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CHAPTER 1
Matrices
1.1 Matrices and Matrix Operations
Definition 1.1
A matrix is a rectangular array of numbers of the form
In the above form the matrix has m rows and n columns, we say that it is of order m X n.
The mn numbers a11, a12, ..., amn are known as the elements of the matrix A. If the elements belong to a field F, we say that A is a matrix over the field F. Since the concept of a field plays an emportant role in development of vector spaces we shall state the axioms for a field at this early stage.
The notation we use is the following
a [member of] S means that a belongs to the set S
a [not member of] S means that a does not belong to the set S
φ denotes the empty set
[there exists] denotes 'there exists'.
A field F consists of a non-empty set of elements and two laws called addition and multiplication for combining elements satisfying the following axioms:
Let a, b, c [member of] F
A1 a + b is a unique element [member of] F
A2 (a+b) + c = a + (b+c)
A3 [there exist]0 [member of] F such that 0 + a = a, all a [member of] F.
A4 For each a [member of] F, [there exist] (-a) [member of] F such that a + (-a) = 0
45 a + b = b + a
M1 ab is a unique element [member of] F
M2 (ab)c = a(bc)
M3 [there exiest]1 [member of] F such that 1a = a for all a [member of] F
M4 If a ≠ 0, [there exist] a-1 [member of] F such that a-1a = 1.
M5 ab = ba
M6 a(b+c) = ab + ac.
Typical examples of fields are: real numbers, rational numbers, and complex numbers.
If we consider the element aij of a matrix the first suffix, i, indicates that the element stands in the ith row, whereas the second suffix, j, indicates that it stands in the jth column.
For example, the element which stands in the second row and the first column is a21.
We usually denote a matrix by a capital letter, say A, or by its (i,j)th element in the form [aij].
Definition 1.2
A square matrix is a matrix of order n X n of the form
[MATHEMATICAL EXPRESSION OMITTED]
The elements a11, a22, ..., ann of A are called diagonal elements.
Example 1.1
The following are matrices:
[MATHEMATICAL EXPRESSION OMITTED]
A is a square matrix of order 3X3.
B is a rectangular matrix of order 2X3.
C and D are matrices of order 3X1 and 1X2 respectively.
Note
C is also known as a column matrix or column vector. D is known as a row matrix or a row vector.
The diagonal elements of A are 1, -1, 1.
The (2,3) element of B is b23 = -2.
Definitions 1.3
(1) The zero matrix of order m X n is the matrix having all its mn elements equal to zero.
(2) The unit or identity matrix I is the square matrix of order n X n whose diagonal elements are all equal to 1 and all remaining elements are 0.
(3) The diagonal matrix A is a square matrix for which aij = 0 whenever i ≠ j. Thus all the off-diagonal elements of a diagonal matrix are zero.
(4) The diagonal matrix for which all the diagonal elements are equal to each other is called a scalar matrix.
Example 1.2
Consider the following matrices:
[MATHEMATICAL EXPRESSION OMITTED]
A is a scalar matrix of order 3X3
B is a diagonal matrix of order 3X3
I is the unit matrix of order 3X3
O is the zero matrix of order 2X3.
Note
We shall use the accepted convention of denoting the zero and unit matrices by O and I respectively. It is of course necessary to state the order of matrices considered unless this is obvious from the text.
Definition 1.4
(1) An upper triangular matrix A is a square matrix whose elements aij = 0 for i j.
(2) A lower triangular matrix A is a square matrix whose elements aij = 0 for i j.
Example 1.3
Consider the two matrices
[MATHEMATICAL EXPRESSION OMITTED]
A is an upper triangular matrix
B is a lower triangular matrix.
Definition 1.5
Two matrices A = [aij] and B = [bij] are said to be equal if
(1) A and B are of the same order, and
(2) the corresponding elements are equal, that is if aij = bij (all i and j).
Operations on Matrices
Definition 1.6
The sum (or difference) of two matrices A = [aij] and B = [bij] of the same order, say m X n, is a matrix C = [cij] also of order m X n such that
cij = aij + bij (i = 1, ... m; j = 1, ... n).
(Or cij = aij - bij if we are considering the difference of A and B.)
Example 1.4
Given
[MATHEMATICAL EXPRESSION OMITTED]
find A+B and A-B.
