math111_logo Composition and Matrix Multiplication


Exercise What does the multiplication of two upper triangular matrices look like? What about lower triangular matrices? diagonal matrices?

Answer Given two upper triangular matrices,

A = [ a1 * . . * ], B = [ b1 * . . * ]
0 a2 . . * 0 b2 . . *
: :   : : :   :
0 0 . . an 0 0 . . bn

their multiplication is still upper triangular, with the diagonal entries simply multiplied together (the other entries are more complicated).

AB = [ a1b1 * . . * ]
0 a2b2 . . *
: :   :
0 0 . . anbn

The similar statement is true for the multiplication of two lower triangular matrices.

For two diagonal matrices,

A = [ a1 0 . . 0 ], B = [ b1 0 . . 0 ]
0 a2 . . 0 0 b2 . . 0
: :   : : :   :
0 0 . . an 0 0 . . bn

their multiplication is still diagonal, with the diagonal entries simply multiplied together.

AB = [ a1b1 0 . . 0 ]
0 a2b2 . . 0
: :   :
0 0 . . anbn