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Notes on Mathematics - 102^1

Peeyush Chandra, A. K. Lal, V. Raghavendra, G. Santhanam

(^1) Supported by a grant from MHRD

Part I

Linear Algebra

10 CHAPTER 1. MATRICES

Example 1.1.4 The linear system of equations 2 x + 3y = 5 and 3 x + 2y = 5 can be identified with the

matrix

[

]

1.1.1 Special Matrices

Definition 1.1.5 1. A matrix in which each entry is zero is called a zero-matrix, denoted by 0. For example,

(^02) × 2 =

[

]

and (^02) × 3 =

[

]

1. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus, its order is m × m (for some m) and is represented by m only.
2. In a square matrix, A = [aij ], of order n, the entries a 11 , a 22 ,... , ann are called the diagonal entries and form the principal diagonal of A.
3. A square matrix A = [aij ] is said to be a diagonal matrix if aij = 0 for i 6 = j. In other words, the non-zero entries appear only on the principal diagonal. For example, the zero matrix (^0) n and

[

]

are a few diagonal matrices. A diagonal matrix D of order n with the diagonal entries d 1 , d 2 ,... , dn is denoted by D = diag(d 1 ,... , dn). If di = d for all i = 1, 2 ,... , n then the diagonal matrix D is called a scalar matrix.

1. A square matrix A = [aij ] with aij =

1 if i = j 0 if i 6 = j is called the identity matrix, denoted by In.

For example, I 2 =

[

]

, and I 3 =

The subscript n is suppressed in case the order is clear from the context or if no confusion arises.

1. A square matrix A = [aij ] is said to be an upper triangular matrix if aij = 0 for i > j. A square matrix A = [aij ] is said to be an lower triangular matrix if aij = 0 for i < j. A square matrix A is said to be triangular if it is an upper or a lower triangular matrix.

For example

 is an upper triangular matrix. An upper triangular matrix will be represented

by

a 11 a 12 · · · a 1 n 0 a 22 · · · a 2 n .. .

0 0 · · · ann

1.2 Operations on Matrices

Definition 1.2.1 (Transpose of a Matrix) The transpose of an m × n matrix A = [aij ] is defined as the n × m matrix B = [bij ], with bij = aji for 1 ≤ i ≤ m and 1 ≤ j ≤ n. The transpose of A is denoted by At.

1.2. OPERATIONS ON MATRICES 11

That is, by the transpose of an m × n matrix A, we mean a matrix of order n × m having the rows of A as its columns and the columns of A as its rows.

For example, if A =

[

]

then At^ =

Thus, the transpose of a row vector is a column vector and vice-versa.

Theorem 1.2.2 For any matrix A, we have (At)t^ = A.

Proof. Let A = [aij ], At^ = [bij ] and (At)t^ = [cij ]. Then, the definition of transpose gives

cij = bji = aij for all i, j

and the result follows. 

Definition 1.2.3 (Addition of Matrices) let A = [aij ] and B = [bij ] be are two m × n matrices. Then the sum A + B is defined to be the matrix C = [cij ] with cij = aij + bij.

Note that, we define the sum of two matrices only when the order of the two matrices are same.

Definition 1.2.4 (Multiplying a Scalar to a Matrix) Let A = [aij ] be an m × n matrix. Then for any element k ∈ R, we define kA = [kaij ].

For example, if A =

[

]

and k = 5, then 5A =

[

]

Theorem 1.2.5 Let A, B and C be matrices of order m × n, and let k, ℓ ∈ R. Then

1. A + B = B + A (commutativity).
2. (A + B) + C = A + (B + C) (associativity).
3. k(ℓA) = (kℓ)A.
4. (k + ℓ)A = kA + ℓA.

Proof. Part 1. Let A = [aij ] and B = [bij ]. Then

A + B = [aij ] + [bij ] = [aij + bij ] = [bij + aij ] = [bij ] + [aij ] = B + A

as real numbers commute. The reader is required to prove the other parts as all the results follow from the properties of real numbers. 

Exercise 1.2.6 1. Suppose A + B = A. Then show that B = 0.

1. Suppose A + B = 0. Then show that B = (−1)A = [−aij ].

Definition 1.2.7 (Additive Inverse) Let A be an m × n matrix.

1. Then there exists a matrix B with A + B = 0. This matrix B is called the additive inverse of A, and is denoted by −A = (−1)A.
2. Also, for the matrix (^0) m×n, A + 0 = 0 + A = A. Hence, the matrix (^0) m×n is called the additive identity.