Solution
[MATHEMATICAL EXPRESSION OMITTED]
Definition 1.7
The multiplication of a matrix A = [aij] of order m X n by a scalar r is the matrix rA such that
rA = [raij] = r[aij] (r = 1, ... m; j = 1, ... n)
Example 1.5
Given
[MATHEMATICAL EXPRESSION OMITTED]
Solution
[MATHEMATICAL EXPRESSION OMITTED]
Note that 3A = A + A + A, and we can evaluate the right-hand side by definition 1.6.
Definition 1.8
The product of the matrix A = [aij] of order m X l and the matrix B = [bij] of order l X n is the matrix C = [cij] of order m X n defined by
[MATHEMATICAL EXPRESSION OMITTED]
We illustrate this definition by showing up the elements of A and B making up the (i,j)th element of C.
[MATHEMATICAL EXPRESSION OMITTED]
Note the following:
(i) The product AB is defined only if the number of columns of A is the same as the number of rows of B. If this is the case, we say that A is conformable to B.
(ii) The (i,j)th element of C is evaluated by using the ith row of A and jth column of B.
(iii) If A is conformable to B the product AB is defined, but it does not follow
that the product BA is defined. Indeed, if A and B are of order m X l and l X n respectively, the product AB is defined but the product BA is not, unless n = m.
(iv) When both AB and BA are defined, AB ≠ BA in general, that is, matrix multiplication is NOT commutative.
(v) (a) If AX = 0, it does not necessarily follow that A = 0 or X = 0.
(b) If AB = AC, it does not necessarily follow that B = C.
(See Sec. 1.2 for further discussion of (iv) and (v) and examples).
Example 1.6
[MATHEMATICAL EXPRESSION OMITTED]
Find (if possible)
(i) AB, (ii) BA, (iii) AC, (iv) CA, (v) BC.
Solution
(i) Since A has 3 columns and B has 3 rows, the product AB is defined.
[MATHEMATICAL EXPRESSION OMITTED]
(ii) Since B has 3 columns and A has 2 rows the product BA is not defined.
(iii) Since A has 3 columns and C has 3 rows, AC is defined
[MATHEMATICAL EXPRESSION OMITTED]
(iv) The product CA is not defined.
(v) [MATHEMATICAL EXPRESSION OMITTED]
Example 1.7
Given the matrices:
[MATHEMATICAL EXPRESSION OMITTED]
write the equations AX = B in full.
Solution
AX = B is the equation:
[MATHEMATICAL EXPRESSION OMITTED]
By Def. 1.8, we can write the above as
[MATHEMATICAL EXPRESSION OMITTED]
By Def. 1.5 the above equation is equivalent to the following system of simultaneous equations:
[MATHEMATICAL EXPRESSION OMITTED]
Definition 1.9
The transpose of a matrix A = [aij] of order m X n is the matrix A' = [bij] of order n X m obtained from A by interchanging the rows and columns of A so that bij = aij (i = 1,2, 1,2, ... m), for example if
[MATHEMATICAL EXPRESSION OMITTED]
The transpose of a column matrix is a row matrix and vice versa, thus if
[MATHEMATICAL EXPRESSION OMITTED]
Note that (A')' = A.
Notation
To denote vectors (see Def. 2.1) we shall make use of several notations.
When considering a one-column matrix or a one-row matrix, we use
[MATHEMATICAL EXPRESSION OMITTED]
Since a one-row matrix is less space-consuming to write than a one-column matrix, we frequently write the column matrix in the transposed form as [1,2,3]'.
Finally, if it is immaterial whether the vectors under discussion are row or column vectors, we use the notation (1,2,3).
1.2 SOME PROPERTIES OF MATRIX OPERATIONS
In this section we shall state without proof (in general) a number of properties of matrix operation. The interested reader will find the proofs in most of the books mentioned in the Bibliography.
Although there are a number of analogies between the algebraic properties of matrices and real numbers, there are also striking differences, some of which will be pointed out.
Let A, B, and C be matrices, each of order m X n
Addition laws
(1) Matrix addition is commutative; that is,
A + B = B + A.
(2) Matrix addition is associative; that is,
A + (B + C) = (A + B) + C.
(3) There exists a unique zero matrix O of order m X n such that
A + 0 = 0 + A = A.
(4) Given the matrix A, there exists a unique matrix B such that
A + B = 0
The above property serves to introduce the subtraction of matrices. Indeed the unique matrix B in the above equation is found to be equal to (-1)A which we write as -A. We then find that A - A = 0 and that -(-A) = A.
(5) Multiplication by scalars. If r and s are scalars, then
(a) r(A+B) = rA + rB
(b) (r+s)A = rA 4- sA
(c) r(A-E) = rA - rB
(d) (r-s)A = rA - sA.
