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Matrix products that involve diagonal factors are especially easy to compute. For example,
In words, to multiply a matrix A on the left by a diagonal matrix D, one can multiply successive rows of A by the successive
diagonal entries of D, and to multiply A on the right by D, one can multiply successive columns of A by the successive diagonal
entries of D.
Triangular Matrices
A square matrix in which all the entries above the main diagonal are zero is called lower triangular, and a square matrix in
which all the entries below the main diagonal are zero is called upper triangular. A matrix that is either upper triangular or
lower triangular is called triangular.
EXAMPLE 2 Upper and Lower Triangular Matrices
Remark Observe that diagonal matrices are both upper triangular and lower triangular since they have zeros below and above
the main diagonal. Observe also that a square matrix in row-echelon form is upper triangular since it has zeros belowthe main
diagonal.
The following are four useful characterizations of triangular matrices. The reader will find it instructive to verify that the
matrices in Example 2 have the stated properties.
A square matrix is upper triangular if and only if the ith row starts with at zeros.
A square matrix is lower triangular if and only if the j th column starts with zeros.
