Lecture 7:Matrix-Vector Product; Matrix of a Linear Transformation; Matrix-Matrix ProductSections 2.1, 2.2.1, 2.2.2

Key Points• The matrix-vector product Ax, where A is a m × n matrix and x is a n-

dimensional column vector, is computed by taking the dot product of each row of A with x. The result is a m-dimensional column vector.

• For a fixed matrix A, the product A x is linear in x:

A(c1x(1) + c2x(2))= c1 A x(1) + c2Ax(2)

• In other words, A acts as a linear transformation, or linear system, which maps n-dimensional vectors to m-dimensional ones.

• Every linear transformation, or linear system, Rn → Rm has a m × n matrix A associated with it. Each column of A is obtained by applying that transformation to the respective standard n-dimensional unit vector.

• If A is m × p and B is p × n, then the product A B is a m × n matrix whose (i, j)th element is the dot product of the ith row of A and the jth column of B.

Review• A m × n matrix consists of entries (or elements) aij, where i and j

are the row and column indices, respectively. The space of all real-valued m × n matrices is denoted by Rm×n .

• A column vector is a matrix consisting of one column only; a row vector is a matrix consisting of one row only. The transpose operator ·T converts row vectors to column vectors and vice versa. By default, a lower-case boldface letter such as a corresponds to a column vector. In situations where the orientation (row or column) of a vector is immaterial, we simply write

a =(a1,...,an)

which is a vector in Rn . • The sum S = A + B of two matrices of the same dimension is

obtained by adding respective entries together: sij = aij + bij

• The matrix cA, where c is a real number, has the same dimensions as A and is obtained by scaling each entry of A by c

Overview

•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)• 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 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 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 & 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 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 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 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 aikbkjk1

n

Example 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 – no dot multiplication

Matrix 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

Verify that is the inverse of 4 111 113 211 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 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 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 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 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.

Matrix-Vector Multiplication• If A is a m × n matrix and x is a n × 1 (column)

vector, then y = Ax

• is an m × 1 vector such that

In other words, the ith entry of y is the dot product of the ith row of A with x.

• We will also view the product y = Ax as a linear combination of the columns of A with coefficients given by the (respective) entries of x.

n

jjiji xay

1

Example

15

9

4

1

2

512

113

Superposition Property• A vector of the form

c1x(1) + c2x(2)

where c1 and c2 are scalars, is known as a linear combination of the vectors x(1) and x(2).

For a fixed matrix A, the product Ax is linear in x, i.e., it has the property that

A(c1x(1) + c2x(2))= c1 A x(1) + c2Ax(2)

for any vectors x(1), x(2) and scalars c1, c2. This is known as the superposition property, and is easily proved by considering the ith entry on each side:

n

jjij

n

jjij

n

jjjij xacxacxcxca

1

)2(2

1

)1(1

1

)2(2

)1(1

Linear Transformation

• An m × n matrix A represents a linear transformation of Rn to Rm . Such a linear transformation is also referred to as a linear system with n-dimensional input vector x and m-dimensional output vector y:

Example• Suppose the 2 × n matrix A and the n-

dimensional column vectors u and v are such that

and

then

4

1Au

2

5Av

8

112v)-A(u

Example• The linear transformation represented by the matrix

is such that

Thus the effect of applying A to an arbitrary vector x is to shift the entries of x up (or down) by two positions in a circular fashion. This linear transformation is an example of a permutation, and all permutations are linear

0010

0001

1000

0100

A

2

1

4

3

4

3

2

1

x

x

x

x

x

x

x

x

A

Extra Credit Activity• You are given an image whose dimensions match

those of a 36 inch (diagonal) display with an aspect ratio of 16 (horizontal) to 9 (vertical). You want to display the image on a 27 inch (diagonal) display with an aspect ratio of 4 (horizontal) to 3 (vertical) such that the image is as large as possible without distortion or cropping. Find the matrix

which accomplsihes this. (Note: I is the identity matrix.)

Iaa

a

0

0

Example• Conversely, every linear transformation A : Rn → Rm has an m × n

matrix associated with it. This can be seen by expressing an arbitrary input vector x as a linear combination of the standard unit vectors:

x = x1 e(1) + . . . + xn e(n)

• By linearity of A( · ), the output vector y = A(x) is given by:y = x1 Ae(1) + . . . + xn Ae(n)

• If we form an m × n matrix A =[aij] using A(e(1)),...,A(e(n)) as its columns (in that order), then the output vector y (above) is, in effect, a linear combination of the columns of A with coefficients x1,...,xn. In other words,

and thus y = A(x) is also given by y = Ax

n

jjiji xay

1

Example• If the linear transformation A( · ): R3 → R3 is such

that

then the matrix A of A(·) is given by

1

1-

1

1

0

0

A

4

5

1-

0

1

0

A

0

1-

3

0

0

1

A and,,

140

151

113

A

Example• Suppose now that A( · ): R2 → R2 represents the projection of

a two-dimensional vector x =(x1,x2) onto the horizontal (i.e., x1) axis. From vector geometry, we know that this is a linear transformation: the projection of a sum of (possibly scaled) vectors is the sum of their projections. We can therefore obtain the matrix A by considering the result of applying A(·) to the two unit vectors (1, 0) and (0, 1). We have

so

01

,00

00A

11A

00

01A

Example• Similarly, the rotation of a two-dimensional vector

through a fixed angle is linear: when two vectors are rotated through the same angle, their (possibly scaled) sum is also rotated through that angle. If B is the matrix representing a counterclockwise rotation by 300, then

so

Question: How were these values obtained?

2/

2/1

1,

2/1

2/

0 3

0B

31B

2/12/1

2/12/3B

Matrix-Matrix Multiplication• If A is m × p and B is p × n, then the product AB

is the m × n matrix whose (i, j)th element is the dot product of the ith row of A and the jth column of B:

The number of columns of A must be the same as the number of rows of B (equal to p in this case, and also referred to as the inner dimension in the product).

p

kkjikij xaAB

1

)(

Example

70

136

51

30

21

,101

412

AB

BA