Matix, Matrices
A matrix is an ordered set of numbers listed rectangular(square) form.
Example: rectangular matrix A 2 x 3
A = |
71 52 43 48 89 63 |
2 x 3 is the dimension of A
A = (aij)nxm - standard notation for a matrix(aij are the elements of the matrix), where 0 ≤ i ≤ n, 0 ≤ j ≤ m.
If the rows of a matrix are equal the columns the matrix is called square matrix.
If A is a sqare matrix, the elements a11, a22,..., ann are called main diagonal of the matrix, and a1n, a2,n-2,..., an1 are the second diagonal.
We say that one matrix is identity matrix if the matrix is a sqare matrix and the elements from the main diagonal are 1 and all other elements are 0. We sign identity matrices with En or with E.
When all the elements of a matrix are 0, we say that the matrix is 0-matrix and write 0 for such a matrix.
The matrix Atnxm is the transpose of
Anxm if we change all rows of A with
their corresponding columns.
For example the transpose matrix of the A is:
$A^t= \left( \begin{array}{cc}
71 & 48 \\
52 & 89 \\
43 & 63 \end{array} \right)$
An animated example:
A matrix is called symmetric matrix if it is square and is equal to its transpose.
If we have two matrix A, and B that all elements of B are equal ot (-1).ai,j
B is called opposite matrix of A and we write B = -A
A = (aij)mxn, B = (bij)mxn the sum of A and B is the matrix:
if λ is a number, scalar product of λA is the matrix:
Examples
Let A = |
1 2 4 8 |
B = |
11 14 10 15 |
A + B = |
1+11 2+14 4+10 8+15 |
= |
12 16 14 23 |
Note: if A and B have different dimensions we can not write A + B
if λ = 10
A = |
1 2 4 8 |
λA = |
10 20 40 80 |
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