Linear AlgebraI Notes4
(
,MATRIX ALGEBRA
Definition:Amatrix Aisarectangulararrayofscalars(numbers),i.e.,acollectionof
numbersorderedbyrowsandcolumns.TheijthcomponentofthematrixA,denotedby aij,
isthenumber inthe intersection ofthei rowandj columnofA.The
th th
components
(elements/entrie)s of a matrix are usually enclosed in parentheses, brackets or braces.The
matrixAissometimeswrittenasA=[aij].Amatrixwithallcomponentsequaltozero
iscalledthezeromatrix.ForeachmatrixA=[aij],therecorrespondsauniquematrix
—A=[—aij],calledtheadditiveinverseofA,obtainedbynegatingeachentryofA.
Definition:AmatrixAissaidtohave order(shape/size)m×n, read“mbyn” ifAhasm
rowsandncolumns.Inthiscase,ifm=n(i.e.Ahassamenumberofrowsandcolumns),
Aiscalledasquarematrix; otherwise,Aissaidtoberectangular.Amatrixconsisting of
asinglerowiscalledarowvector,whileamatrixconsistingofasinglecolumniscalleda
columnvector.Conventionally,a1×1matrixisascalarandviceversa.
Definition:TwomatricesA=[aij]andB=[bij]areequaliftheyhavethesameshapeand
correspondingentriesareequal,i.e.,aij=bijforeachi=1,2,···,mandj=1,2,···,n.
MatrixAddition
LetA=[aij] and B=[bij] be twom×n matrices.Then the sumof A and Bis the m×n
matrixA+Bobtained by adding corresponding entries.Matrixsubtractionis done in a
similarmannerexceptthatentriesaresubtractedinsteadofbeingadded.Fortwomatrices ofthe
same shape, the differenceA—Bis the matrix A—B =A+(—B).
Example:
Let 1—2 3
A= —1 0
andB= .Then
4 5 — 2
6
—5—45
1 —2 3 —1 0 2
A 0 —2 5
4 5 —6 —5 —4 5 = —1 1 —1
+
+B=
and 1 —2 3 —1 0 2
4 5 —6 —5 —4 5 2—2 1
A — = 9 9 —11
.
—B=
PropertiesofMatrixAddition
LetA,B,andCbem×nmatrices.Then:
(i) A+Bisanm×nmatrix(closureproperty).
(ii)(A+B)+C=A+(B+C) (associative law formatrix addition).
(iii)A+B=B+A(commutativelawformatrixaddition).
(iv) Thereisanm×nmatrixdenotedby0consistingofallzeros(thezeromatrix)such
thatA+0=0+A=A (existence of additive identity).
(v) Thereisanm×nmatrix(—A)suchthatA+(—A)=(—A)+A=0where0isthe zero
matrix (existence of additive inverses).
, Scalar Multiplication
LetAbeanm ×nmatrix.TheproductofascalarαtimesamatrixA,denotedbyαA,is the matrix
obtained by multiplying each entry of A by α.
Example:
—5 0 2
LetA= 3 4 — andlet α=—1/2.Then
3
1 —1 1
—5 0 2 5/2 0 —1
αA=— 1 3 4 — = —3/2 —2 3/2 .
2 3 —1/2 1/2 —1/2
1 —1 1
Properties of ScalarMultiplication
LetA and Bbe a m×n matrices and let α and βbe scalars.Then: (i)
αA is an m×n matrix (closure property).
(ii) (αβ)A=α(βA)(associativelawforscalarmultiplication).
(iii) α(A+B)=αA+αB(scalarmultiplicationisdistributiveovermatrix addition). (iv)
(α+β)A=αA+βA(scalarmultiplicationisdistributiveoverscalaraddition). (v)
1A=A(identity property-thenumber 1isanidentity element underscalar
multiplication).
Tr anspose
LetAbeanm×nmatrix.ThetransposeofA,denotedbyAT, isthen×mmatrixobtained by
interchanging rows and columns ofA, i.e., thei row ofAis thei column ofA and the
th th T
jthcolumn of A is the jthrow of AT.
