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:
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*
det(
)
- 2 *
det(
)
+ 3 *
det(
)
det(
)
=
5*9-6*8 = -3.
*
=
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
*
=
*
=
Author: Wolfgang Schreiner
Last Modification: October 4, 1999