3.7 Transpose
3.7.2 Symmetric matrices
Definition 3.10 (Symmetric matrix) A matrix A issymmetric if it is equal to its transpose, A=AT.
Only square matrices can be symmetric. If A is symmetric, then aij =aji. That is, entries diagonally opposite to each other must be equal: the matrix is symmetric about its diagonal.
Activity 3.20 Fill in the missing numbers if the matrixA is symmetric:
A=
1 4
2
5 3
=
1
−7
=AT
If Dis a diagonal matrix, then dij = 0 =dji for all i6=j. So as we saw before in Activity 3.16, DT =D; that is, all diagonal matrices are symmetric.
R Read the remaining parts of Chapter 1, Sections 1.1–1.7.
Overview
In this chapter we have looked at the terminology associated with matrices and the operations defined for matrices, when they are defined and the properties they satisfy.
We have seen how to manipulate matrices algebraically.
You will be working with matrices throughout this course so it is important for you to gain a facility with these definitions and operations – you should be able to use them with ease.
Learning outcomes
At the end of this chapter and the relevant reading you should be able to:
define a matrix and explain the terminology used with matrices, such as row, column, size, square matrix, diagonal matrix, equality of matrices, transpose of a matrix, symmetric matrix
define and use matrix addition, scalar multiplication and matrix multiplication appropriately (know when and how these operations are defined)
manipulate matrices algebraically
define what is meant by the inverse of a square matrix, know and use the properties of the inverse of a matrix and find the inverse of a 2×2 matrix define what is meant by An where A is a square matrix and n is an integer;
demonstrate and use the fact that (An)−1 = (A−1)n.
state and use the properties of the transpose, use transpose in combination with the other operations defined on matrices
3
3.7. Test your knowledge and understanding
Test your knowledge and understanding
Work Exercises 1.1–1.7 in the text A-H. The solutions to all exercises in the text can be found at the end of the textbook.
Work Problems 1.5–1.7 in the text A-H. You will find the solutions on the VLE.
In addition, work the exercises below. The solutions are at the end of this chapter.
Exercises
Exercise 3.1
For fixed real numbers a, b, c, d and real numbers x, y, z, w assume that AB =
a b c d
x y z w
=
1 0 0 1
=I.
Write down the four linear equations in x, y, z, w that you obtain by first multiplying the matrices on the left and then equating the entries of the product to the entries of the identity matrix. Then solve these equations for x, y, z, w.
You should find that solution is only possible if ad−bc6= 0, and that the solution is
B = 1
ad−bc
d −b
−c a
. Compare this with the result you were given on page 40.
Exercise 3.2
What is meant by the statement that A is a symmetric matrix?
If B is an m×k matrix, show that the matrixBTB is a k×k symmetric matrix.
Comments on selected activities
Feedback to activity 3.1
This is the entry in the third row and first column, so a31= 7.
Feedback to activity 3.2
D=
a 0 0
0 e 0
0 0 i
.
Feedback to activity 3.3
A−B =
0 2 −4
2 1 6
.
3
Feedback to activity 3.4 1 2
1 4
+
−2 5
2 0
=
−1 7
3 4
.
Feedback to activity 3.5 AB is 2×2 andBA is 3×3,
AB =
10 5
6 2
BA=
7 5 10
2 1 3
3 3 4
Feedback to activity 3.6
AB=
1 3 3 7
BA=
4 6 3 4
. Feedback to activity 3.7
If A ism×n and B is n×p, then AB is an m×p matrix. The size of a matrix is not changed by scalar multiplication, so both λ(AB) and (λA)B are m×p. Looking at the (i, j) entries of each,
(λ(AB))ij = λ(ai1b1j +ai2b2j+. . .+ainbnj)
= λai1b1j +λai2b2j +. . .+λainbnj
= (λai1)b1j + (λai2)b2j+. . .+ (λain)bnj
= ((λA)B)ij, so these two matrices are equal.
Feedback to activity 3.8
For the first two rules, the 0 matrix must be a 2×3 matrix, 0 =
0 0 0
0 0 0
.
For the last set of rules, ifA is 2×3, then for 0A= 0, the 0 matrix multiplying A must be p×2 wherep is any positive integer, and then 0A is equal to the 0 matrix of size p×3. Similarly, forA0 = 0, the 0 matrix multiplying A must be 3×q where q is any positive integer, and then A0 is equal to the 0 matrix of size 2×q.
Feedback to activity 3.9 In this case I is m×m.
Feedback to activity 3.10
Look at how this is done in Example 1.17 of the text A-H.
Feedback to activity 3.12
AB=
1 2 3 4
−2 1
3 2 −12
=
1 0 0 1
3
3.7. Comments on selected activities
and
BA=
−2 1
3 2 −12
1 2 3 4
=
1 0 0 1
.
