ES 240: Scientific and Engineering Computation. Chapter 8 Chapter 8: Linear Algebraic Equations and Matrices Uchechukwu Ofoegbu Temple University.

ES 240: Scientific and Engineering Computation. Chapter 8

Chapter 8: Linear Algebraic Equations and Matrices

Uchechukwu Ofoegbu

Temple University


OverviewOverview

Matrix:– rectangular array of elements represented by a single

symbol (e.g. [A]).

Element– An individual entry of a matrix

– example: a23 – arow column


Overview (cont)Overview (cont)

A horizontal set of elements is called a row and a vertical set of elements is called a column.

The first subscript of an element indicates the row while the second indicates the column.

The size of a matrix is given as m rows by n columns, or simply m by n (or m x n).

1 x n matrices are row vectors.

m x 1 matrices are column vectors.


Special MatricesSpecial Matrices

Matrices where m=n are called square matrices. There are a number of special forms of square matrices:

Symmetric

A 5 1 2

1 3 7

2 7 8

Diagonal

A a11

a22

a33

Identity

A 1

1

1

Upper Triangular

A a11 a12 a13

a22 a23

a33

Lower Triangular

A a11

a21 a22

a31 a32 a33

Banded

A

a11 a12

a21 a22 a23

a32 a33 a34

a43 a44


Matrix OperationsMatrix Operations

Equal Matrices– Two matrices are considered equal if and only if every element in the first

matrix is equal to every corresponding element in the second. – Both matrices must be the same size.

Matrix addition and subtraction– performed by adding or subtracting the corresponding elements. – Matrices must be the same size.


Example Addition & SubtractionExample Addition & Subtraction

2 1 3 1 4 7

4 0 5 8 3 2

1 3 10

12 3 3

1 82 1 3

4 34 0 5

7 2

2 1 3 1 4 7

4 0 5 8 3 2

3 5 4

4 3 7

is not defined.


Matrix MultiplicationMatrix Multiplication

Scalar matrix multiplication is performed by multiplying each element by the same scalar.

If A is a row matrix and B is a column matrix, then we can form the product AB provided that the two matrices have the same length.

The product AB is a 1x1 matrix obtained by multiplying corresponding entries of A and B and then forming the sum.

1

21 2 1 1 2 2n n n

n

b

ba a a a b a b a b

b


Example Multiplying Row to ColumnExample Multiplying Row to Column

3

2 1 3 2

5

2 3 1 2 3 5 7

3

4 0 2 1 2

5

is not defined.


Matrix MultiplicationMatrix Multiplication

If A is an mxn matrix and B is an nxq matrix, then we can form the product AB.

The product AB is an mxq matrix whose entries are obtained by multiplying the rows of A by the columns of B.

The entry in the ith row and jth column of the product AB is formed by multiplying the ith row of A and jth column of B.

c ij aikbkj

k1

n


Example Matrix MultiplicationExample Matrix Multiplication

7 12 -5

-19 0 2

3 2 02 1 3

2 1 23 0 2

5 3 1

is not defined.

3 2 02 1 3

2 1 23 0 2

5 3 1

Matlab command: A*B Matlab command: A*B – no dot multiplication– no dot multiplication


Matrix Inverse and TransposeMatrix Inverse and Transpose

The inverse of a square matrix A, denoted by A-1, is a square matrix with the property

A-1A = AA-1 = I,where I is an identity matrix of the same size. – Matlab command: inv(A), A^-1

The transpose of a matrix involves transforming its rows into columns and its columns into rows.

– (aij)T=aji– Matlab command: a’ or transpose(a)


Example Example

Verify that is the inverse of 4 111 11

3 211 11

2 1.

3 4

4 1 2 1 1 011 113 3 4 0 1211 11

4 12 1 1 011 1133 4 0 1211 11

checks

checks


Representing Linear AlgebraRepresenting Linear Algebra

Matrices provide a concise notation for representing and solving simultaneous linear equations:

a11x1 a12x2 a13x3 b1

a21x1 a22x2 a23x3 b2

a31x1 a32x2 a33x3 b3

a11 a12 a13

a21 a22 a23

a31 a32 a33

x1

x2

x3

b1

b2

b3

[A]{x} {b}


Solving a Matrix EquationSolving a Matrix Equation

Solving a Matrix Equation – If the matrix A has an inverse, then the solution of the matrix equation

AX = B is given by X = A-1B.


Example Solving a Matrix EquationExample Solving a Matrix Equation

Use a matrix equation to solve 2 4 2

3 7 7.

x y

x y

The matrix form of the equation is

2 4 2.

3 7 7

x

y

1 7 22 4 2 2 72

3 7 7 3 7 412

x

y


Solving With MATLABSolving With MATLAB

MATLAB provides two direct ways to solve systems of linear algebraic equations [A]{x}={b}:– Left-divisionx = A\b

– Matrix inversionx = inv(A)*b

Disadvantages of the matrix inverse method:– less efficient than left-division – only works for square, non-singular systems.


LabLab

Ex 8.9 (see pgs 193/194)

ES 240: Scientific and Engineering Computation. Chapter 8 Chapter 8: Linear Algebraic Equations and Matrices Uchechukwu Ofoegbu Temple University.

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