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Search references for INTEGER MATRIX. Phrases containing INTEGER MATRIX
See searches and references containing INTEGER MATRIX!INTEGER MATRIX
Matrix whose entries are integers
integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix
Integer_matrix
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there
Unimodular_matrix
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
On when an integer positive definite quadratic form represents all positive integers
quadratic form arising from an integer matrix represents all positive integers up to 15, then it represents all positive integers. Conway and Schneeberger chose
15_and_290_theorems
Algorithmic runtime requirements for matrix multiplication
floating point numbers, but not necessarily for integers). Strassen's algorithm improves on naive matrix multiplication through a divide-and-conquer approach
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
Method to solve optimization problems
notably problems where the constraint matrix is totally unimodular and the right-hand sides of the constraints are integers or – more general – where the system
Linear_programming
Natural number
number 73 is the largest member of Bhargava's definite quadratic 17–integer matrix Φ s ( P ) {\displaystyle \Phi _{s}(P)} representative of all prime numbers:
21_(number)
Matrices named after Élie Cartan
(symmetrizable) generalized Cartan matrix is a square matrix A = ( a i j ) {\displaystyle A=(a_{ij})} with integer entries such that For diagonal entries
Cartan_matrix
Commutative group (mathematics)
ring of integers ℤ in a unique way. In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication
Abelian_group
Natural number
number 59), an integer-matrix quadratic form that represents all prime numbers when it represents the sequence of seventeen integers {2, ..., 47, 67
53_(number)
Natural number
equation. 7 is one of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: {1, 3, 5, 7, 11, 15, 33}. There are
7
Arithmetic operation
with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely
Division_(mathematics)
Natural number
33 is the last of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: {1, 3, 5, 7, 11, 15, 33}. Islamic
33_(number)
Natural number
definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems
15_(number)
In mathematics, invariant of square matrices
of a matrix belongs to the smallest ring that contains the entries of the matrix. The determinant of a matrix of integers is always an integer, and,
Determinant
dimension (size) of matrix. The matrix entries will be denoted aij. The table below uses the Kronecker delta δij for two integers i and j which is 1 if
List_of_named_matrices
Natural number
total number of Leibniz terms (signed products) evaluated for a 16×16 integer matrix (n = 16) 20,922,789,888,136 : 16th factoriangular number 21,246,581
10,000,000,000,000
Matrix normal form
obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always
Smith_normal_form
Theorem about admissible crystal symmetries
a 6-fold rotation matrix in the equilateral triangle basis is an integer matrix with order 6. Let OrdN denote the set of integers that can be the order
Crystallographic restriction theorem
Crystallographic_restriction_theorem
Image dithering algorithm
of the map into a floating point format, rather than the traditional integer matrix format shown above. For this, the following formula can be used: Mpre(i
Ordered_dithering
Matrix form in linear algebra
equivalent by using a different unimodular matrix U {\displaystyle U} . A unimodular matrix is a square integer matrix whose determinant is 1 or −1 (and hence
Hermite_normal_form
Matrix operation generalizing exponentiation of scalar numbers
}\left(I+{\frac {X}{k}}\right)^{k}} for integer-valued k, where I is the n × n identity matrix. Equivalently, the matrix exponential is provided by the solution
Matrix_exponential
Algorithm to multiply matrices
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms
Matrix multiplication algorithm
Matrix_multiplication_algorithm
Hardware acceleration unit for artificial intelligence tasks
Ootomo, Hiroyuki; Ozaki, Katsuhisa; Yokota, Rio (July 2024). "DGEMM on integer matrix multiplication unit". The International Journal of High Performance
Neural_processing_unit
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Gives a functional equation satisfied by the generating function of any rational cone
n} integer matrix B (i.e., C is defined as the nonnegative span of the n columns of B). The integer-point generating function (also called integer-point
Stanley's_reciprocity_theorem
Mathematical problem
an n×n {1, −1} matrix is an integer multiple of 2n−1. Let R be an n by n {1, −1} matrix. The Gram matrix of R is defined to be the matrix G = RRT. From
Hadamard's maximal determinant problem
Hadamard's_maximal_determinant_problem
Concept in linear algebra
non-unital rings), and let n > 0 {\displaystyle n>0} be a positive integer. Then any matrix-valued function P : K → M a t ( K , n × n ) {\displaystyle P\colon
