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.
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.
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).
If A belongs to Km,l and B belongs to Kl,n then:
C = A * B belonging to Km,n
ci,j = ∑ ai,k*bk,j, 1 ≤ i ≤ m, 1 ≤ j ≤ n.
Distinct Matrix Formats.
n x n - Square Matrix Kn,n,
m x 1 - one column Matrix called vector. Km,1 = Km.
See also: Groups & Bodies, Equations Matrix.