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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.
 +: 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

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