Matrix_pencil
Mathematical structure used in graph theory
Wilson matrix is the following 4×4 matrix having integers as elements: W=[5765710876810957910] This is the coefficient matrix of the following system of
Wilson_matrix
Generalization of additive and multiplicative inverses
defined. For example, in the case of matrix multiplication, there is one n×n identity matrix for every positive integer n, and two identity matrices of different
Inverse_element
Finite extension of the rationals
to x is a monic polynomial with integer coefficients. Suppose that the matrix A that represents an element x has integer entries in some basis e. By the
Algebraic_number_field
Array of numbers
that a matrix (in the usual sense) can have as long as they are positive integers. A matrix with m rows and n columns is called an m × n matrix, or m-by-n
Matrix_(mathematics)
Mathematical proof about the permanent of matrices
of a matrix with entries only 0 or 1. This will prove that 01-Permanent is #P-hard as well. Using modular arithmetic, convert an integer matrix A {\displaystyle
♯P-completeness of 01-permanent
♯P-completeness_of_01-permanent
For a square matrix, the transpose of the cofactor matrix
classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally
Adjugate_matrix
Matrix with a multiplicative inverse
{\displaystyle k} is a positive integer. Singular matrix Binomial inverse theorem LU decomposition Matrix decomposition Matrix square root Minor (linear algebra)
Invertible_matrix
Algorithmic runtime requirements for common math procedures
algorithm. This table lists the complexity of mathematical operations on integers. On stronger computational models, specifically a pointer machine and consequently
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Mathematical concept in algebra
linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} . The
Nilpotent_matrix
(Mathematical) decomposition into a product
matrix U, and a permutation matrix P; this is a matrix formulation of Gaussian elimination. By the fundamental theorem of arithmetic, every integer greater
Factorization
Mathematical operation in linear algebra
columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number
Matrix_multiplication
Complex matrix whose conjugate transpose equals its inverse
where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix. For any nonnegative integer n, the set of all n × n
Unitary_matrix
bases and its application to integer programming is described by Shmuel Onn. The Graver basis of an m × n integer matrix A is the finite set G(A) of minimal
Graver_basis
Mathematics concept
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose
Hadamard_matrix
the choice of an integer matrix φ, constrained as above. There is a countable number of such surfaces. Let n be a positive integer, and Λ n {\displaystyle
Inoue_surface
Norm on a vector space of matrices
Consider the matrix as a rectangular array of numbers; then a matrix norm may be defined as a function of the entries of the matrix. Such matrix norms are
Matrix_norm
Matrix decomposition
(also known as eigenvalue decomposition or EVD) is a factorization of a matrix A {\displaystyle A} into a canonical form given by A = Q D Q − 1 {\displaystyle
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Math operation methods
In linear algebra, the inverse square matrix A {\displaystyle A} is another square matrix A − 1 {\displaystyle A^{-1}} such that the product A − 1 A {\displaystyle
Methods_of_matrix_inversion
Square matrix used to represent a graph or network
computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices
Adjacency_matrix
Analytic function on the upper half-plane with a certain behavior under the modular group
columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. The so-called
Modular_form
Dimension of the column space of a matrix
integer k such that A can be factored as A = C R {\displaystyle A=CR} , where C is an m × k matrix and R is a k × n matrix. In fact, for all integers
Rank_(linear_algebra)
Rational numbers with root 5 added
} , integer primes of the form p = 5 n ± 2 {\displaystyle p=5n\pm 2} where n {\displaystyle n} is an integer, and the factors of integer primes
Golden_field
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
science, the matrix mortality problem (or mortal matrix problem) is a decision problem that asks, given a set of size m of n×n matrices with integer coefficients
Matrix_mortality_problem
Square matrix where a[i,j]=1/(i+j-1)
{i+j-2}{i-1}}^{2},} where n is the order of the matrix. It follows that the entries of the inverse matrix are all integers, and that the signs form a checkerboard
Hilbert_matrix
Characterizes sets of lattices that are bounded in a certain sense
absolute value of the determinant of a matrix – this is constant on the cosets, since an invertible integer matrix has determinant 1 or −1. Mahler's compactness
Mahler's_compactness_theorem
Lattice group in Euclidean space whose points are integer n-tuples
