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## 1/22/14

### Matrix (over body K).

Matrix (over body K) is a rectangular table:

| a1,1  a1,2  ...  a1,n   |
|                                   |
| a2,1  a2,2  ...  a2,n   |
A = |                                   |
| ...       ...     ...   ...       |
|                                   |
| am,1  am,2  ...am,n  |

where ai,j belongs to K, 1 ≤ i ≤ m, 1 ≤ j ≤ n.

We'll say that A is matrix in format m x n, i.e. matrix with m rows and n columns.

Set of all such matrices we'll note as Km,n.

Operations:

Multiplication by number.

for u that belongs to K, A that belongs to Km,n => B = u * A occurs: bi,j = u * ai,j.

for A,B that belong to Km,n => C = A + B occurs: ci,j = ai,j + bi,j.

Transposition.

for A that belongs to Km,n => B = AT occurs: bj,i = ai,j

Element-wise absolute value.

for A that belongs to Cm,n => B = |A| occurs: bi,j = |ai,j|

(more will be added soon perhaps).

Multiplication.

If A belongs to Km,l and B belongs to Kl,n then:

C = A * B belonging to Km,n

where:

l
ci,j = ∑ ai,k*bk,j, 1 ≤ i ≤ m, 1 ≤ j ≤ n.
k=1

Distinct Matrix Formats.

n x n - Square Matrix Kn,n,
m x 1 - one column Matrix called vector. Km,1 = Km.

Source: [1].