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MATH 304

Linear Algebra

Lecture 4:

Matrix multiplication.

Diagonal matrices.

Inverse matrix.

Matrices

Definition.Anm-by-n matrixis a rectangular

array of numbers that hasmrows andncolumns: (a

11a12...a1n

a

21a22...a2n............

a m1am2...amn))))) if the dimensions are known.

Matrix algebra: linear operations

Addition:two matrices of the same dimensions

can be added by adding their corresponding entries.

Scalar multiplication:to multiply a matrixAby

a scalarr, one multiplies each entry ofAbyr.

Zero matrixO:all entries are zeros.

Negative:-Ais defined as (-1)A.

Subtraction:A-Bis defined asA+ (-B).

As far as the linear operations are concerned, the m×nmatrices can be regarded asmn-dimensional vectors.

Properties of linear operations

(A+B) +C=A+ (B+C)

A+B=B+A

A+O=O+A=A

A+ (-A) = (-A) +A=O

r(sA) = (rs)A r(A+B) =rA+rB (r+s)A=rA+sA 1A=A 0A=O

Dot product

Definition.Thedot productofn-dimensional

vectorsx= (x1,x2,...,xn) andy= (y1,y2,...,yn) is a scalar x·y=x1y1+x2y2+···+xnyn=n? k=1x kyk.

The dot product is also called thescalar product.

Matrix multiplication

The product of matricesAandBis defined if the

number of columns inAmatches the number of rows inB.

Definition.LetA= (aik) be anm×nmatrix and

B= (bkj) be ann×pmatrix. TheproductABis

defined to be them×pmatrixC= (cij) such that cij=?nk=1aikbkjfor all indicesi,j.

That is, matrices are multipliedrow by column:

A=(((((a

11a12...a1na21a22...a2n............

am1am2...amn))))) =(((((v 1 v 2... v m)))))

B=(((((b

11 b12...b1p b 21
b22...b2p............ b n1 bn2...bnp))))) = (w1,w2,...,wp) =?AB=(((((v

1·w1v1·w2...v1·wp

v

2·w1v2·w2...v2·wp............

v m·w1vm·w2...vm·wp)))))

Examples.

(x1,x2,...,xn)((((y 1 y 2... y n)))) = (?nk=1xkyk), y 1 y 2... y n)))) (x1,x2,...,xn) =((((y

1x1y1x2...y1xn

y

2x1y2x2...y2xn............

y nx1ynx2...ynxn))))

Example.

1 1-1

0 2 1?

(0 3 1 1 -2 5 6 0

1 7 4 1))

=?-3 1 3 0 -3 17 16 1?

0 3 1 1

-2 5 6 0

1 7 4 1))

?1 1-1

0 2 1?

is not defined

System of linear equations:???????a

11x1+a12x2+···+a1nxn=b1

a

21x1+a22x2+···+a2nxn=b2

a m1x1+am2x2+···+amnxn=bm

Matrix representation of the system:

(a

11a12...a1n

a

21a22...a2n............

a m1am2...amn))))) (x 1 x 2... x n))))) =(((((b 1 b 2... b m))))) ???????a

11x1+a12x2+···+a1nxn=b1

a

21x1+a22x2+···+a2nxn=b2

a m1x1+am2x2+···+amnxn=bm??Ax=b, where

A=(((((a

11a12...a1n

a

21a22...a2n............

a m1am2...amn))))) ,x=(((((x 1 x 2... x n))))) ,b=(((((b 1 b 2... b m)))))

Properties of matrix multiplication:

(AB)C=A(BC) (associative law) (A+B)C=AC+BC(distributive law #1)

C(A+B) =CA+CB(distributive law #2)

(rA)B=A(rB) =r(AB)

Any of the above identities holds provided that

matrix sums and products are well defined.

IfAandBaren×nmatrices, then bothABandBA

are well definedn×nmatrices.

However, in general,AB?=BA.

Example.LetA=?2 00 1?

,B=?1 10 1?

ThenAB=?2 20 1?

,BA=?2 10 1?

IfABdoes equalBA, we say that the matricesA

andBcommute.

Problem.LetAandBbe arbitraryn×nmatrices.

Is it true that (A-B)(A+B) =A2-B2?

(A-B)(A+B) = (A-B)A+ (A-B)B = (AA-BA) + (AB-BB) =A2+AB-BA-B2

Hence (A-B)(A+B) =A2-B2if and only if

Acommutes withB.

