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2

2.1

2012 Pearson Education, Inc.

Matrix Algebra

MATRIX OPERATIONS

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Slide 2.1- 2 2012 Pearson Education, Inc.

MATRIX OPERATIONS

IfAis an matrixthat is, a matrix with mrowsand ncolumnsthen the scalar entry in the ith row

andjth column ofAis denoted by aijand is called the

(i,j)-entry ofA. See the figure below.

Each column ofAis a list of mreal numbers, whichidentifies a vector in .

m n

m

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MATRIX OPERATIONS

The columns are denoted by a1, , an, and the matrixAis written as

.

The number aijis the ith entry (from the top) of thejth

column vector aj.

The diagonal entriesin an matrix are

a11, a22, a33, , and they form the main diagonalofA.

A diagonal matrixis a sequence matrix whose

nondiagonal entries are zero.

An example is the identity matrix,In.

1 2a a a

nA

m n ij

A a

n m

n n

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SUMS AND SCALAR MULTIPLES

An matrix whose entries are all zero is a zeromatrixand is written as 0.

The two matrices are equalif they have the same size

(i.e., the same number of rows and the same numberof columns) and if their corresponding columns areequal, which amounts to saying that theircorresponding entries are equal.

IfAandBare matrices, then the sum isthe matrix whose columns are the sums of thecorresponding columns inAandB.

m n

m n A Bm n

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Since vector addition of the columns is done

entrywise, each entry in is the sum of thecorresponding entries inAandB.

The sum is defined only whenAandBare the

same size.

Example 1:Let

and . Find and .

A B

A B

4 0 5 1 1 1, ,

1 3 2 3 5 7A B

2 3

0 1C

A B A C

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7/21Slide 2.1- 7 2012 Pearson Education, Inc.


b. c.

d.

e.

f.

Each quantity in Theorem 1 is verified by showingthat the matrix on the left side has the same size as

the matrix on the right and that correspondingcolumns are equal.

( ) ( )A B C A B C 0A A

( )r A B rA rB ( )r s A rA sA

( ) ( )r sA rs A

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MATRIX MULTIPLICATION

When a matrixBmultiplies a vector x, it transforms x

into the vectorBx.

If this vector is then multiplied in turn by a matrixA,

the resulting vector isA(Bx). See the Fig. below.

ThusA (Bx) is produced from x by a composition of

mappingsthe linear transformations.

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Our goal is to represent this composite mapping as

multiplication by a single matrix, denoted byAB, so

that . See the figure below.

IfAis ,Bis , and xis in , denote the

columns ofBby b1, , bpand the entries in xby

x1, , xp.

( x)=(AB)xA B

m n n p p

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Then

By the linearity of multiplication byA,

The vectorA (Bx) is a linear combination of thevectorsAb1, ,Abp, using the entries in xas weights.

In matrix notation, this linear combination is writtenas

.

1 1x b ... bp pB x x

1 1

1 1

( x) ( b ) ... ( b )

b ... bp p

p p

A B A x A x

x A x A

1 2( x) b b b x

pA B A A A

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Thus multiplication by

transforms xintoA (Bx).

Definition:IfAis an matrix, and ifBis an

matrix with columns b1, , bp, then the productABisthe matrix whose columns areAb1, ,Abp.

That is,

Multiplication of matrices corresponds to composition

of linear transformations.

1 2

b b bp

A A A

m n n p

m p

1 2 1 2b b b b b b

ppAB A A A A

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Example 2:ComputeAB, where and

.

Solution:Write , and compute:

2 31 5

A 4 3 9

1 2 3

B

1 2 3b b bB

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Each column ofABis a linear combination of thecolumns ofAusing weights from the correspondingcolumn ofB.

Rowcolumn rule for computingAB

If a productABis defined, then the entry in row iandcolumnjofABis the sum of the products ofcorresponding entries from row iofAand columnjofB.

If (AB)ijdenotes the (i,j)-entry inAB, and ifAis an

matrix, then

.m n

1 1( ) ...

ij i j in njAB a b a b

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PROPERTIES OF MATRIX MULTIPLICATION

Theorem 2:LetAbe an matrix, and letBand

Chave sizes for which the indicated sums and

products are defined.

a. (associative law of

multiplication)b. (left distributive law)

c. (right distributive law)

d. for any scalar re. (identity for matrix

multiplication)

m n

( ) ( )A BC AB C

( )A B C AB AC

( )B C A BA CA

( ) ( ) ( )r AB rA B A rB m n

I A A AI

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Proof:Property (a) follows from the fact that matrix

multiplication corresponds to composition of linear

transformations (which are functions), and it is

known that the composition of functions is

associative.

Let

By the definition of matrix multiplication,

1c c

pC

1

1

c c

( ) ( c ) ( c )

p

p

BC B B

A BC A B A B

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The definition ofABmakes for all

x, so

The left-to-right order in products is critical because

ABandBAare usually not the same.

Because the columns ofABare linear combinations

of the columns ofA, whereas the columns ofBAare

constructed from the columns ofB. The position of the factors in the productABis

emphasized by saying thatAis right-multipliedbyB

or thatBis left-multipliedbyA.

( x) ( )xA B AB

1( ) ( )c ( )c ( )

pA BC AB AB AB C

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If , we say thatAandBcommutewithone another.

Warnings:1. In general, .

2. The cancellation laws do nothold for matrix

multiplication. That is, if , then it is

nottrue in general that .

3. If a productABis the zero matrix, you cannot

conclude in general that either or .

AB BA

AB BA

AB ACB C

0A 0B

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POWERS OF A MATRIX

IfAis an matrix and if kis a positive integer,thenAkdenotes the product of kcopies ofA:

IfAis nonzero and if xis in , thenAkxis the resultof left-multiplying xbyArepeatedly ktimes.

If , thenA0xshould be xitself.

ThusA

0

is interpreted as the identity matrix.

n n

k

k

A A A

n

0k

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THE TRANSPOSE OF A MATRIX

Given an matrixA, the transposeofAis the

matrix, denoted byAT, whose columns are

formed from the corresponding rows ofA.

Theorem 3:LetAandBdenote matrices whose sizes

are appropriate for the following sums and

products.

a. b.

c. For any scalar r,

d.

m nn m

( )T T

A A( )T T TA B A B ( )T TrA rA

( )

T T T

AB B A

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Slide 2 1- 21 2012 Pearson Education Inc

THE TRANSPOSE OF A MATRIX

The transpose of a product of matrices equals the

product of their transposes in the reverseorder.

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