Therefore A−1 =
−2 1
3 2 −12
. Feedback to activity 3.13
We will show one way, you should show that A−1A=I. AA−1 =
a b c d
1 ad−bc
d −b
−c a
= 1
ad−bc
ad−bc −ab+ba cd−dc −bc+ad
=
1 0 0 1
. Feedback to activity 3.15
We will do the last property and leave the others to you. The inverse of Ar is a matrix B such that ArB =BAr =I. So show that the matrix B = (A−1)r works:
Ar(A−1)r = (A A . . . A
| {z }
r times
)(A−1A−1. . . A−1
| {z }
r times
).
Removing the brackets (matrix multiplication is associative) and replacing each central AA−1 =I, the resultant will eventually be AIA−1 =AA−1 =I. In the same way,
(A−1)rAr= (A−1A−1. . . A−1
| {z }
r times
)(A A . . . A
| {z }
r times
) =I.
Therefore (Ar)−1 = (A−1)r. Feedback to activity 3.16
IfD= (dij) is a diagonal n×n matrix, thendij = 0 for alli6=j. Therefore dji = 0 =dij for all i6=j. And dii =dii does not change, so DT =D.
Feedback to activity 3.17 ForA =AT, we must have b=c.
Feedback to activity 3.18
Given the sizes of A and B, the matrix AB is m×p, so (AB)T isp×m. Also, AT is n×m and BT isp×n, so the only way these matrices can be multiplied is asBTAT (unless m=p).
Feedback to activity 3.19
The (i, j) entry of BTAT is obtained by taking row i of BT, which is column i of the matrix B and multiplying each of the n terms by the corresponding entry of column j of AT, which is row j of the matrix A, and then summing the products.
You can also write the entries as:
(AB)T
ij =aj1b1i+aj2b2i+. . .+ajnbni. and BTAT
ij =b1iaj1+b2iaj2+. . .+bniajn.
3
Since multiplication of real numbers is commutative, these two expression are the same real number.
Feedback to activity 3.20 The matrix is
A =
1 4 5
4 2 −7
5 −7 3
=AT
Comments on exercises
Solution to exercise 3.1 The equations are
(ax+bz= 1
cx+dz = 0 and
(ay+bw= 0 cy+dw= 1
To begin you can solve the first set by multiplying the top equation by d and the bottom equation by b and then subtracting one equation from the other to eliminate the terms in z. You will obtain (ad−bc)x= 1. Then provided ad−bc6= 0,
x= d ad−bc.
Repeat the steps, this time eliminating the terms in x and solve for z =−c/(ad−bc).
Then solve the second set of equations in the same way.
Solution to exercise 3.2
A matrix A is symmetric if AT =A.
Since BT is a k×m matrix, BTB is k×k. (BTB)T =BT(BT)T =BTB which shows that it is symmetric.
4
Chapter 4 Vectors
Introduction
Matrices lead us to a study of vectors, which can be viewed as n×1 matrices, but which have far reaching applications viewed as elements of a Euclidean space,Rn. To understand this, we develop our geometric intuition by looking atR2 and R3, and use vectors to obtain equations of familiar geometric objects, namely lines and planes.
Aims
The aims of this chapter are to:
Define a vector and define Rn, Euclidean n-space
Define the inner product of two vectors and establish the properties satisfied by this operation
Develop geometric insight by looking at vectors in R2 and R3
Become familiar with forming lines and planes in R2 and R3 using linear combinations of vectors
Extend these ideas to lines and hyperplanes in Rn.
Essential reading
R Anthony, M. and M. Harvey. Linear Algebra: Concepts and Methods. Chapter 1, Sections 1.8–1.12
This chapter of the subject guide closely follows the second half of Chapter 1 of the textbook. You should read the corresponding sections of the textbook and work through all the activities there while working through the sections of this subject guide.
Further reading
If you would like to have another source containing the material covered in this chapter, you can look up the key concepts and definitions (as listed in the synopsis which
follows) in any elementary linear algebra textbook, such as the two listed in Chapter 1, using either the table of contents or the index.
4
Synopsis
We define a vector, what we mean by a linear combination of vectors and say what we mean by Euclideann-space, Rn. Then we define the inner product of two vectors inRn and look at its properties. We pause to establish a fundamental relationship between the vector Ax, whereA is an m×n matrix and x∈Rn, and the column vectors of A.
We then focus on developing geometric insight, beginning with R2, looking at what we mean by the length and direction of a vector, and the angle between two vectors, and extend these concepts to R3. We then study lines in R2, learning how to describe them using Cartesian equations or vector equations and how to switch from one description to the other. We then extend these ideas to vectors in R3, where the extra dimension increases the possibilities of how lines can interact. We next extend the idea of linear combinations of vectors to planes in R3 and look at vector equations and Cartesian equations of planes. We give several examples of determining the interactions of planes and of lines and planes. Finally we extend these concepts toRn, to lines and
hyperplanes.