### Vector and Matrix

##### 3. Some matrix notations and terminologies

In theoretical discussions about matrices, we sometimes use the notation **A** = (`a`_{ij})
to denote a matrix, where `a`_{ij} is the entry at the `i`-th row and `j`-th column
(the `ij`-entry). Sometimes we even indicate the size by writing
**A** = (`a`_{ij})_{m×n}. In terms of this compact notation, we have

(`a`_{ij}) + (`b`_{ij}) = (`a`_{ij} + `b`_{ij}),
`c`(`a`_{ij}) = (`ca`_{ij}).

Example

(`i` + `j`)_{3×3} = [ |
1 + 1 |
1 + 2 |
1 + 3 |
] = [ |
2 |
3 |
4 |
] |

2 + 1 |
2 + 2 |
2 + 3 |
3 |
4 |
5 |

3 + 1 |
3 + 2 |
3 + 3 |
4 |
5 |
6 |

(`ij`)_{4×5} = [ |
1×1 |
1×2 |
1×3 |
1×4 |
1×5 |
] = [ |
1 |
2 |
3 |
4 |
5 |
] |

2×1 |
2×2 |
2×3 |
2×4 |
2×5 |
2 |
4 |
6 |
8 |
10 |

3×1 |
3×2 |
3×3 |
3×4 |
3×5 |
3 |
6 |
9 |
12 |
15 |

4×1 |
4×2 |
4×3 |
4×4 |
4×5 |
4 |
8 |
12 |
16 |
20 |

(`j`^{2})_{4×3} = [ |
1^{2} |
2^{2} |
3^{2} |
] = [ |
1 |
4 |
9 |
] |

1^{2} |
2^{2} |
3^{2} |
1 |
4 |
9 |

1^{2} |
2^{2} |
3^{2} |
1 |
4 |
9 |

1^{2} |
2^{2} |
3^{2} |
1 |
4 |
9 |

For an `n` by `n` square matrix **A** = (`a`_{ij}), the terms
`a`_{11}, `a`_{22}, ..., `a`_{nn} are the
diagonal entries of the matrix. **A** is a
diagonal matrix if all the off-diagonal entries are zero.
In other words, `a`_{ij} = 0 for `i` ≠ `j`.
The following are all the 2 by 2 and 3 by 3 diagonal matrices.

**A** is an upper triangular matrix
if all the entries below the diagonal are zero. In other words, `a`_{ij} = 0 for
`i` > `j`. The following are all the 2 by 2 and 3 by 3 upper triangular
matrices.

Similarly, **A** is a lower triangular matrix
if all the entries above the diagonal are zero. In other words, `a`_{ij} = 0 for
`i` < `j`. The following are all the 2 by 2 and 3 by 3 upper triangular matrices.

Clearly, **A** is upper and lower triangular at the same time if and only if it is diagonal.