[PDF] Some Linear Algebra Notes

Two classical theorems on commuting matrices

AM= MA Then either M = ° or M is nonsingular Furthermore if A = A (so that M commutes with each element of ~l) then M is scalar :l PROOF Suppose that the rank of M is r, and write (II" M = P ° where P, Q are nonsingular and II' is the r X r identity matrix Then for each AE~l, (2) (II' (P-IAP) ° ° 0) _ (QAQ-I) Put (All P-IAP= A~I

EXERCISES OF MATRICES OPERATIONS Question 1

EXERCISES OF MATRICES OPERATIONS 3 (24) If A,Bare both n×nmatrices and ABis singular, then Ais singular or Bis singular (25) Ais diagonalizable, then Ais non-singular (26) Ais symmetric, then Ais non-singular

2 ALGEBRE` - Major-Prépa

6 La matrice triangulaire A admet deux valeurs propres r´eelles 1 et −1 : elle est diagonalisable a) Si M ∈ R, alors AM = M3 = MA et M ∈ C Par la question pr´ec´edente, M est donc de la forme αI + βA b) Comme A2 = I, on a (αI +βA)2 = (α2 +β2)I +2αβA et cette matrice vaut A lorsque α2 + β2 = 0 et 2αβ = 1, on a donc β

TD 03 : Matrices

(c)Montrer qu’il existe une base B de R3 dans laquelle la matrice de u est égale à 0 1 0 0 0 0 0 0 2 (d)Déterminer les endomorphismes qui commutent avec u 6 / Montrer que 0 0 0 1 1 −1 2 2 −2 est semblable à −1 0 0 0 0 0 0 0 0 7 / Montrer qu’une matrice A ∈M n(K) est semblable à la matrice dont

Rank of Matrix by Determinant

Example 1 Find the Rank of Matrix using Determinant ????= 1 2 3 2 4 7 3 6 10 ????=140−42−220−21+312−12 =1−2−2−1+30

Some Linear Algebra Notes

ij] is row (column) equivalent to a unique ma-trix in reduced (column) row echelon form The uniqueness proof is involved, see Ho man and Kunze, Linear Algebra, 2nd ed Note: the row echelon form of a matrix is not unique Why? Theorem 2 3 Let Ax= band Cx= dbe two linear systems, each of mequations in nunknowns If

Rank of Matrix by Normal Form - WordPresscom

R3 R3 –R2, ????≅ 1 0 0 2 0 0 3 1 0 C2 C2-2C1, ????≅ 1 0 0 0 0 0 0 1 0 As, Normal Form of given matrix A is having Identity Matrix of Order 2 rank (A)= r(A) = 2 ????≅ 1

64 Hermitian Matrices - Naval Postgraduate School

Ch 6: Eigenvalues 6 4 Hermitian Matrices We consider matrices with complex entries (a i;j 2C) versus real entries (a i;j 2R) 1 in R the length of a real number xis jxj= the length from the origin to the number

TD - Matrices

Exercice 17 Soit A ∈Mn (R)une matrice nilpotente d’ordre p >1 On pose B =In −A 1 Montrer que B est inversible et exprimer son inverse à l’aide de A (penser à la factorisation de I −Ap) 2 Application : B = 1 −1 0 0 0 1 −1 0 0 0 1 −1 0 0 0 1

Matrix Multiplication - University of Plymouth

Table of Contents 1 Introduction 2 Matrix Multiplication 1 3 Matrix Multiplication 2 4 The Identity Matrix 5 Quiz on Matrix Multiplication Solutions to Exercises

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Some Linear Algebra Notes

Anmxnlinear systemis a system ofmlinear equations innunknownsxi,i= 1;:::;n: a

11x1+a12x2++a1nxn=b1

a

21x1+a22x2++a2nxn=b2

a m1x1+am2x2++amnxn=bm Thecoecientsaijgive rise to the rectangular matrixA= (aij)mxn(the rst subscript is the row, the second is the column). This is a matrix withmrows andncolumns: A=2 6 664a

11a12a1n

a

21a22a2n......

a m1am2amn3 7 775:
Asolutionto the linear system is a sequence of numberss1;s2;;sn, which has the property that each equation is satised whenx1=s1;x2=s2;;xn=sn. If the linear system has a nonzero solution it isconsistent, otherwise it isinconsistent. If the right hand side of the linear system constant 0, then it is called ahomogeneouslinear sys- tem. The homogeneous linear system always has thetrivial solutionx= 0. Two linear systems areequivalent, if they both have exactly the same solutions. Def 1.1: AnmxnmatrixAis a rectangular array ofmnreal or complex numbers arranged inm (horizontal) rows andn(vertical) columns. Def 1.2: TwomxnmatricesA= (aij) andB= (bij) areequal, if they agree entry by entry. Def 1.3: ThemxnmatricesAandBare added entry by entry. Def 1.4: IfA= (aij) andris a real number, then thescalarmultiple ofrandAis the matrix rA= (raij). IfA1;A2;:::;Akaremxnmatrices andc1;c2;:::;ckare real numbers, then an expression of the form c

