Square matrix

A matrix with the same number of rows and columns. Its order is the shared number of rows and columns.

Matrix multiplication: An operation that combines matrices by multiplying rows of one matrix by columns of another. A square matrix can be multiplied by itself, enabling powers and polynomial expressions.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. A square matrix represents a linear transformation from a finite-dimensional space to itself.
Rectangular matrix: A matrix whose numbers of rows and columns differ. It contrasts directly with the defining dimension equality of a square matrix.
Eigenvector: A nonzero vector whose direction is unchanged by a linear transformation, apart from scaling. Eigenvectors identify directions on which the matrix acts by simple scalar multiplication.
Linear algebra: The branch of mathematics studying vector spaces, linear maps, and systems of linear equations. Square matrices are central representations of linear maps between finite-dimensional spaces.
Determinant: A scalar associated with a square matrix that encodes properties including invertibility and signed volume scaling. Only square matrices have determinants, which test whether their transformations are invertible.
System of linear equations: A collection of linear equations sharing the same variables. When equations and variables are equally numerous, their coefficients form a square matrix.
Rank (linear algebra): The dimension of the row space or column space of a matrix. Unlike determinant and inverse, rank is defined for square and rectangular matrices alike.
Diagonalization: The expression of a matrix as similar to a diagonal matrix. It simplifies powers and functions of a square matrix when enough independent eigenvectors exist.
Endomorphism: A map from a mathematical object to itself that preserves its structure. A square matrix represents an endomorphism after choosing a basis for a finite-dimensional vector space.

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