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4.7.4 Matrix Operations

We define on matrices over R the following arithmetic operations.


Definition 60 (Matrices over the Reals) The domain of  real matrices of dimension m x n is defined as follows:
Mm, n := Nm x Nn -> R.


Definition 61 (Matrix Operations) For every m in N, n in N, and p in N we define the following operations on real matrices.
Constants

 null matrix (Nullmatrix):

0: Nm x Nn -> R,
0i, j := 0.
 unity matrix (Einheitsmatrix):
1: Nn x Nn -> R,
1i, j := if i = j then 1 else 0.
Addition
+: Mm, n x Mm, n -> Mm, n
A+B := such C in Mm, n:
   forall i in Nm, j in Nn: Ci, j = Ai, j + Bi, j.
short: (A+B)i, j := Ai, j + Bi, j.
Scalar Product
*: R x Mm, n -> Mm, n
c*A := such C in Mm, n:
   forall i in Nm, j in Nn: Ci, j = c*Ai, j.
short: (c*A)i, j := c*Ai, j.
Matrix Product
+: Mm, n x Mn, p -> Mm, p
A*B := such C in Mm, p:
   forall i in Nm, j in Np: Ci, j = (sum0 <= k < n Ai,k*Bk,j).
short: (A+B)i, j := (sum0 <= k < n Ai,k*Bk,j).
Determinant
| . |: Mn, n -> R,
if n = 1:
   |A| := A0, 0,
if n > 1:
   |A| := (sum0 <= j < n A0, j*(-1)j*|B|)
      where B = such B in Mn-1,n-1:
         forall k in Nn-1, l in Nn-1:
            Bk, l = (if l<j then Ak+1, l else Ak+1, l+1).

The determinant of an n×n matrix A is defined by the determinants of a number of (n-1)×(n-1) matrices B that are constructed from A by deleting the first row and some column j.


Example 
det(
1 2 3
4 5 6
7 8 9
) = 1* det(
5 6
8 9
) - 2 * det(
4 6
7 9
) + 3 * det(
4 5
7 8
)
det(
5 6
8 9
) = 5*9-6*8 = -3.
1 2 3
4 5 6
*
a b
c d
e f
=
1a + 2c + 3e 1b + 2d + 3f
4a + 5c + 6e 4b + 5d + 6f

The matrix operations satisfy many equations that also hold for numbers, e.g., A*(B+C) = A*B+A*C. However, unlike in number domains, matrix multiplication is not commutative.


Example 
1 2
3 4
*
3 1
2 4
=
7 9
17 19
3 1
2 4
*
1 2
3 4
=
6 10
14 20


Author: Wolfgang Schreiner
Last Modification: October 4, 1999

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