1A1+c2A2++ckAk

is alinear combinationof theA's with coecientsc1;c2;:::;ck. Def 1.5: Thetransposeof themxnmatrixA= (aij) is thenxmmatrixAT= (aji). Def 1.6: Thedot productorinner productof then-vectorsa= (ai) andb= (bi) is ab=a1b1+a2b2+:::+anbn 1 Example: Determine the values of x and y so thatvw= 0 andvu= 0, wherev=2 4x 1 y3 5 ;w=2 42
2 13 5 ;andu=2 41
8 23
5 Def 1.7: IfA= (aij) is anmxpmatrix andB= (bij) apxnmatrix they can be multiplied and theijentry of themxnmatrixC=AB: c ij= (ai)T(bj) . Example: Let A=2

41 2 3

2 1 4 21 53
5 andB=2 41 2
0 4 3 53 5

If possible, ndAB,BA,A2,B2.

Which matrix rows/columns do you have to multiply in order to get the 3;1 entry of the matrixAB? Describe the rst row ofABas the product of rows/columns ofAandB. The linear system (see beginning) can thus be written in matrix formAx=b.

Write it out in detail.

A is called thecoecient matrixof the linear system and the matrix 2 6

66664a

11a12a1n...b1

a

21a22a2n...b2.........

a m1am2amn...bn3 7

77775:

is called theaugmented matrixof the linear system. Note:Ax=bis consistent if and only ifbcan be expressed as a linear combination of the columns of

Awith coecientsxi.

Theorem 1.1LetA;B, andCbemxnmatrices, then

(a)A+B=B+A (b)A+ (B+C) = (A+B) +C (c) there is a uniquemxnmatrixOsuch that for anymxnmatrixA:A+O=A (d) for each mxn matrixA, there is aunique mxn matrixDsuch that A+D=O

D=Ais the negative ofA.

Theorem1.2 LetA;B, andCbe matrices of the appropriate sizes, then (a)A(BC) = (AB)C 2 (b) (A+B)C=AC+BC (c)C(A+B) =CA+CB

Prove part (b)

Theorem 1.3Letr;sbe real numbers andA;Bmatrices of the appropriate sizes, then (a)r(sA) = (rs)A (b) (r+s)A=rA+sA (c)r(A+B) =rA+rB (d)A(rB) =r(AB) = (rA)B Theorem 1.4Letrbe a scalar,A;Bmatrices of appropriate sizes, then (a) (AT)T=A (b) (A+B)T=AT+BT (c) (AB)T=BTAT (d) (rA)T=rAT prove part (c) Note: (a)ABneed not equalBA. (b)ABmay be the zero matrixOwithAnot equalOandBnot equalO. (c)ABmay equalACwithBnot equalC.

Find two dierent 2 x 2 matricesAsuch thatA2= 0.

Find three dierent 2 x 2 matricesA,BandCsuch thatAB=AC,A6= 0 andB6=C.

Def 1.8:

indent A matrixA= [aij] is adiagonal matrixifaij= 0 fori6=j. Ascalar matrixis a diagonal matrix whose diagonal entries are equal. The scalar matrixIn=dij, wheredii= 1 anddij= 0 fori6=jis called thenxnidentity matrix. Example: If square matricesAandBsatisfy thatAB=BA, then (AB)p=ApBp.

Def 1.9:

A matrix A with real enties issymmetricifAT=A.

A matrix with real entries isskewsymmetricifAT=A.

LetB=2

41 2
0 4 3 53 5 computeBBTandBTB. What can you say about them? AnnxnmatrixAisupper triangularifaij= 0 fori > j,lower triangularifa= 0 fori < j. Given anmxnmatrixA= [aij]. If we cross out some, but not all of it's rows and columns, we get a 3 submatrixofA. A matrix can bepartitionedinto submatrices by drawing horizontal lines between rows and vertical lines between columns. Def 1.10: An nxn matrixAisnonsingularorinvertible, if there exists annxnmatrixBsuch that

AB=BA=In

Bwould then be the inverse ofA

Otherwise A issingularornoninvertible.

Remark: At this point, we have not shown that ifAB=In, thenBA=In, this will be done in

Theorem 2.11. In the mean time we assume it.

IfD=2

41=4 0 0

02 0

0 0 33

5 . FindD1. Theorem 1.5The inverse of a matrix, if it exists is unique.

Prove it.

