integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix...

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...

integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers...

algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it...

(symmetrizable) generalized Cartan matrix is a square matrix A = ( a i j ) {\displaystyle A=(a_{ij})} with integer entries such that For diagonal entries...

floating point numbers, but not necessarily for integers). Strassen's algorithm improves on naive matrix multiplication through a divide-and-conquer approach...

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...

by a unimodular matrix (that is, an invertible integer matrix whose inverse is also an integer matrix). Changing the generating set of the kernel of M...

definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated...

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...

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...

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...

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...

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...

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...

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element...

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...

_{i=0}^{\ell }\lambda ^{i}A_{i}.} The degree of the matrix pencil is defined as the largest integer 0 ≤ k ≤ ℓ {\displaystyle 0\leq k\leq \ell } such that...

number 59), an integer-matrix quadratic form that represents all prime numbers when it represents the sequence of seventeen integers {2, ..., 47, 67...

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...

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 = [...

\infty }\left(I+{\frac {X}{k}}\right)^{k}} for integer-valued k, where I is the n × n identity matrix. Equivalently, given by the solution to the differential...

contained (divisor) need not be integers. The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number...

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...

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...

Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms...

positive definite quadratic integer matrix represents all odd numbers when it contains at least the set of seven integers: { 1 , 3 , 5 , 7 , 11 , 15 ...

classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that...

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...

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...