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3. Some matrix notations and terminologies

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

(aij) + (bij) = (aij + bij), c(aij) = (caij).

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
(j2)4×3 = [ 12 22 32 ] = [ 1 4 9 ]
12 22 32 1 4 9
12 22 32 1 4 9
12 22 32 1 4 9

For an n by n square matrix A = (aij), the terms a11, a22, ..., ann are the diagonal entries of the matrix. A is a diagonal matrix if all the off-diagonal entries are zero. In other words, aij = 0 for ij. The following are all the 2 by 2 and 3 by 3 diagonal matrices.

[ a 0 ]
0 b
[ a 0 0 ]
0 b 0
0 0 c

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

[ a b ]
0 c
[ a b c ]
0 d e
0 0 f

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

[ a 0 ]
b c
[ a 0 0 ]
b c 0
d e f

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


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