1.3. SOME MORE SPECIAL MATRICES 13

Therefore,

( A(BC)

ij =

∑^ n k=

aik

BC

kj =

∑^ n k=

aik

( ∑p ℓ=

bkℓcℓj

∑^ n k=

∑^ p ℓ=

aik

bkℓcℓj

∑^ n k=

∑^ p ℓ=

aikbkℓ

cℓj

∑^ p

ℓ=

( ∑n k=

aikbkℓ

cℓj =

∑^ t

ℓ=

AB

iℓcℓj

(AB)C

ij. Part 5. For all j = 1, 2 ,... , n, we have

(DA)ij =

∑^ n k=

dik akj = diaij

as dik = 0 whenever i 6 = k. Hence, the required result follows. The reader is required to prove the other parts. 

Exercise 1.2.12 1. Let A and B be two matrices. If the matrix addition A + B is defined, then prove that (A + B)t^ = At^ + Bt. Also, if the matrix product AB is defined then prove that (AB)t^ = BtAt.

1. Let A = [a 1 , a 2 ,... , an] and B =

b 1 b 2 .. . bn

. Compute the matrix products AB and BA.

1. Let n be a positive integer. Compute An^ for the following matrices: [ 1 1 0 1

]

Can you guess a formula for An^ and prove it by induction?

1. Find examples for the following statements.

(a) Suppose that the matrix product AB is defined. Then the product BA need not be defined. (b) Suppose that the matrix products AB and BA are defined. Then the matrices AB and BA can have different orders. (c) Suppose that the matrices A and B are square matrices of order n. Then AB and BA may or may not be equal.

1.3 Some More Special Matrices

Definition 1.3.1 1. A matrix A over R is called symmetric if At^ = A and skew-symmetric if At^ = −A.

1. A matrix A is said to be orthogonal if AAt^ = AtA = I.

Example 1.3.2 1. Let A =

 and^ B^ =

.^ Then^ A^ is a symmetric matrix and

B is a skew-symmetric matrix.

14 CHAPTER 1. MATRICES

1. Let A =

√^1 3 √^1 3 √^1 1 3 √ 2 − √^12 √^1 6 √^1 6 −^ √^2 6

.^ Then^ A^ is an orthogonal matrix.

1. Let A = [aij ] be an n × n matrix with aij =

1 if i = j + 1 0 otherwise

. Then An^ = 0 and Aℓ^6 = 0 for 1 ≤ ℓ ≤

n − 1. The matrices A for which a positive integer k exists such that Ak^ = 0 are called nilpotent matrices. The least positive integer k for which Ak^ = 0 is called the order of nilpotency.

1. Let A =

[

]

. Then A^2 = A. The matrices that satisfy the condition that A^2 = A are called idempotent matrices.

Exercise 1.3.3 1. Show that for any square matrix A, S = 12 (A + At) is symmetric, T = 12 (A − At) is skew-symmetric, and A = S + T.

1. Show that the product of two lower triangular matrices is a lower triangular matrix. A similar statement holds for upper triangular matrices.
2. Let A and B be symmetric matrices. Show that AB is symmetric if and only if AB = BA.
3. Show that the diagonal entries of a skew-symmetric matrix are zero.
4. Let A, B be skew-symmetric matrices with AB = BA. Is the matrix AB symmetric or skew-symmetric?
5. Let A be a symmetric matrix of order n with A^2 = 0. Is it necessarily true that A = 0?
6. Let A be a nilpotent matrix. Show that there exists a matrix B such that B(I + A) = I = (I + A)B.

1.3.1 Submatrix of a Matrix

Definition 1.3.4 A matrix obtained by deleting some of the rows and/or columns of a matrix is said to be a submatrix of the given matrix.

For example, if A =

[

]

, a few submatrices of A are

[1], [2],

[

]

, [1 5],

[

]

, A.

But the matrices

[

]

and

[

]

are not submatrices of A. (The reader is advised to give reasons.)

Miscellaneous Exercises

Exercise 1.3.5 1. Complete the proofs of Theorems 1.2.5 and 1.2.11.

1. Let x =

[

x 1 x 2

]

, y =

[

y 1 y 2

]

, A =

[

]

and B =

[

cos θ − sin θ sin θ cos θ

]

. Geometrically interpret y = Ax and y = Bx.

