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...

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...

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

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

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

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...

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...

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...

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

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...

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...

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...

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...

_{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...

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...

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...

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...

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...

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...

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

\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...

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...

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 ...

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...

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...

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...

matrix U, and a permutation matrix P; this is a matrix formulation of Gaussian elimination. By the fundamental theorem of arithmetic, every integer greater...