Linear Algebra Matrices Vocabulary

Flashcards covering key vocabulary and definitions related to matrices and linear algebra.

Matrix

An array with m rows and n columns of real numbers.

aij

The (ij)-entry in a matrix, located at the i-th row and j-th column.

Submatrix

A matrix obtained by deleting some rows and/or columns of A.

Transpose Matrix

An n × m matrix A^T such that a^Tij = aji for all i, j. Obtained by exchanging rows with columns.

Square Matrix

An n × n matrix.

Main Diagonal

The vector (a11, …, aii, …, ann) of a square matrix.

Symmetric Matrix

A matrix where A = A^T.

Triangular Matrix

A matrix where aij = 0 for all j < i (upper) or j > i (lower).

Diagonal Matrix

A matrix where aij = 0 for all i != j.

Identity Matrix (In)

An n × n diagonal matrix with aii = 1 for all i.

Commuting Matrices

Two square matrices A, B commute when AB = BA.

Zero Divisor

A matrix A is a left (right) zero divisor if there exists another matrix B != 0 such that AB = 0 (BA = 0).

Determinant

A real number associated with a square matrix A, denoted by |A| or det A.

Cofactor

For any entry aij, its cofactor Cij(A) = (-1)^(i+j) |A(ij)|, where A(ij) is the (n-1)x(n-1) submatrix of A obtained by deleting the i-th row and j-th column.

Inverse Matrix

A matrix A−1 such that A A−1 = A−1 A = In. Not all square matrices have an inverse.

Rank of a Matrix

The maximum number of linearly independent column (or row) vectors in the matrix.