IfAis a nonsingular matrix whose inverse is2 1

4 1 , ndA. Theorem 1.6IfAandBare both nonsingularnxnmatrices thenABis nonsingular and (AB)1=B1A1:

Prove it.

Corollary 1.1IfA1;A2;:::;Arare nonsingularnxnmatrices, thenA1A2:::Aris nonsingular and (A1A2Ar)1=A1rA12A11 .Theorem 1.7IfAis a nonsingular matrix, thenA1is nonsingular and (A1)1=A: why? Theorem 1.8IfAis a nonsingular matrix, thenATis nonsingular and (A1)T= (AT)1: Show that if the matrixAis symmetric and nonsingular, thenA1is symmetric.

Note: IfAis a nonsingularnxnmatrix. Then

(a) the linear systemAx=bhas the unique solutionx=A1b. Why? (b) the homogeneous linear system Ax = 0 has the unique solution x = 0. Why?

Consider the linear systemAx=b, whereA=2 1

4 1 4

Find a solution ifb=3

4

SupposeAanmxnmatrix,xann-vector, i.e inRn.

ThenAx=yis anm-vector,yinRm. SoArepresents amatrix transformationf, f:Rn!Rm, x!y=Ax.

ForuinRn:f(u) =Auis theimageofu.

ff(u) =Auju2Rngis therangeoff. For the given matrix transformationsfand vectorsu, ndf(u).

Geometrically (draw pictures), what doesfdo?

(a)A=1 0 01 ;u=2 7 (b)A=3 0 0 3 ;u=1 2 (c)A=cossin sincos ;u 1=1 0 ;u 2=0 1 (d)A=0 1 0 0 0 1 ;u=2 41
2 33
5 (a) was are ectionon thex-axis. (b) wasdilationby a factor of 3. If the factor is<1, it's called acontraction. (c) was a rotation around the origin by angle. (d) was a verticalprojectiononto theyz-plane.

Letf:Rn!Rm,

x!y=Ax.

Show that:

(a)f(u+v) =f(u) +f(v), foru;v2Rn. (b)f(cu) =cf(u), wherec2R,u2Rn. Def 2.1 Anmxn, matrix is said to be inreduced row echelon formif it satises the following properties: (a) all zero rows, if there are any, are at the bottom of the matrix.

(b) the rst nonzero entry from the left of a nonzero row is a 1.This entry is called a leading one of

its row. (c) For each nonzero row, the leading one appears to the right and below any leading ones in preced- ing rows. (d) If a column contains a leading one, then all other entries in that column are zero. Anmxn, matrix is inrow echelon form, if it satises properties (a), (b), and (c).

Similar denition for column echelon form.

5

What can you say about these matrices?

(1)A=2

40 1 0 0 1 0

0 0 0 0 0 0

0 0 1 1 1 03

5 (2)A=2

40 1 0 0 1 0

0 0 1 2 1 0

0 0 0 0 1 03

5 (3)A=2 6

640 1 01 0 0

0 0 1 2 0 0

0 0 0 0 1 0

0 0 0 0 0 03

7 75.
Def 2.2 Anelementary row (column) operationon a matrixAis one of these: (a) interchange of two rows (b) multiply a row by a nonzero number (c) add a multiple of one row to another.

For the matrixA=2

41 11 0 3

3 4 1 1 10

4642143

5 . Find (a) a row-echelon form (b) the reduced row-echelon form Def 2.3 Anmxn, matrixBisrow (column) equivalentto anmxn, matrixA, ifBcan be produced by applying a nite sequence of elementary row (column) operations toA. Theorem 2.1Every nonzeromxnmatrixA= [aij] is row (column) equivalent to a matrix in row (column) echelon form. Theorem 2.2Every nonzeromxn, matrixA= [aij] is row (column) equivalent to a unique ma- trix in reduced (column) row echelon form. The uniqueness proof is involved, see Homan and Kunze, Linear Algebra, 2nd ed. Note: the row echelon form of a matrix is not unique. Why? Theorem 2.3LetAx=bandCx=dbe two linear systems, each ofmequations innunknowns. If

the augmented matrices [Ajb] and [Cjd] are row equivalent, then the linear systems are equivalent, i.e.

they have exactly the same solutions. Corollary 2.1IfAandCare row equivalentmxnmatrices, then the homogeneous systemsAx= 0 andCx= 0 are equivalent. Find the solutions (if they exist) for these augmented matrices: 6 (a)A=2 6

641 11 0... 3

0 1 0 0... 2

0 0 0 1...13

7 75
(b)A=2 6

641 11 0... 3

0 1 0 0... 2

0 0 0 0...13

7 75
(c)A="

1 12 0... 3

0 0 0 1...1#

Theorem 2.4Ahomogeneous system ofmlinear equations innunknowns always has a nontrivial solution ifm < n, that is, if the number of unknowns exceeds the number of equations.

