Problems and Solutions in Matrix Calculus

1 Problems and SolutionsinMatrix CalculusbyWilli-Hans SteebInternational School for Scientific ComputingatUniversity of Johannesburg, South AfricaPrefaceThe manuscript supplies a collection of Problems in introductory and ad-vanced Matrix book: Problems and Solutions in Introductory and Advanced Matrix Calculus ,2nd editionbyWilli-Hans Steeb and Yorick HardyWorld Scientific Publishing, Singapore 2016vContentsNotationx1 Basic Operations12 Linear Equations93 Determinants and Traces124 Eigenvalues and Eigenvectors225 Commutators and Anticommutators366 Decomposition of Matrices407 Functions of Matrices468 Linear Differential Equations549 Kronecker Product5810 Norms and Scalar Products6711 Groups and Matrices7212 Lie Algebras and Matrices8613 Graphs and Matrices9214 Hadamard Product9415 Differentiation9616 Integration9717 Numerical Methods99vii18 Miscellaneous106 Bibliography143 Index146viiiNotation.

2 =is defined as belongs to (a set)/ does not belong to (a set) intersection of sets union of sets empty setNset of natural numbersZset of integersQset of rational numbersRset of real numbersR+set of nonnegative real numbersCset of complex numbersRnn-dimensional Euclidean spacespace of column vectors withnreal componentsCnn-dimensional complex linear spacespace of column vectors withncomplex componentsHHilbert spacei 1<zreal part of the complex numberz=zimaginary part of the complex numberz|z|modulus of complex numberz|x+iy|= (x2+y2)1/2, x,y RT SsubsetTof setSS Tthe intersection of the setsSandTS Tthe union of the setsSandTf(S)image of setSunder mappingff gcomposition of two mappings (f g)(x) =f(g(x))xcolumn vector inCnxTtranspose ofx(row vector)0zero (column) vector . normx y x yscalar product (inner product) inCnx yvector product inR3A,B,Cm nmatricesdet(A)determinant of a square matrixAtr(A)trace of a square matrixArank(A)rank of matrixAATtranspose of matrixAxAconjugate of matrixAA conjugate transpose of matrixAA conjugate transpose of matrixA(notation used in physics)A 1inverse of square matrixA(if it exists)Inn nunit matrixIunit operator0nn nzero matrixABmatrix product ofm nmatrixAandn pmatrixBA BHadamard product (entry-wise product)ofm nmatricesAandB[A,B] :=AB BAcommutator for square matricesAandB[A,B]+:=AB+BAanticommutator for square matricesAandBA BKronecker product of matricesAandBA BDirect sum of matricesAandB jkKronecker delta with jk= 1 forj=kand jk= 0 forj6=k eigenvalue real parameterttime variable HHamilton operatorThe Pauli spin matrices are used extensively in the book.

3 They are givenby x:=(0 11 0), y:=(0 ii0), z:=(100 1).In some cases we will also use 1, 2and 3to denote x, yand 1 Basic OperationsProblem a column vector inRnandx6=0. LetA=xxTxTxwhereTdenotes the transpose, a row vector. the 8 8 Hadamard matrixH= 111111111111 1 1 1 111 1 1 1 11111 1 111 1 11 1 111 1 111 1 11 111 11 11 1 11 111 11 11 11 1 .(i) Do the 8 column vectors in the matrixHform a basis inR8? Prove ordisprove.(ii) CalculateHHT, whereTdenotes transpose. Compare the results from(i) and (ii) and ,Bben nmatrices such thatABAB= 0n. Can weconclude thatBABA= 0n?12 Problems and SolutionsProblem square matrixAoverCis calledskew-hermitianifA= A . Show that such a Matrix isnormal, , we haveAA =A ann nskew-hermitian Matrix overC, = A. LetUbe ann nunitary Matrix , ,U =U 1. Show thatB:=U AUis a skew-hermitian ,X,Yben nmatrices. Assume thatXA=In, AY=InwhereInis then nunit Matrix .

