of of

S is a which has the same number of rows and columns, that is, m = n. For example,

(1.8)

is a square n x n matrix. Our interest will be devoted entirely to square matrices. The leftto-right downward-sloping line of elements from all to ann is called the major diagonal of the matrix. A diagonal matrix D is a square matrix with all elements equal to zero except the elements on the major diagonal. For example,

all

(1.9)

is a 4 x 4 diagonal matrix. The identity matrix I is a diagonal matrix with unity diagonal elements. The identity matrix is the matrix equivalent of the scalar number unity. The matrix

(1.10)

is the 4 x 4 identity matrix. A triangular matrix is a square matrix in which all of the elements on one side of the

major diagonal are zero. The remaining elements may be zero or nonzero. An upper triangular matrix U has all zero elements below the major diagonal. The matrix

al2 al3 an a23 o a33 o 0

(1.11 )

is a 4 x 4 upper triangular matrix. A lower triangular matrix L has all zero elements above the major diagonal. The matrix

["" 0 0 nL = a21 an 0 (1.12)a31 a32 a33a41 a42 a43 is a 4 x 4 lower triangular matrix.