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Chapter 2. Matrices Gareth Williams J & B, 滄海書局. 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices 2.2 Properties of Matrix Operations 2.3 Symmetric Matrices 2.4 Inverse of a Matrix and Cryptography 2.5 Leontief Input-Output Model

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Chapter 2

Chapter 2

Matrices

Gareth Williams

J & B, 滄海書局


Chapter 2 4814834

  • 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices

  • 2.2 Properties of Matrix Operations

  • 2.3 Symmetric Matrices

  • 2.4 Inverse of a Matrix and Cryptography

  • 2.5 Leontief Input-Output Model

  • 2.6 Markov Chains, Population Movements, and Genetics

  • 2.7 A Communication Model and Graph Relationships in Sociology


2 1 addition scalar multiplication and multiplication of matrices

2.1 Addition, Scalar Multiplication, and Multiplication of Matrices

Example 1

Determine the elements a12, a23, and a34 for the following matrix A. Given the main diagonal of A.

Solution


Chapter 2 4814834

Equality of Matrices

Definition

The matrices are equal if they are of the same size and if their corresponding elements are equal. Thus A = B if aij = bij.


Addition of matrices

Addition of Matrices

Definition

Let A and B be matrices of the same size.

Their sumA + B is the matrix obtained by adding together the corresponding elements of A and B.

The matrix A + B will be of the same size as A and B.

If A and B are not of the same size, they cannot be added, and we say that the sum does not exist.

Thus if C = A + B, then cij = aij+ bij.


Example 2

Determine A + B, and A + C, if the sum exist.

Example 2

Solution


Scalar multiplication of matrices

Scalar Multiplication of Matrices

Definition

Let A be a matrix and c be a scalar. The scalar multiple of A by c, denoted cA, is the matrix obtained by multiplying every element of A by c. The matrix cA will be the same size as A.


Example 3

Example 3

Solution


Negation and subtraction

Negation and Subtraction

Definition

We now define subtraction of matrices in such a way that makes it compatible with addition, scalar multiplication, and negative. Let

A – B = A + (–1)B


Negation and subtraction1

Negation and Subtraction

Example

Solution


Multiplication of matrices

Multiplication of Matrices

Definition

Let the number of columns in a matrix A be the same as the number of rows in a matrix B. The product AB then exists.

The element in row i and column j of AB is obtained by multiplying the corresponding elements of row i of A and column j of B and adding the products.

If the number of columns in A does not equal the number of row B, we say that the product does not exist.


Example 4

Example 4


Example 5

Example 5


Example 6

Let C = AB,

Example 6

Solution


Size of a product matrix

A

m r

Size of a Product Matrix

If A is an m r matrix and B is an r n matrix, then AB will be an m n matrix.

B

= AB

r n

m n


Example 7

Example 7

If A is a 5  6 matrix and B is an 6 7 matrix. What is the size of the product of AB?

Solution


Chapter 2 4814834

Definition

A zero matrix is a matrix in which all the elements are zeros.

A diagonal matrix is a square matrix in which all the elements not on the main diagonal are zeros.

An identity matrix is a diagonal matrix in which every diagonal element is 1.


Theorem 2 1

Theorem 2.1

Let A be m n matrix and 0mn be the zero m n matrix. Let B be an n n square matrix. 0n and In be the zero and identity n n matrices. Then

A + 0mn = 0mn + A = A

B0n = 0nB = 0n

BIn = InB = B

Proof


Theorem 2 11

Theorem 2.1


Example 8

Example 8

Solution


Matrix notation and systems of equations

Matrix Notation and Systems of Equations

A system of m linear equations in n variables as follows

Let

We can write the system of equations in the matrix form

AX = B


Matrix multiplication in terms of columns

Matrix Multiplication in Terms of Columns

(a)Let A be an m×n matrix and B be an n×r matrix. Let the column of B be the matrices b1, b2, …, br, . Write B=[b1, b2, …, br]. Thus

(b)Let A be an m×n matrix and B be an n×r matrix. Let the row of A be the matrices a1, a2, …, am, .