Example:
—1 2 3
—1 0 2 5 T
0 4 —1
LetA= 2 4 —3 2 .ThenA .
= 2 —3 0
3 —1 0 —
5 2 —1
1
PropertiesoftheTranspose
LetA and Bbe a m×n matrices and let α be a scalar.Then: (i)
T
AT = A.
(ii) (A+B)T=AT+BT.
(iii) (αA)T=αAT.
Sym m etry
LetA=[aij]beann×nmatrix.ThenAissaidtobeasymmetricmatrixifA=AT,i.e.,
aij=aji.Inotherwords,AissymmetriciftheentriesofAaresymmetricallylocatedabout 2
, themaindiagonal(thelinefromtheupper-left-handcornertothelower-right-handcorner).
Example:
1 —2 0
LetA= —2 4 — .ThenAissymmetricsinceA=AT.
3
0 —3 0
Tr ace
LetA=[aij] be ann×nmatrix.ThetraceofA,denotedbytrace(A),is thesumof the entries on the
main diagonal.That is,
n
Σ
trace(A)=a11 +a22+···+ann= aii.
i=1
Example:
—3 2
1
LetA= 4 —1—2 .Thentrace(A) =(—3)+(—1)+0=—4.
5 0 0
Linear Com binations
LetA1,A2,···,Anbem×nmatricesandletα1,α2,···,αnbescalars.Then,theexpression
n
Σ
α1A1+α2A2+···+αnAn= αiAi
i=1
iscalledalinearcombinationoftheA1,A2,···,An.
Example:
Let 4—1 —3 —6 0
A= ,B= ,andC= .Letα=2,β=—1,and
6
5 0 1 2 —2 —4
λ= .Then
1
2
αA 4—1
+βB+λC=2 —3 —6 10 6
+(—1) +
5 0 1 2 2 —2—4
11 7
= 8 —
4
=D
So,DisalinearcombinationofA,B,andC.
(
,MATRIX ALGEBRA
Definition:Amatrix Aisarectangulararrayofscalars(numbers),i.e.,acollectionof
numbersorderedbyrowsandcolumns.TheijthcomponentofthematrixA,denotedby aij,
isthenumber inthe intersection ofthei rowandj columnofA.The
th th
components
(elements/entrie)s of a matrix are usually enclosed in parentheses, brackets or braces.The
matrixAissometimeswrittenasA=[aij].Amatrixwithallcomponentsequaltozero
iscalledthezeromatrix.ForeachmatrixA=[aij],therecorrespondsauniquematrix
—A=[—aij],calledtheadditiveinverseofA,obtainedbynegatingeachentryofA.
Definition:AmatrixAissaidtohave order(shape/size)m×n, read“mbyn” ifAhasm
rowsandncolumns.Inthiscase,ifm=n(i.e.Ahassamenumberofrowsandcolumns),
Aiscalledasquarematrix; otherwise,Aissaidtoberectangular.Amatrixconsisting of
asinglerowiscalledarowvector,whileamatrixconsistingofasinglecolumniscalleda
columnvector.Conventionally,a1×1matrixisascalarandviceversa.
Definition:TwomatricesA=[aij]andB=[bij]areequaliftheyhavethesameshapeand
correspondingentriesareequal,i.e.,aij=bijforeachi=1,2,···,mandj=1,2,···,n.
MatrixAddition
LetA=[aij] and B=[bij] be twom×n matrices.Then the sumof A and Bis the m×n
matrixA+Bobtained by adding corresponding entries.Matrixsubtractionis done in a
similarmannerexceptthatentriesaresubtractedinsteadofbeingadded.Fortwomatrices ofthe
same shape, the differenceA—Bis the matrix A—B =A+(—B).
Example:
Let 1—2 3
A= —1 0
andB= .Then
4 5 — 2
6
—5—45
1 —2 3 —1 0 2
A 0 —2 5
4 5 —6 —5 —4 5 = —1 1 —1
+
+B=
and 1 —2 3 —1 0 2
4 5 —6 —5 —4 5 2—2 1
A — = 9 9 —11
.
