N×Nmatriisissa

N×Nmatriisissa denotes the setting of an N by N matrix, where N is a positive integer. An N×Nmatriisissa is a two‑dimensional array with N rows and N columns, each entry typically taken from a field such as the real or complex numbers. The size of such a matrix is specified by the pair (N,N), and its dimension—in the sense of linear algebra—is N. In many contexts the term implicitly indicates that the matrix is square, which allows for the definition of determinants, eigenvalues, and many special matrix classes such as symmetric, orthogonal, and diagonal matrices.

The determinant of an N×Nmatriisissa, denoted det(A) for a matrix A, can be computed recursively via cofactor expansion, or more efficiently using LU decomposition. For N=2 and N=3 the determinant has closed‑form formulas involving the entries of the matrix, while for larger N numeric algorithms are employed. The inverse of an N×Nmatriisissa exists precisely when det(A) is non‑zero; in this case the inverse can be expressed in terms of the adjugate matrix divided by the determinant.

Eigenvalue problems for N×Nmatriisissa are central to many scientific fields. The characteristic polynomial is of degree N, and its roots give the eigenvalues. The number of distinct eigenvalues depends on the algebraic and geometric multiplicities of the roots. When N×Nmatriisissa has distinct eigenvalues it is diagonalizable over the underlying field.

In computational applications, the storage requirement for an N×Nmatriisissa is N² elements. Therefore, sparse matrix techniques are often employed when N is large and many entries are zero. Even so, operations involving N×Nmatriisissa such as matrix multiplication, solving linear systems, and computing eigenvalues remain fundamental to fields ranging from physics and engineering to data science and machine learning.