n-tuples of integers. The two-dimensional integer lattice is also called the square lattice (or grid lattice) and the three-dimensional integer lattice is
Integer_lattice
Algebra of formal sums
as well: if M {\displaystyle M} is a d × d {\displaystyle d\times d} integer matrix with determinant ± 1 {\displaystyle \pm 1} , then the rows of M {\displaystyle
Free_abelian_group
Amount of resources to perform an algorithm
arithmetic complexity of the computation of the determinant of a n×n integer matrix is O ( n 3 ) {\displaystyle O(n^{3})} for the usual algorithms (Gaussian
Computational_complexity
Type of mathematical expression
addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of
Polynomial
Polynomial with all terms of degree two
symmetric matrix with integer coefficients twos out a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients
Quadratic_form
Matrix whose conjugate transpose is its negative (additive inverse)
square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That
Skew-Hermitian_matrix
Polynomial equation whose integer solutions are sought
AX=C, where A is an m × n matrix of integers, X is an n × 1 column matrix of unknowns and C is an m × 1 column matrix of integers. The computation of the
Diophantine_equation
Matrix B such that B² equals a given matrix A
nonnegative integer must either be another integer or an irrational number, excluding non-integer rationals. Contrast that to a matrix of integers, which can
Square_root_of_a_matrix
Four finite groups derived from the Leech lattice
transitive on Λ2, and indeed he found a new matrix, not monomial and not an integer matrix. Let η be the 4-by-4 matrix 1 2 ( 1 − 1 − 1 − 1 − 1 1 − 1 − 1 − 1
Conway_group
Type of similarity matrix in computational biology
probability matrix: f ( j ) M ( i , j ) = f ( i ) M ( j , i ) {\displaystyle f(j)M(i,j)=f(i)M(j,i)} In fact, this relationship holds for all positive integer powers
Point_accepted_mutation
Concept in topology
C_{k}} are free. Then ∂ k {\displaystyle \partial _{k}} , which is an integer matrix, restricts to an invertible morphism which can thus be diagonalized
H-cobordism
Natural number
functions in matrix of order 6 with separate row and column permutations. 298 is a number where 6n+1 and 6n-1 are both prime. "Facts about the integer". mathworld
298_(number)
Algorithm for determinants of integers
calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed
Bareiss_algorithm
Natural number
the Germanic concept of 1200 as a long thousand. It is the first 4-digit integer. The decimal representation for one thousand is 1000—a one followed by
1000_(number)
Natural number
(151 + 157). It is the totient sum of the first 41 integers. 309 = 3 × 103. It is a Blum integer and a centered icosahedral number. 310 = 2 × 5 × 31
300_(number)
Matrix equal to its transpose
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{
Symmetric_matrix
Concepts from linear algebra
characteristic polynomial, that is, the largest integer k such that (λi − λ)k evenly divides that polynomial. Suppose a matrix A has order n and d ≤ n distinct eigenvalues
Eigenvalues_and_eigenvectors
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
Arithmetic operation
involving two numbers: the base, b, and the exponent, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that
Exponentiation
In mathematics, vector subspace
characteristic that the given integer matrix has the appropriate rank in it. All fields include integers, but some integers may equal to zero in some fields
Linear_subspace
Square matrix in which each ascending skew-diagonal from left to right is constant
. Given an integer n > 0 {\displaystyle n>0} , define the corresponding ( n × n ) {\displaystyle (n\times n)} -dimensional Hankel matrix B n {\displaystyle
Hankel_matrix
Lossy compression technique
Dividing the DCT coefficient matrix element-wise with this quantization matrix, and rounding to integers results in: [ − 26 − 3 − 6 2 2 − 1 0
Quantization (image processing)
Quantization_(image_processing)
Integer side lengths of a right triangle
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known
Pythagorean_triple
Formula for systems of linear equations
prove that an integer programming problem whose constraint matrix is totally unimodular and whose right-hand side is integer, has integer basic solutions
Cramer's_rule
Array of nonnegative integers in combinatorics
is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} (with positive integer indices i and j) that is nonincreasing in
Plane_partition
Matrix that shows the relationship between two classes of objects
thus, a general matrix of non-negative integers describes a hypergraph. The incidence matrix of an incidence structure C is a p × q matrix B (or its transpose)
Incidence_matrix
entrywise to the matrix B ⊗ 4 {\displaystyle B^{\otimes 4}} . The resulting 16 × 16 {\displaystyle 16\times 16} integer matrix is a generator matrix for the Barnes–Wall
Barnes–Wall_lattice
Algorithm for solving systems of linear equations
