M {\displaystyle M} is represented
Partial_inverse_of_a_matrix
Families of matrices in mathematics, physics, and quantum information
and the shift matrix is just the translation operator (a cyclic permutation matrix) in that cyclic vector space, so the exponential of the momentum.
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Mathematical operation on vectors in 3D space
under a rotation of the basis but is changed into its opposite by an odd permutation of the basis vectors. Therefore, the cross product is a pseudovector
Cross_product
Matrix whose entries are integers
of signed permutation matrices. The characteristic polynomial of an integer matrix has integer coefficients. Since the eigenvalues of a matrix are the roots
Integer_matrix
Group of symmetries of the square
The group composition operation is represented as matrix multiplication. Larger signed permutation matrices represent in the same way the hyperoctahedral
Dihedral_group_of_order_8
Problem in computer science
(rectangular) matrix, where it involves a complex permutation of the data elements, with many cycles of length greater than 2. In contrast, for a square matrix (N
In-place_matrix_transposition
Matrix manipulation algorithm
matrix. The Cholesky factor L will typically suffer 'fill in', that is have more non-zeros than the upper triangle of A. We seek a permutation matrix
Minimum_degree_algorithm
Measure of linear correlation
are a permutation of the set {1,...,n}. The permutation i′ is selected randomly, with equal probabilities placed on all n! possible permutations. This
Pearson correlation coefficient
Pearson_correlation_coefficient
Concept in linear algebra
}}} Let P {\textstyle P} be an n × n {\textstyle n\times n} permutation matrix such that A P = ( C , D ) {\textstyle AP=(C,D)} in block partitioned
Rank_factorization
Type of product of matrices
\ast \mathbf {D} )]} , where P {\displaystyle \mathbf {P} } is a permutation matrix. ( A ∙ B ) ( C ⊗ D ) = ( A C ) ∙ ( B D ) {\displaystyle (\mathbf
Khatri–Rao_product
Method of encryption
In cryptography, a transposition cipher (also known as a permutation cipher) is a method of encryption which scrambles the positions of characters (transposition)
Transposition_cipher
Choice of reference for distinguishing an object and its mirror image
This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Similarly, let A be a nonsingular linear
Orientation_(vector_space)
equal to mN(Ii)/N(Ij). The Brandt matrix B(m) is the H×H matrix with entries Bij. Up to conjugation by a permutation matrix it is independent of the choice
Brandt_matrix
its action on C ( X ) {\displaystyle \mathbb {C} (X)} is given by a permutation matrix [ c ] {\displaystyle [c]} , and the trace of [ c ] d {\displaystyle
Cyclic_sieving
Musical work by Johann Sebastian Bach
(1931); Zoltán Göncz (1992, which makes use of what he refers to as a permutational matrix of the 4 subjects); Yngve Jan Trede (1995); and Thomas Daniel (2010)
The_Art_of_Fugue
Branch of mathematics that studies the properties of groups
being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through
Group_theory
Mathematical concept
permutation of basis vectors. Sometimes the notation J {\displaystyle J} is used instead of Ω {\displaystyle \Omega } for the skew-symmetric matrix.
Symplectic_matrix
Matrices important in quantum mechanics and the study of spin
Pauli matrix is Hermitian, and together with the identity matrix I {\displaystyle \mathbb {I} } (sometimes considered as the zeroth Pauli matrix σ 0 {\displaystyle
Pauli_matrices
Type of group in abstract algebra
irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed
Symmetric_group
Shape in the geometry of numbers
\textstyle c_{n}\neq 0} ) and W {\displaystyle \textstyle W} is a permutation matrix. Assuming that V {\displaystyle \textstyle V} has been triangulated
Klein_polyhedron
PERMUTATION MATRIX
PERMUTATION MATRIX
Girl/Female
Hindu
Achievement, Omnipresence, Permeation
Girl/Female
Tamil
Vyaapti | வà¯à®¯à®¾à®ªà®¤à¯€
Achievement, Omnipresence, Permeation
Vyaapti | வà¯à®¯à®¾à®ªà®¤à¯€
Boy/Male
Hindu, Indian, Jain, Marathi, Sanskrit, Sindhi, Tamil
Lines on Any Particular Raaga from Sanskrit; Permutations and Combinations of Parents; Aarya Cost King Ashoka's Birth
PERMUTATION MATRIX
PERMUTATION MATRIX
Girl/Female
Arabic, Muslim
Beginning
Girl/Female
Muslim/Islamic
Clever
Girl/Female
Muslim
Sweetheart
Girl/Female
Muslim
Honorable
Surname or Lastname
English and Irish
English and Irish : variant of Higgins, resulting from a misdivision of a name such as Tom Higgins.
Girl/Female
Arabic
Soft and Gentle
Boy/Male
Muslim/Islamic
Just
Boy/Male
Indian, Sanskrit
Endless; Limitless
Boy/Male
Indian
One who surpasses, Excels
Girl/Female
Hindu, Indian, Marathi, Sanskrit
Sweet Natured
PERMUTATION MATRIX
PERMUTATION MATRIX
PERMUTATION MATRIX
PERMUTATION MATRIX
PERMUTATION MATRIX
v. t.
The white fibrous matter forming the matrix from which fungi.
v. t.
Alteration in the order of a series; permutation.
n.
The arrangement of any determinate number of things, as units, objects, letters, etc., in all possible orders, one after the other; -- called also alternation. Cf. Combination, n., 4.
a.
Proof against penetration or permeation by water; impervious to water; as, a waterproof garment; a waterproof roof.
n.
Permutation.
n.
Barter; exchange.
n.
Long continuance.
n.
The lifeless portion of tissue, either animal or vegetable, situated between the cells; the intercellular substance.
n.
The earthy or stony substance in which metallic ores or crystallized minerals are found; the gangue.
n.
Any one of such possible arrangements.
n.
The five simple colors, black, white, blue, red, and yellow, of which all the rest are composed.
n.
The womb.
n.
A rectangular arrangement of symbols in rows and columns. The symbols may express quantities or operations.
n.
The act of drinking excessively; a drinking bout.
n.
The act of permeating, passing through, or spreading throughout, the pores or interstices of any substance.
n.
The substitution of one root vowel for another, thus indicating a corresponding modification of use or meaning; vowel permutation; as, get, gat, got; sing, song; hang, hung.
n.
Alt. of Perduration
n.
The cavity in which anything is formed, and which gives it shape; a die; a mold, as for the face of a type.
n.
The act of permuting; exchange of the thing for another; mutual transference; interchange.
n.
Hence, that which gives form or origin to anything