1. Consider the two coordinate transformations x 1 = a 11 y 1 + a 12 y 2 x 2 = a 21 y 1 + a 22 y 2

and y 1 = b 11 z 1 + b 12 z 2 y 2 = b 21 z 1 + b 22 z 2

16 CHAPTER 1. MATRICES

Theorem 1.3.6 is very useful due to the following reasons:

1. The order of the matrices P, Q, H and K are smaller than that of A or B.
2. It may be possible to block the matrix in such a way that a few blocks are either identity matrices or zero matrices. In this case, it may be easy to handle the matrix product using the block form.
3. Or when we want to prove results using induction, then we may assume the result for r × r submatrices and then look for (r + 1) × (r + 1) submatrices, etc.

For example, if A =

[

]

and B =

a b c d e f

 , Then

AB =

[

] [

a b c d

]

[

]

[e f ] =

[

a + 2c b + 2d 2 a + 5c 2 b + 5d

]

If A =

 ,^ then^ A^ can be decomposed as follows:

A =

 ,^ or^ A^ =

 ,^ or

A =

 and so on.

Suppose A =

m 1 m 2 n 1 n 2

[

P Q

R S

]

and B =

s 1 s 2 r 1 r 2

[

E F

G H

]

. Then the matrices P, Q, R, S and

E, F, G, H, are called the blocks of the matrices A and B, respectively. Even if A + B is defined, the orders of P and E may not be same and hence, we may not be able

to add A and B in the block form. But, if A + B and P + E is defined then A + B =

[

P + E Q + F

R + G S + H

]

Similarly, if the product AB is defined, the product P E need not be defined. Therefore, we can talk of matrix product AB as block product of matrices, if both the products AB and P E are defined. And

in this case, we have AB =

[

P E + QG P F + QH

RE + SG RF + SH

]

That is, once a partition of A is fixed, the partition of B has to be properly chosen for purposes of block addition or multiplication.

Exercise 1.3.7 1. Compute the matrix product AB using the block matrix multiplication for the matrices

A =

  

1 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1

   and^ B^ =

  

1 2 2 1 1 1 2 1 1 1 1 1 − 1 1 − 1 1

  .

1. Let A =

[

P Q

R S

]

. If P, Q, R and S are symmetric, what can you say about A? Are P, Q, R and S symmetric, when A is symmetric?

1.4. MATRICES OVER COMPLEX NUMBERS 17

1. Let A = [aij ] and B = [bij ] be two matrices. Suppose a 1 , a 2 ,... , an are the rows of A and b 1 , b 2 ,... , bp are the columns of B. If the product AB is defined, then show that

AB = [Ab 1 , Ab 2 ,... , Abp] =

a 1 B a 2 B .. . anB

[That is, left multiplication by A, is same as multiplying each column of B by A. Similarly, right multiplication by B, is same as multiplying each row of A by B.]

1.4 Matrices over Complex Numbers

Here the entries of the matrix are complex numbers. All the definitions still hold. One just needs to look at the following additional definitions.

Definition 1.4.1 (Conjugate Transpose of a Matrix) 1. Let A be an m×n matrix over C. If A = [aij ] then the Conjugate of A, denoted by A, is the matrix B = [bij ] with bij = aij.

For example, Let A =

[

1 4 + 3i i 0 1 i − 2

]

. Then

A =

[

1 4 − 3 i −i 0 1 −i − 2

]

1. Let A be an m × n matrix over C. If A = [aij ] then the Conjugate Transpose of A, denoted by A∗, is the matrix B = [bij ] with bij = aji.

For example, Let A =

[

1 4 + 3i i 0 1 i − 2

]

. Then

A∗^ =

4 − 3 i 1 −i −i − 2

1. A square matrix A over C is called Hermitian if A∗^ = A.
2. A square matrix A over C is called skew-Hermitian if A∗^ = −A.
3. A square matrix A over C is called unitary if A∗A = AA∗^ = I.
4. A square matrix A over C is called Normal if AA∗^ = A∗A.

Remark 1.4.2 If A = [aij ] with aij ∈ R, then A∗^ = At.

Exercise 1.4.3 1. Give examples of Hermitian, skew-Hermitian and unitary matrices that have entries with non-zero imaginary parts.

1. Restate the results on transpose in terms of conjugate transpose.
2. Show that for any square matrix A, S = A+A ∗ 2 is Hermitian,^ T^ =^ A−A∗ 2 is skew-Hermitian, and A = S + T.
3. Show that if A is a complex triangular matrix and AA∗^ = A∗A then A is a diagonal matrix.

Chapter 2

Linear System of Equations

2.1 Introduction

Let us look at some examples of linear systems.

1. Suppose a, b ∈ R. Consider the system ax = b.

(a) If a 6 = 0 then the system has a unique solution x = b a. (b) If a = 0 and i. b 6 = 0 then the system has no solution. ii. b = 0 then the system has infinite number of solutions, namely all x ∈ R.

