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 |
a_{1} |
* | . . | * | ], = [B |
b_{1} |
* | . . | * | ] |

0 | a_{2} |
. . | * | 0 | b_{2} |
. . | * | |||

: | : | : | : | : | : | |||||

0 | 0 | . . | a_{n} |
0 | 0 | . . | b_{n} |

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

= [AB |
a_{1}b_{1} |
* | . . | * | ] |

0 | a_{2}b_{2} |
. . | * | ||

: | : | : | |||

0 | 0 | . . | a_{n}b_{n} |

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

For two diagonal matrices,

= [A |
a_{1} |
0 | . . | 0 | ], = [B |
b_{1} |
0 | . . | 0 | ] |

0 | a_{2} |
. . | 0 | 0 | b_{2} |
. . | 0 | |||

: | : | : | : | : | : | |||||

0 | 0 | . . | a_{n} |
0 | 0 | . . | b_{n} |

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

= [AB |
a_{1}b_{1} |
0 | . . | 0 | ] |

0 | a_{2}b_{2} |
. . | 0 | ||

: | : | : | |||

0 | 0 | . . | a_{n}b_{n} |