each matrix entry generated by this variant is the determinant of a submatrix of the original matrix. In particular, if one starts with integer entries
Gaussian_elimination
Tool for digital signal processing
while in multi-dimensional systems the decimators are D × D nonsingular integer matrix. it considers only those samples that are on the lattice generated by
Filter_bank
Possible form of a matrix
Bézout's identity. The reduced echelon form of a matrix with integer entries generally contains non-integer entries, because of the need to divide by its
Row_echelon_form
Principal square root of minus 1
result is some integer times the imaginary unit, an imaginary integer; any such numbers can be added and the result is also an imaginary integer: a i + b i
Imaginary_unit
Expression for sums of powers of integers
Since these matrix equations hold for every order, it is possible to obtain the polynomials of the sums of powers of successive integers without resorting
Faulhaber's_formula
Quotient of two integers
integers, a numerator p and a nonzero denominator q. For example, 3 7 {\displaystyle {\tfrac {3}{7}}} is a rational number, as is every integer (for
Rational_number
Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix K(N) is an (N + 1) × (N + 1) matrix. The first few Krawtchouk matrices are:
Krawtchouk_matrices
Natural number
two-hundred and ninety integers. Alternatively, a more precise version states that an integer quadratic matrix represents all positive integers when it contains
29_(number)
Matrices similar to diagonal matrices
linear algebra, a square matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists
Diagonalizable_matrix
for generating primitive triples using the sequence of consecutive odd integers 1 , 3 , 5 , 7 , 9 , 11 , … {\displaystyle 1,3,5,7,9,11,\ldots } and the
Formulas for generating Pythagorean triples
Formulas_for_generating_Pythagorean_triples
Formulation of quantum mechanics
quantity, the phase, is only well-defined up to an integer multiple of 2π. Only when the unitary symmetry matrix is part of a family that comes arbitrarily close
Matrix_mechanics
Natural number
number, as the solution to x − ϕ ( x ) {\displaystyle x-\phi (x)} for the integers 95, 119, 143, and 529. 23 is the second Smarandache–Wellin prime in base
23_(number)
Matrix of geometric progressions
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row:
Vandermonde_matrix
Matrix decomposition
compared to nm integers. This same idea extends to color images by applying this operation to each channel or stacking the channels into one matrix. Since the
Singular_value_decomposition
Determinant of a subsection of a square matrix
{\displaystyle C_{2,3}=(-1)^{2+3}(M_{2,3})=-13.} Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A, also called minor
Minor_(linear_algebra)
Most widely known generalized inverse of a matrix
column vectors. For any positive integer n {\displaystyle n} , the n × n {\displaystyle n\times n} identity matrix is denoted I n ∈ K n × n {\displaystyle
Moore–Penrose_inverse
Numbers obtained by adding the two previous ones
sequence based upon the Fibonacci sequence Wythoff array – Infinite matrix of integers derived from the Fibonacci sequence International Conference on Fibonacci
Fibonacci_sequence
Matrix-valued random variable
probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled
Random_matrix
Type of matrix factorization
factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition)
LU_decomposition
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INTEGER MATRIX
INTEGER MATRIX
Boy/Male
German, Norse, Swedish
Guarded by Ing; Ing's Beauty
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
Boy/Male
Arabic, Muslim
To Wait
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
Girl/Female
American, Australian, Danish, Finnish, German, Scandinavian, Swedish, Teutonic
Guarded by Ing; Ing is Beautiful; Daughter of Hero; Enclosure
Boy/Male
Norse
Son's army.
Girl/Female
Scandinavian Teutonic Danish Swedish
Ing's abundance. Feminine of Ing who was Norse mythological god of the earth's fertility.
Boy/Male
Muslim
To wait
INTEGER MATRIX
INTEGER MATRIX
INTEGER MATRIX
INTEGER MATRIX
INTEGER MATRIX
INTEGER MATRIX
INTEGER MATRIX
n.
A complete entity; a whole number, in contradistinction to a fraction or a mixed number.
n.
One who makes an entrance or beginning.
v. t.
To deposit, as a dead body, in the earth; to bury; to inter.
n.
One who intends.
a.
Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.
v. t.
To inter with funeral rites; to bury.
v. t.
To deposit or inter in a chapel; to enshrine.
n.
That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided.
n.
One who makes an index.
v. t.
To inter.
v. t.
To inter again.
n.
One who gathers the vintage.
v. t.
To deposit and cover in the earth; to bury; to inhume; as, to inter a dead body.
v. t.
To bury; to inter; to entomb; as, obscurely sepulchered.
v. t.
To place in a tomb; to bury; to inter; to entomb.
v. t.
To inhume; to bury; to inter.
p. pr. & vb. n.
of Inter
imp. & p. p.
of Inter
n.
One who inters.
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