Partitioning of matrices

(1)

(2)

Partitioning of Matrices


Example 9

Example 9


Example 91

Example 9


2 2 properties of matrix operations

2.2 Properties of Matrix Operations

Theorem 2.2

Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed.

Properties of Matrix Addition and scalar Multiplication

1. A + B = B + ACommutative property of addition

2. A + (B + C) = (A + B) + CAssociative property of addition

3. A + 0 = 0 + A = A(where0 is the appropriate zero matrix)

4. c(A + B) = cA + cBDistributive property of addition

5. (a + b)C = aC + bCDistributive property of addition

6. (ab)C = a(bC)


Theorem 2 2

Theorem 2.2

Proof


Chapter 2 4814834

Theorem 2.2

Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed.

Properties of Matrix Multiplication

1. A(BC) = (AB)CAssociative property of multiplication

2. A(B + C) = AB + AC Distributive property of multiplication

3. (A + B)C = AC + BCDistributive property of multiplication

4. AIn = InA = A (whereIn is the appropriate identity matrix)

5. c(AB) = (cA)B = A(cB)

Note: AB BA in general.Multiplication of matrices is not commutative.


Theorem 2 21

Theorem 2.2

Proof


Example 1

Example 1

Solution


Example 21

Example 2

Solution


Example 31

Example 3

By Example 2, compare the number of multiplications involved in the two ways (AB)C and A(BC) of computing the product ABC.

Solution


Chapter 2 4814834

Note

If A is an m r matrix and B is r n matrix, the number of scalar multiplications involved in computing the product AB is mrn.

If A is an m r matrix and B is r n matrix, the number of additions involved in computing the product AB is m(r-1)n.


Chapter 2 4814834

Caution

  • In algebra we know that the following cancellation laws apply.

  • If ab = ac and a  0 then b = c.

  • If pq = 0 then p = 0 or q = 0.

  • However the corresponding results are not true for matrices.

  • AB = AC does not imply that B = C.

  • PQ = 0 does not imply that P = 0 or Q = 0.


Powers of matrices

Powers of Matrices

Theorem 2.3

If A is an n n square matrix and r and s are nonnegative integers, then

1. ArAs = Ar+s.

2. (Ar)s = Ars.

3. A0 = In (by definition)

Proof


Example 41

Example 4

Solution


Example 51

Example 5

Simplify the following matrix expression.

Solution


2 3 symmetric matrices

2.3 Symmetric Matrices

Definition

The transpose of a matrix A, denoted At, is the matrix whose columns are the rows of the given matrix A.


Example 11

Example 1

Solution


Theorem 2 4 properties of transpose

Theorem 2.4 Properties of Transpose

Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed.

1. (A + B)t = At + BtTranspose of a sum

2. (cA)t = cAtTranspose of a scalar multiple

3. (AB)t = BtAtTranspose of a product

4. (At)t = A

Proof


Chapter 2 4814834

match

match

Definition

A symmetric matrix is a matrix that is equal to its transpose.


Example 32

Example 3

Let A and B be symmetric matrices of the same size. Prove that the product AB is symmetric if and only if AB = BA.

*We have to show (a) if AB is symmetric, then AB = BA, and the converse, (b) AB = BA, then AB is symmetric.

Proof

(a) Let AB be symmetric, then

AB= by definition of symmetric matrix

= by the transpose of a product

= since A and B are symmetric matrices

(b) Let AB = BA, then

(AB)t = by the transpose of a product

= since A and B are symmetric matrices

=


Chapter 2 4814834

Definition

Let A be a square matrix. The trace A, denoted tr(A) is the sum of the diagonal elements of A. Thus if A is an n n matrix.

tr(A) = a11 + a22 + … + ann


Chapter 2 4814834

Example 4

Determine the trace of the matrix

Solution


Theorem 2 5 properties of trace

Theorem 2.5 Properties of Trace

Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed.

1. tr(A + B) = tr(A) + tr(B)

2. tr(AB) = tr(BA)

3. tr(cA) = trc(A)

4. tr(At) = tr(A)

Proof


Theorem 2 5

Theorem 2.5


Matrices with complex elements

Equality:

Addition:

Subtraction:

Multiplication:

Matrices with Complex Elements

The element of a matrix may be complex numbers. A complex number is of the form

z = a + bi

Where a and b are real numbers and a is called the real part and b the imaginary part of z.