Example 1.8
Given
[MATHEMATICAL EXPRESSION OMITTED]
and
[MATHEMATICAL EXPRESSION OMITTED]
(a) show that (i) A + B = B + A (ii) A + (B+C) = (A + B) + C.
(b) find the matrix X such that A + X = 0.
Solution
[MATHEMATICAL EXPRESSION OMITTED]
Also, using result (i) above,
[MATHEMATICAL EXPRESSION OMITTED]
Multiplication laws
(6) If A is m X l and B = l X m, both AB and BA are defined, but matrix multiplication is not commutative; that is, in general AB ≠ BA. Note that this is an important difference between the multiplication of matrices and real numbers. If AB = BA, we say that A and Bcommute or are commutative.
Example 1.9
Given
[MATHEMATICAL EXPRESSION OMITTED]
find AB, BA, CD and DC.
Solution
[MATHEMATICAL EXPRESSION OMITTED]
hence AB ≠ BA.
[MATHEMATICAL EXPRESSION OMITTED]
hence CD = DC.
Example 1.10
Given
[MATHEMATICAL EXPRESSION OMITTED]
(i) evaluate AX, and
(ii) verify that AB = AC.
Comment on the results.
Solution
(i) [MATHEMATICAL EXPRESSION OMITTED]
(ii) [MATHEMATICAL EXPRESSION OMITTED]
In the above example it is shown that
(i) AX = 0 does not necessarily imply that A = 0 or X = 0, and
(ii) AB = AC does not necessarily imply that B = C.
These are just two more of the striking differences between operations with matrices and real numbers.
(7) Matrix multiplication is
(i) distributive with respect to addition; that is,
A(B+C) = AB + AC, and
(B+C)A = BA + CA.
provided that the matrices are conformable.
(ii) associative, that is,
A(BC) = (AB)C = ABC
provided that expressions on either side of the equations are defined
Definition 1.10
The rth power of a square matrix A of order n X n is denoted by Ar and defined by
Ar = Ar-1 A for r ≥ 1
and
A0 = I (I being the unit matrix of order n X n).
Thus A2 = AA
and A3 = AAA.
The powers of a matrix are commutative, that is
ArAs = Ar+s = AsAr (r,s ≥ 0)
Example 1.11
Given
[MATHEMATICAL EXPRESSION OMITTED]
evaluate the matrix polynomial
f(A) = A3 + 4A2 + 6A + AI.
Solution
We find that
[MATHEMATICAL EXPRESSION OMITTED]
and
[MATHEMATICAL EXPRESSION OMITTED]
so that
[MATHEMATICAL EXPRESSION OMITTED]
Definition 1.11
If A is a matrix of order n X n and there exists another matrix B of order n X n such that
AB = BA = I
then B is called the inverse of A and is denoted by A-1, and A is said to be non-singular. If A does not posess an inverse, we say that it is singular.
If the inverse of a matrix exists, it is unique.
Example 1.12
Find (if they exist) the inverses of
(i) [MATHEMATICAL EXPRESSION OMITTED]
Solution
We must find [MATHEMATICAL EXPRESSION OMITTED] such that
AA-1 = I = A-1A.
(i) [MATHEMATICAL EXPRESSION OMITTED]
On equating the elements (using the first product)
[MATHEMATICAL EXPRESSION OMITTED]
so that u = -2, v = 1, w = 3/2 and z = - 1/2.
In this case A is non-singular.
(ii) [MATHEMATICAL EXPRESSION OMITTED]
This system of equations is inconsistent, hence there is no solution. In this case A is a singular matrix.
Theorem 1.1
If A = [aij] is a matrix of order m X n, and B = [bij] is a matrix of order n X r, then
(AB)' = B'A'.
Proof
Let AB = C = [cij]
The (i,j)th element of (AB)' is cji.
But [MATHEMATICAL EXPRESSION OMITTED]
where A' = [a'ij and B' = [bij].
Also [MATHEMATICAL EXPRESSION OMITTED], where [MATHEMATICAL EXPRESSION OMITTED] is the (i,j)th element of B'A'.
The result follows.
We can generalise the above result and show that
(A1A2 ... An)' = A'n ... A'2A'1
provided that the products are defined.
(Continues…)
Excerpted from "Matrix Theory & Applications for Scientists & Engineers"
by .
Copyright © 2018 Alexander Graham.
Excerpted by permission of Dover Publications, Inc..
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