—B=
PropertiesofMatrixAddition
LetA,B,andCbem×nmatrices.Then:
(i) A+Bisanm×nmatrix(closureproperty).
(ii)(A+B)+C=A+(B+C) (associative law formatrix addition).
(iii)A+B=B+A(commutativelawformatrixaddition).
(iv) Thereisanm×nmatrixdenotedby0consistingofallzeros(thezeromatrix)such
thatA+0=0+A=A (existence of additive identity).
(v) Thereisanm×nmatrix(—A)suchthatA+(—A)=(—A)+A=0where0isthe zero
matrix (existence of additive inverses).
, Scalar Multiplication
LetAbeanm ×nmatrix.TheproductofascalarαtimesamatrixA,denotedbyαA,is the matrix
obtained by multiplying each entry of A by α.
Example:
—5 0 2
LetA= 3 4 — andlet α=—1/2.Then
3
1 —1 1
—5 0 2 5/2 0 —1
αA=— 1 3 4 — = —3/2 —2 3/2 .
2 3 —1/2 1/2 —1/2
1 —1 1
Properties of ScalarMultiplication
LetA and Bbe a m×n matrices and let α and βbe scalars.Then: (i)
αA is an m×n matrix (closure property).
(ii) (αβ)A=α(βA)(associativelawforscalarmultiplication).
(iii) α(A+B)=αA+αB(scalarmultiplicationisdistributiveovermatrix addition). (iv)
(α+β)A=αA+βA(scalarmultiplicationisdistributiveoverscalaraddition). (v)
1A=A(identity property-thenumber 1isanidentity element underscalar
multiplication).
Tr anspose
LetAbeanm×nmatrix.ThetransposeofA,denotedbyAT, isthen×mmatrixobtained by
interchanging rows and columns ofA, i.e., thei row ofAis thei column ofA and the
th th T
jthcolumn of A is the jthrow of AT.
Example:
—1 2 3
—1 0 2 5 T
0 4 —1
LetA= 2 4 —3 2 .ThenA .
= 2 —3 0
3 —1 0 —
5 2 —1
1
PropertiesoftheTranspose
LetA and Bbe a m×n matrices and let α be a scalar.Then: (i)
T
AT = A.
(ii) (A+B)T=AT+BT.
(iii) (αA)T=αAT.
Sym m etry
LetA=[aij]beann×nmatrix.ThenAissaidtobeasymmetricmatrixifA=AT,i.e.,
aij=aji.Inotherwords,AissymmetriciftheentriesofAaresymmetricallylocatedabout 2
, themaindiagonal(thelinefromtheupper-left-handcornertothelower-right-handcorner).
Example:
1 —2 0
LetA= —2 4 — .ThenAissymmetricsinceA=AT.
3
0 —3 0
Tr ace
LetA=[aij] be ann×nmatrix.ThetraceofA,denotedbytrace(A),is thesumof the entries on the
main diagonal.That is,
n
Σ
trace(A)=a11 +a22+···+ann= aii.
i=1
Example:
—3 2
1
LetA= 4 —1—2 .Thentrace(A) =(—3)+(—1)+0=—4.
5 0 0
Linear Com binations
LetA1,A2,···,Anbem×nmatricesandletα1,α2,···,αnbescalars.Then,theexpression
n
Σ
α1A1+α2A2+···+αnAn= αiAi
i=1
iscalledalinearcombinationoftheA1,A2,···,An.
Example:
Let 4—1 —3 —6 0
A= ,B= ,andC= .Letα=2,β=—1,and
6
5 0 1 2 —2 —4
λ= .Then
1
2
αA 4—1
+βB+λC=2 —3 —6 10 6
+(—1) +
5 0 1 2 2 —2—4
11 7
= 8 —
4
=D
So,DisalinearcombinationofA,B,andC.