Let be the complex numbers. The rules of arithmetic for complex numbers are as follows.


Conjugate of a complex number

Conjugate of a Complex Number

Definition

The conjugate of a complex number z = a+bi is defined and written = a-bi


Example 52

Consider the complex numbers

Compute

Example 5

Solution


Example 61

Compute A + B, 2A, and AB.

Example 6

Solution


Conjugate transpose of a matrix

Conjugate Transpose of a Matrix

Definition

(1) The conjugate of a matrix A is denoted and is obtained by taking the conjugate of each element of the matrix.

(2) The conjugate transpose of A is written and denoted by

(3) A square matrix C is said be Hermitian if C = C*.


Example

Example

Compute C*. Is C Hermitian?


Theorem 2 6 properties of conjugate transpose

Theorem 2.6 Properties of Conjugate Transpose

Let A and B be matrices with complex elements and let z be a complex number.

1. (A + B)* = A* + B* Conjugate transpose of a sum

2. (zA)* = zA* Conjugate transpose of a scalar multiple

3. (AB)* = B*A* Conjugate transpose of a product

4. (A*)* = A


2 4 the inverse of a matrix

2.4 The Inverse of a Matrix

Definition

Let A be an n n matrix. If a matrix B can be found such that AB = BA = In, then A is said to be invertible and B is called inverse of A. If such a matrix B does not exist, then A has no inverse.


Example 12

Example 1

Prove that the matrix has the inverse

Proof


Theorem 2 7

Theorem 2.7

The inverse of an invertible matrix is unique.

Proof


Notation

Notation

AA-1=A-1A=In

A-k=A-1A-1…A-1

k times


Determine the inverse of a matrix

Determine the Inverse of a Matrix

AA-1=In

A-1=X=[x1, x2, …,xn] and In=[e1, e2, …,en]

Where

AX=A[x1, x2, …,xn]= [Ax1, Ax2, …,Axn] =[e1, e2, …,en]

Thus Ax1=e1, Ax2=e2, … ,Axn=en.

[A : In] →……. →[In : A-1]


Gauss jordan elimination for finding the inverse of a matrix

Gauss-Jordan Elimination for Finding the Inverse of a Matrix

Let A be an n n matrix.

1. Adjoin the identity n n matrix In to A to form the matrix [A : In].

2. Compute the reduced echelon form of [A : In].

If the reduced echelon form is of the type [In : B], then B is the inverse of A.

If the reduced echelon form is not of the type [In : B], in that the first n n submatrix is not In, then A has no inverse.


Theorem

Theorem

An n n matrix A is invertible if and only if it is row equivalent to In.


Example 22

Determine the inverse of the matrix

Example 2

Solution


Example 33

Example 3

Determine the inverse of the following matrix, if it exist.

Solution


Properties of matrix inverse

Properties of Matrix Inverse

Let A and B be inverse matrices and c a nonzero scalar, Then

Proof


Properties of matrix inverse1

Properties of Matrix Inverse


Example 42

Example 4

Solution


Theorem 2 8

Let AX = B be a system of n linear equation in n variable. If A–1 exists, the solution is unique and is given by X = A–1B.

Theorem 2.8

Proof


Example 53

Solve the system of equation

Example 5

Solution


Elementary matrices

Elementary Matrices

Definition

An elementary matrix is one that can be obtained from the identity matrix In through a single elementary row operation.


Elementary matrices1

Elementary Matrices


Illustrated example

Illustrated Example

Let

Interchange row 2 and row 3 of A:

Multiple row 2 by k:

Add k row 1 to row 2:


Theorem1

Theorem

Each elementary matrix is square and invertible.

Interchange row 2 and row 3 of A:

Multiple row 2 by a:

Add c row 1 to row 2:


Theorem2

Theorem

If A and B are row equivalent matrices and A is invertible, then B is invertible.

Proof


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