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Properties of Matrices.
1. Equality. Two m x n matrices A and B are said to be equal if all corresponding elements are equal, that is, if aij = bj for each i and j.
2. Zero. The symbol 0 will be used to denote the matrix (or vector), each of whose elements is zero.
3. Addition. The sum of two m x n matrices A and B is defined as the matrix obtained by adding corresponding elements:
A + B = (aj + (bj) = (au + bj). (2)
With this definition it follows that matrix addition is commutative and associative, so that
A + B = B + A, A + (B + C) = (A + B) + C. (3)
4. Multiplication by a Number. The product of a matrix A by a complex number
a is defined as follows:
aA = a(aij) = (aaij). (4)
The distributive laws
a(A + B) = aA + aB, (a + â) A = aA + â A (5)
are satisfied for this type of multiplication. In particular, the negative of A, denoted by -A, is defined by
-A = (-1)A. (6)
5. Subtraction. The difference A - B of two m x n matrices is defined by
A - B = A + (-B). (7)
A - B = (aj - (bij) = (au - bj), (8)
which is similar to Eq. (2).
6. Multiplication. The product AB of two matrices is defined whenever the number
of columns of the first factor is the same as the number of rows in the second. If A and B are m x n and n x r matrices, respectively, then the product C = AB is an m x r matrix. The element in the ith row and jth column of C is found by multiplying each
element of the ith row of A by the corresponding element of the jth column of B and
then adding the resulting products. Symbolically,
Chapter 7. Systems of First Order Linear Equations
By direct calculation it can be shown that matrix multiplication satisfies the associative law
(AB)C = A(BC) (10)
and the distributive law
A(B + C) = AB + AC. (11)
However, in general, matrix multiplication is not commutative. For both products AB and BA to exist and to be of the same size, it is necessary that A and B be square
matrices of the same order. Even in that case the two products are usually unequal, so
that, in general
AB = BA. (12)
To illustrate the multiplication of matrices, and also the fact that matrix multiplication is not necessarily commutative, consider the matrices
1 -2 1'
0 2 -1 12 1 1
'2 1 -1' B = I 1 -1 0
From the definition of multiplication given in Eq. (9) we have
2-2+2 AB = I 0 + 2 - 2 412
1 + 2 - 1 -1 + 0 + 1'
0 - 2 + 1 0 + 0 - 1
2-1-1 -2 + 0+1.
Similarly, we find that
0 3 0
II 4 2
4 in 4
Clearly, AB = BA.
7. Multiplication of Vectors. Matrix multiplication also applies as a special case if the matrices A and B are 1 x n and n x 1 row and column vectors, respectively. Denoting these vectors by xT and y we have
xT y = Ò,õ³Ó³. (13)
This is the extension to n dimensions of the familiar dot product from physics and calculus. The result of Eq. (13) is a (complex) number, and it follows directly from Eq. (13) that
xTy = yTx, xT (y + z) = xT y + xTz, (ax)Ty = a(xTy) = xT (ay). (14)
7.2 Review of Matrices
There is another vector product that is also defined for any two vectors having the same number of components. This product, denoted by (x, y), is called the scalar or inner product, and is defined by
(x, y) = ? Õ³Ó³ ,
The scalar product is also a (complex) number, and by comparing Eqs. (13) and (15) we see that
(x, y) = x y.
Thus, if all of the elements of y are real, then the two products (13) and (15) are identical. From Eq. (15) it follows that
(x, y) = (y, x), (ax, y) = a(x, y),
(x, y + z) = (x, y) + (x, z), (x,ay) = a(x, y).
Note that even if the vector x has elements with nonzero imaginary parts, the scalar product of x with itself yields a nonnegative real number,