A matrix is a rectangular array of elements (or entries) set out in rows and columns.

 

The entries of a matrix carry double subscripts; aij belongs in the ith row of the jth column,with i being the row index and j the column index. A matrix with m rows and n columns is an mxn, or m-by-n, matrix.

 

The expression mxn is the dimension, or order, of a matrix. Matrices with n=m are square matrices of order m, or n, denoted by [aij]m, or [aij]n.

 

The first non-zero entry in a row of a matrix is called the leading entry, or leading element, of that row.

 

The entries a11 a22 a33, ..., ann, that make up the principal diagonal, or main diagonal--from upper left to lower right--are diagonal entries.

 

If either m or n is unity, the matrix is a vector, or algebraic vector, to distinguish it from vectors representing directed line segments.

 

A matrix with m=1consists of a single n-dimensional row vector; if n=1, we have an m-dimensional column vector. If n=m=1, the matrix is reduced to a single entry and becomes a scalar.

 

A matrix in which all entries above the main diagonal are zero is a lower triangular matrix; if all entries below the main diagonal are zero, the matrix is an upper triangular matrix.

 

When all entries above and below the main diagonal are zero, and all non-zero entries are diagonal, the matrix is a diagonal matrix.

 

The sum of the diagonal entries of a square matrix is the trace of the matrix.

 

A diagonal matrix with a11=a22= a33=...=ann=k, where k is a constant, is a scalar matrix. If k=1, the matrix is a unit matrix, or identity matrix, usually denoted by I.

 

A matrix in which all entries are zero is a null matrix. Analogously, a vector with all entries equal to zero is a null vector.