4 Show thatX= ,Bben nmatrices. Assume thatAis nonsingular, 1exists. Show that ifBA= 0n, thenB= ,Bben nmatrices andA+B=In, AB= thatA2=AandB2= :=xxT+yyT(1)wherex=(cos( )sin( )),y=(sin( ) cos( ))and R. FindxTx,yTy,xTy,yTx. Find the a 2 2 matrixAoverRsuch thatA(10)=1 2(11), A(01)=1 2(1 1).Problem the vector spaceR4. Find all pairwise orthogonalvectors (column vectors)x1,..,xp, where the entries of the column vectorscan only be +1 or 1. Calculate the matrixp j=1xjxTjand find the eigenvalues and eigenvectors of this Operations3 Problem 22 222 2 2 26 .(i) LetXbe anm nmatrix. Thecolumn rankofXis the maximumnumber of linearly independent columns. Therow rankis the maximumnumber of linearly independent rows. The row rank and the column rankofXare equal (called therankofX). Find the rank ofAand denote it byk.(ii) Locate ak ksubmatrix ofAhaving rankk.(iii) Find 3 3 permutation matricesPandQsuch that in the matrixPAQthe submatrix from (ii) is in the upper left portion 2 2 matricesA,Bsuch thatAB= 0nandBA6= anm nmatrix andBbe ap qmatrix.

5 Thenthedirect sumofAandB, denoted byA B, is the (m+p) (n+q) Matrix defined byA B:=(A00B).LetA1,A2bem mmatrices andB1,B2ben nmatrices. Calculate(A1 B1)(A2 B2).Problem ann nmatrix overR. Find all matrices thatsatisfy the equationATA= matrixAfor whichAp= 0n, wherepis a positive integer,is callednilpotent. Ifpis the least positive integer for whichAp= 0nthenAis said to be nilpotent of indexp. Find all 2 2 matrices over the realnumbers which are nilpotent withp= 2, that ann nmatrixAis involutary if and only if(In A)(In+A) = ann nsymmetric Matrix overR. LetPbe anarbitraryn nmatrix overR. Show thatPTAPis ann nskew-symmetric Matrix overR, A. LetPbe an arbitraryn nmatrix overR. Show thatPTAPis and SolutionsProblem an invertiblen nmatrix overCandBbe ann nmatrix overC. We define then nmatrixD:=A , wheren= 2,3,..Problem ,B,C,Dben nmatrices overR. Assume thatABTandCDTare symmetric andADT BCT=In, whereTdenotestranspose.

6 Show thatATD CTB= nmatrixP= (pij) is called astochastic matrixifeach of its rows is a probability vector, , if each entry ofPis nonnegativeand the sum of the entries in each row is 1. LetAandBbe two stochasticn nmatrices. Is the Matrix productABalso a stochastic Matrix ?Problem ann nmatrix overC. Thefield of valuesofAis defined as the setF(A) :={z Az:z Cn,z z= 1}.Let RandA= 100000001 100000001 100000001 100000001 100000001 100000001 100000001 100000001 .(i) Show that the setF(A) lies on the real axis.(ii) Show that|z Az| + ann nmatrix overCandF(A) the field ofvalues. LetUbe ann nunitary Matrix .(i) Show thatF(U AU) =F(A).(ii) Apply the theorem to the two matricesA1=(0 11 0), A2=(100 1)Basic Operations5which are unitarily one find a unitary matrixUsuch thatU (0cd0)U=(0cei de i 0)wherec,d Cand R?Problem a symmetric matrixAoverRA= a11a12a13a14a12a22a23a24a13a23a33a34a14a 24a34a44 and the orthonormal basis (so-calledBell basis)x+=1 2 1001 ,x =1 2 100 1 y+=1 2 0110 ,y =1 2 01 10.