1. We now consider a system with 2 equations in 2 unknowns. Consider the equation ax + by = c. If one of the coefficients, a or b is non-zero, then this linear equation represents a line in R^2. Thus for the system

a 1 x + b 1 y = c 1 and a 2 x + b 2 y = c 2 ,

the set of solutions is given by the points of intersection of the two lines. There are three cases to be considered. Each case is illustrated by an example.

(a) Unique Solution x + 2y = 1 and x + 3y = 1. The unique solution is (x, y)t^ = (1, 0)t. Observe that in this case, a 1 b 2 − a 2 b 1 6 = 0. (b) Infinite Number of Solutions x + 2y = 1 and 2x + 4y = 2. The set of solutions is (x, y)t^ = (1 − 2 y, y)t^ = (1, 0)t^ + y(− 2 , 1)t with y arbitrary. In other words, both the equations represent the same line. Observe that in this case, a 1 b 2 − a 2 b 1 = 0, a 1 c 2 − a 2 c 1 = 0 and b 1 c 2 − b 2 c 1 = 0. (c) No Solution x + 2y = 1 and 2x + 4y = 3. The equations represent a pair of parallel lines and hence there is no point of intersection. Observe that in this case, a 1 b 2 − a 2 b 1 = 0 but a 1 c 2 − a 2 c 1 6 = 0.

1. As a last example, consider 3 equations in 3 unknowns. A linear equation ax + by + cz = d represent a plane in R^3 provided (a, b, c) 6 = (0, 0 , 0). As in the case of 2 equations in 2 unknowns, we have to look at the points of intersection of the given three planes. Here again, we have three cases. The three cases are illustrated by examples.

20 CHAPTER 2. LINEAR SYSTEM OF EQUATIONS

(a) Unique Solution Consider the system x+ y + z = 3, x+ 4y + 2z = 7 and 4x+ 10y − z = 13. The unique solution to this system is (x, y, z)t^ = (1, 1 , 1)t; i.e. the three planes intersect at a point. (b) Infinite Number of Solutions Consider the system x + y + z = 3, x + 2y + 2z = 5 and 3x + 4y + 4z = 11. The set of solutions to this system is (x, y, z)t^ = (1, 2 − z, z)t^ = (1, 2 , 0)t^ + z(0, − 1 , 1)t, with z arbitrary: the three planes intersect on a line. (c) No Solution The system x + y + z = 3, x + 2y + 2z = 5 and 3x + 4y + 4z = 13 has no solution. In this case, we get three parallel lines as intersections of the above planes taken two at a time. The readers are advised to supply the proof.

2.2 Definition and a Solution Method

Definition 2.2.1 (Linear System) A linear system of m equations in n unknowns x 1 , x 2 ,... , xn is a set of equations of the form

a 11 x 1 + a 12 x 2 + · · · + a 1 nxn = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 nxn = b 2 .. .

am 1 x 1 + am 2 x 2 + · · · + amnxn = bm

where for 1 ≤ i ≤ n, and 1 ≤ j ≤ m; aij , bi ∈ R. Linear System (2.2.1) is called homogeneous if b 1 = 0 = b 2 = · · · = bm and non-homogeneous otherwise.

We rewrite the above equations in the form Ax = b, where

A =

a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n .. .

am 1 am 2 · · · amn

, x =

x 1 x 2 .. . xn

, and b =

b 1 b 2 .. . bm

The matrix A is called the coefficient matrix and the block matrix [A b] , is the augmented matrix of the linear system (2.2.1).

Remark 2.2.2 Observe that the ith row of the augmented matrix [A b] represents the ith equation and the jth column of the coefficient matrix A corresponds to coefficients of the jth variable xj. That is, for 1 ≤ i ≤ m and 1 ≤ j ≤ n, the entry aij of the coefficient matrix A corresponds to the ith equation and jth variable xj ..

For a system of linear equations Ax = b, the system Ax = 0 is called the associated homogeneous system.

Definition 2.2.3 (Solution of a Linear System) A solution of the linear system Ax = b is a column vector y with entries y 1 , y 2 ,... , yn such that the linear system (2.2.1) is satisfied by substituting yi in place of xi.

That is, if yt^ = [y 1 , y 2 ,... , yn] then Ay = b holds. Note: The zero n-tuple x = 0 is always a solution of the system Ax = 0 , and is called the trivial solution. A non-zero n-tuple x, if it satisfies Ax = 0 , is called a non-trivial solution.