Diagonal matrices

IfA= (aij) is a square matrix, then the entriesaii are calleddiagonal entries. A square matrix is calleddiagonalif all non-diagonal entries are zeros.

Example.

(7 0 00 1 00 0 2)) , denoteddiag(7,1,2).

ThenA+B=diag(s1+t1,s2+t2,...,sn+tn),

rA=diag(rs1,rs2,...,rsn).

Example.((7 0 00 1 00 0 2))

(-1 0 00 5 00 0 3)) =((-7 0 00 5 00 0 6))

TheoremLetA=diag(s1,s2,...,sn),

B=diag(t1,t2,...,tn).

ThenA+B=diag(s1+t1,s2+t2,...,sn+tn),

rA=diag(rs1,rs2,...,rsn).

AB=diag(s1t1,s2t2,...,sntn).

In particular, diagonal matrices always commute

(i.e.,AB=BA).

Example.((7 0 00 1 00 0 2))

(a

11a12a13

a

21a22a23

a

31a32a33))

=((7a117a127a13 a

21a22a23

2a312a322a33))

TheoremLetD=diag(d1,d2,...,dm) andAbe

anm×nmatrix. Then the matrixDAis obtained fromAby multiplying theith row bydifor i= 1,2,...,m:

A=(((((v

1 v 2... v m))))) =?DA=(((((d 1v1 d

2v2...

d mvm)))))

Example.((a

11a12a13

a

21a22a23

a

31a32a33))

(7 0 00 1 00 0 2)) =((7a11a122a13

7a21a222a23

7a31a322a33))

TheoremLetD=diag(d1,d2,...,dn) andAbe

anm×nmatrix. Then the matrixADis obtained fromAby multiplying theith column bydifor i= 1,2,...,n:

A= (w1,w2,...,wn)

=?AD= (d1w1,d2w2,...,dnwn)

Identity matrix

Definition.Theidentity matrix(orunit matrix) is

a diagonal matrix with all diagonal entries equal to 1.

Then×nidentity matrix is denotedInor simplyI.

I

1= (1),I2=?1 00 1?

,I3=((1 0 00 1 00 0 1))

In general,I=(((1 0...0

0 1...0............

0 0...1)))

Theorem.LetAbe an arbitrarym×nmatrix.

ThenImA=AIn=A.

Inverse matrix

LetMn(R) denote the set of alln×nmatrices with

real entries. We canadd,subtract, andmultiply elements ofMn(R). What aboutdivision?

Definition.LetAbe ann×nmatrix. Suppose

there exists ann×nmatrixBsuch that

AB=BA=In.

Then the matrixAis calledinvertibleandBis

called theinverseofA(denotedA-1).

A non-invertible square matrix is calledsingular.

AA-1=A-1A=I

Examples

A=?1 10 1?

,B=?1-1 0 1? ,C=?-1 0 0 1?

AB=?1 10 1??

1-1 0 1? =?1 00 1?

BA=?1-1

0 1?? 1 1 0 1? =?1 00 1? C

2=?-1 0

0 1?? -1 00 1? =?1 00 1?

ThusA-1=B,B-1=A, andC-1=C.

Inverting diagonal matrices

TheoremA diagonal matrixD=diag(d1,...,dn)

is invertible if and only if all diagonal entries are

IfDis invertible thenD-1=diag(d-11,...,d-1n).

(d

10...0

0d2...0............

0 0...dn)))))

-1 =(((((d -110...0

0d-12...0............

0 0...d-1n)))))

Inverting 2×2 matrices

Definition.Thedeterminantof a 2×2 matrix

A=?a b

c d? is detA=ad-bc.

TheoremA matrixA=?a b

c d? is invertible if and only if detA?= 0.

If detA?= 0 then?a b

c d? -1 =1 ad-bc? d-b -c a?

System ofnlinear equations innvariables:???????a

11x1+a12x2+···+a1nxn=b1

a

21x1+a22x2+···+a2nxn=b2

a n1x1+an2x2+···+annxn=bn??Ax=b, where

A=(((((a

11a12...a1n

a

21a22...a2n............

a n1an2...ann))))) ,x=(((((x 1 x 2... x n))))) ,b=(((((b 1 b 2... b n)))))

TheoremIf the matrixAis invertible then the

system has a unique solution, which isx=A-1b.quotesdbs_dbs20.pdfusesText_26