7 The Bell basis forms an orthonormal basis inR4. Let Adenote the matrixAin the Bell basis. What is the condition on the entriesaijsuch that thematrixAis diagonal in the Bell basis?Problem matrixis ann nmatrixHwith entriesin{ 1,+1}such that any two distinct rows or columns ofHhave innerproduct 0. Construct a 4 4 Hadamard Matrix starting from the columnvectorx1= (1 1 1 1) Hadamard matrixis ann nmatrixM(wherenis even) with entries in{0,1}such that any two distinct rows or columnsofMhaveHamming distancen/2. The Hamming distance between twovectors is the number of entries at which they differ. Find a 4 4 binaryHadamard and SolutionsProblem a normalized column vector inRn, matrixTis called aHouseholder matrixifT:=In nmatrixPis aprojection matrixifP =P, P2=P.(i) LetP1andP2be projection matrices . IsP1+P2a projection Matrix ?(ii) LetP1andP2be projection matrices .

8 IsP1P2a projection Matrix ?(iii) LetPbe a projection Matrix . IsIn Pa projection Matrix ? CalculateP(In P).(iv) IsP=13 1 1 11 1 11 1 1 a projection Matrix ?Problem thatA=A1+iA2is a nonsingularn nmatrix, whereA1andA2are realn thatA1is also nonsingular. Find the inverse ofAusing the nmatrices overR. Assume thatA6=B,A3=B3andA2B=B2A. IsA2+B2invertible?Problem a positive definiten nmatrix overR. Letx thatA+xxTis also positive ,Bben nmatrices overC. The matrixAis calledsimilarto the matrixBif there is ann ninvertible matrixSsuch thatA=S similar toB, thenBis also similar toA, sinceB=SAS 1.(i) Consider the two matricesA=(1 02 1), B=(1 00 1).Basic Operations7 Are the matrices similar?(ii) Consider the two matricesC=(100 1), D=(0 11 0).Are the matrices similar?Problem the vector inR2v=( 1 + sin( ) 1 sin( )).Then find a normalized vector inR2which is orthonormal to this ann nmatrix overCwithA3=A.

9 AssumethatAis invertible.(i) Show thatA 1=A.(ii) Show that (A+In)(A In) = 0n.(iii) Show that rank(A) = tr(A2).Problem ann nmatrix overC. Show thattr(A A) = 0implies thatA= (cyclic) 4 4 permutation matrixC= 0 1 0 00 0 1 00 0 0 11 0 0 0 andJbe the counter diagonal identity Matrix . Show thatCJCT=(0 11 0) (0 11 0)where is the direct sum andCT=C R. Consider the matricesK( ) = 1 + 1 + 1 + , S= 100 1100 1 1 .8 Problems and Cn(column vector) andv6=0. Is =In 1v vv v a projection Matrix ?Problem all 3 3 invertible matricesAoverRsuch thatA 110 = 011 .Chapter 2 Linear EquationsProblem (112 1),b=(15).Find the Solutions of the system of linear equationsAx= (1 12 2),b=(3 )where R. What is the condition on so that there is a solution of theequationAx=b?Problem 3.(i) Find all Solutions of the system of linear equations (cos( ) sin( ) sin( ) cos( ))(x1x2)=(x1x2), R.

10 (ii) What type of equation is this?Problem Rn nandx,b Rn. Consider the linear equationAx=b. Show that it can be written asx=Tx, , the system of linear equationsAx=badmits no solutionwe call the equations inconsistent. If there is a solution, the equations are910 Problems and Solutionscalled consistent. LetAx=bbe a system ofmlinear equations innunknowns and suppose that the rank ofAism. Show that in this caseAx=bis the overdetermined linear systemAx=b. Findan xsuch that A x b 2= minx Ax b 2 minx r(x) 2with theresidual vectorr(x) :=b Axand . 2denotes the that solving the system of nonlinear equations withthe unknownsx1,x2,x3,x4(x1 1)2+ (x2 2)2+x23=a2(x4 b1)2(x1 2)2+x22+ (x3 2)2=a2(x4 b2)2(x1 1)2+ (x2 1)2+ (x3 1)2=a2(x4 b3)2(x1 2)2+ (x2 1)2+x23=a2(x4 b4)2leads to a linear underdetermined system. Solve this system with respecttox1, anm nmatrix overR.