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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


  • 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


Equality of Matrices 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 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 MatricesA + B, and A + C, if the sum exist.

Example 2

Solution


Scalar multiplication of matrices
Scalar Multiplication of Matrices 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 Matrices

Solution


Negation and subtraction
Negation and Subtraction Matrices

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 Matrices

Example

Solution


Multiplication of matrices
Multiplication of Matrices 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 Matrices


Example 5
Example 5 Matrices


Example 6

Let MatricesC = AB,

Example 6

Solution


Size of a product matrix

A Matrices

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 Matrices

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

Solution


Definition Matrices

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 Matrices

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 Matrices


Example 8
Example 8 Matrices

Solution


Matrix notation and systems of equations
Matrix Notation and Systems of Equations Matrices

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 Matrices

(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) Matrices

(2)

Partitioning of Matrices


Example 9
Example 9 Matrices


Example 91
Example 9 Matrices


2 2 properties of matrix operations
2.2 Properties of Matrix Operations Matrices

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 + A Commutative property of addition

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

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

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

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

6. (ab)C = a(bC)


Theorem 2 2
Theorem 2.2 Matrices

Proof


Theorem 2.2 Matrices

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)C Associative 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 Matrices

Proof


Example 1
Example 1 Matrices

Solution


Example 21
Example 2 Matrices

Solution


Example 31
Example 3 Matrices

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

Solution


Note Matrices

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.


Caution Matrices

  • 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 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 Matrices

Solution


Example 51
Example 5 Matrices

Simplify the following matrix expression.

Solution


2 3 symmetric matrices
2.3 Symmetric Matrices 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 Matrices

Solution


Theorem 2 4 properties of transpose
Theorem 2.4 Properties of Transpose Matrices

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 + Bt Transpose of a sum

2. (cA)t = cAt Transpose of a scalar multiple

3. (AB)t = BtAt Transpose of a product

4. (At)t = A

Proof


match Matrices

match

Definition

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


Example 32
Example 3 Matrices

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

=


Definition Matrices

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


Example 4 Matrices

Determine the trace of the matrix

Solution


Theorem 2 5 properties of trace
Theorem 2.5 Properties of Trace Matrices

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


Matrices with complex elements

Equality: Matrices

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 Matrices

Definition

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


Example 52

Consider the complex numbers Matrices

Compute

Example 5

Solution


Example 61

Compute MatricesA + B, 2A, and AB.

Example 6

Solution


Conjugate transpose of a matrix
Conjugate Transpose of a Matrix Matrices

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 Matrices

Compute C*. Is C Hermitian?


Theorem 2 6 properties of conjugate transpose
Theorem 2.6 MatricesProperties 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 Matrices

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 Matrices

Prove that the matrix has the inverse

Proof


Theorem 2 7
Theorem 2.7 Matrices

The inverse of an invertible matrix is unique.

Proof


Notation
Notation Matrices

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 Matrices

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 Matrices

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 Matrices

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 Matrices

Example 2

Solution


Example 33
Example 3 Matrices

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

Solution


Properties of matrix inverse
Properties of Matrix Inverse Matrices

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

Proof



Example 42
Example 4 Matrices

Solution


Theorem 2 8

Let MatricesAX = 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 Matrices

Example 5

Solution


Elementary matrices
Elementary Matrices Matrices

Definition

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



Illustrated example
Illustrated Example Matrices

Let

Interchange row 2 and row 3 of A:

Multiple row 2 by k:

Add k row 1 to row 2:


Theorem1
Theorem Matrices

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 Matrices

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

Proof


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