GivenAx= 0 withA=2

41 02 0 0 3

0 1 0 0 11

0 0 0 1 2 43

5

Find the solution set forAx= 0.

Gaussian elimination: transform [Ajb] to [Cjd], where [Cjd] is in row echelon form. Gauss Jordan reduction: transform [Ajb] to [Cjd], where [Cjd] is in reduced row echelon form. Find an equation relatinga,bandc, so that the linear system: x+ 2y3z=a

2x+ 3y+ 3z=b

5x+ 9y6z=c

is consistent for any values ofa,bandc, that satisfy that equation.

LetAx=b,b6= 0, be a consistent linear system.

Show that ifxpis a particular solution to the given nonhomogeneous system andxhis a solution to the associated homogeneous systemAx= 0, thenxp+xhis a solution to the given systemAx=b. Ethane is a gas similar to methane that burns in oxygen to give carbon dioxide gas and steam. The steam condenses to form water droplets. The chemical equation for this reaction is: C

2H6+O2!CO2+H2O

Balance this equation.

Def 2.4 anelementary matrixis a matrix obtained from the identity matrix by performing a single elementary row operation. 7 Find the matrixEotained from the identity matrixI3by the row manipulation (3)-2(1)!(3) (i.e. the third row is replaced by row 3 - 2 * row 1).

For the matrixA=2

42 1 0

1 01 11 33 5 (a) left multiplyAwithE1=2

40 0 1

0 1 0

1 0 03

5 (b) left multiply the matrix you got from part (a) withE2=2

41 0 0

1 1 0

0 0 13

5 (c) left multiply the matrix you got from part (b) withE3=2

41 0 0

01 0

0 0 13

5 (d) left multiply the matrix you got from part (c) withE4=2

41 0 0

0 1 0

2 0 13

5 give a further sequence of elementary matrices that will transformAto - row-echelon form, - reduced row echelon form Theorem 2.5Perform an elementary row operation (with matrixE) onmxn, matrixAto yield matrixB. ThenB=EA.

Theorem 2.6LetA,Bbemxn, matrices. Equivalent:

(a)Ais row equivalent toB. (b) There exist elementary matricesE1;E2;:::;Ek, such that

B=EkEk1:::E1A:

Theorem 2.7An elementary matrixEis nonsingular, and its inverse is an elementary matrix of the same type. Lemma 2.1LetAbe annxnmatrix and suppose the homogeneous systemAx= 0 has only the trivial solutionx= 0. ThenAis row equivalent toIn. Theorem 2.8Ais nonsingular if and only ifAis the product of elementary matrices. Corollary 2.2Ais nonsingular if and only ifAis row equivalent toIn.

Theorem 2.9Equivalent:

(a) The homogeneous system ofnlinear equations innunknownsAx= 0 has a nontrivial solution. (b)Ais singular.

Theorem 2.10Equivalent:

8 (a)nxnmatrixAis singular. (b)Ais row equivalent to a matrix that has a row of zeroes. Theorem 2.11LetA, B benxnmatrices such thatAB=In, thenBA=InandB=A1. Prove: IfAandBarenxnmatrices andABnonsingular, thenAandBare each also nonsingu- lar. Def 2.5A,B mxn, matrices.AisequivalenttoB, if we can obtainBfromAby a nite sequence of elementary row and column operations. Theorem 2.12IfAis a nonzeromxn, matrix, thenAis equivalent to a partitioned matrix of the form:

IrOr;nr

O mr;rOmr;nr

Theorem 2.13LetA, B bemxn, matrices. Equivalent:

(a)Ais equivalent to B. (b)B=PAQ(P= product of elementary row matrices,Q= product of elementary column matri- ces).

Theorem 2.14LetAbe annxnmatrix. Equivalent:

(a)Ais nonsingular. (b)Ais equivalent toIn. Def 3.1 LetS= 1;2;:::;nin this order. Arearrangementj1j2:::jnof the elements ofSis a per- mutation ofS. How many permutations of the setS= 1;2;3 are there? How many permutations of the setS= 1;2;:::;nare there? A permutationj1j2:::jnis said to have aninversionif a larger integerjrprecedes a smaller one, j s.

A permutation is even if the total number of inversions iseven, oroddif the total number of inversions

in it is odd.

Is the permutation 43512 even or odd?

Def 3.2 LetA= [aij] be annxnmatrix. Thedeterminantfunctiondetis dened by det(A) =X()a1j1a2j2:::anjn , where the summation is over all permutationsj1j2:::jnof the setS. The sign is taken as + or - according to whether the permutationj1j2:::jnis even or odd. 9

Compute the determinant of

A=2 4a

11a12a13

a

21a22a23

a

31a32a333

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