Advanced Topics in Digital Communications Spezielle Methoden der digitalen Datenübertragung

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1 Advaced Topics i Digital Commuicatios Spezielle Methode der digitale Dateübertragug Dr.-Ig. Dirk Wübbe Istitute for Telecommuicatios ad igh-frequecy Techiques Departmet of Commuicatios Egieerig Room: N300, Phoe: 041/ wuebbe@at.ui-breme.de Lecture Thursday, 08:30 10:00 i S170 Eercise Wedesday, 14:00 16:00 i N40 Dates for eercises will be aouced durig lectures. Tutor Yidog Lag Room: N350 Phoe lag@at.ui-breme.de

2 Aim of Course ad Requiremets Bridgig the gap betwee courses ad theses Course focuses o state-of-the-art topics beig subect of curret research Iteractive eercises Eecuted i small groups Solve little problems with Matlab autoomously Presetatio ad discussio durig eercises Requiremets for course attedace (recommeded) Commuicatios Techology I + II Chael Codig I + II Basics of Digital Sigal Processig I Outlie

3 Outlie Eigevalues ad eigevectors, pseudo iverse Decompositios (QR, uitary matrices, sigular value, Cholesky ) Part : Basics ad Prelimiaries Motivatig systems with Multiple Iputs ad Multiple Outputs (multiple access techiques) Geeral classificatio ad descriptio of MIMO systems (SIMO, MISO, MIMO) Mobile Radio Chael Part 3: Iformatio Theory for MIMO Systems Repetitio of IT basics, chael capacity for SISO AWGN chael Etesio to SISO fadig chaels Geeralizatio for the MIMO case Part 4: Multiple Atea Systems SIMO: diversity gai, beamformig at receiver MISO: space-time codig, beamformig at trasmitter MIMO: BLAST with detectio strategies Ifluece of chael (correlatio) Part 5: Relayig Systems Basic relayig structures Relayig protocols ad eemplary cofiguratios Outlie 3

4 Liear Algebra Notatios ad defiitios Vectors ad matrices Special Matrices Elemetary operatios, Matri multiplicatio, Traspose, ermitia Traspose Determiats, Vector ad Matri orm Liear combiatio (rage, ull space) Liear equatio systems Cramer s rule, Gaussia elimiatio, iterative methods Iverse matri, matri iversio lemma, iverse of a block matri Matri factorizatios LU, Cholesky, QR (Gram-Schmidt, ouseholder, Gives) Eigevalues ad eigevectors Sigular Value Decompositio SVD (pseudo-iverse, coditio umber) Least squares 4

5 Notatios ad Defiitios (1) Vectors Colum vectors (preferred): boldface lower case Row vectors: uderlied boldface lower case 1 1 Matrices Boldface capital letters (m matri) a1,1 a1, a1, a1 a,1 a, a, a A a1 a a am,1 am, a m, am Colum vectors are ust m 1matrices Row vectors are ust 1 matrices 5

6 Notatios ad Defiitios () Some special matrices Idetity matri ad zero matri I Diagoal, lower ad upper triagular matrices d , d 0 D,1, 0 L 0 0 d,1,, U Eplicit dimesios: I m : m m idetity matri 0 m, : m zero matri u u u 0 u u 0 0 u 1,1 1, 1,,,, 6

7 Let A, B, C be m matrices ad, be scalars Additio ad scalar multiplicatio are defied elemet-wise Properties Basic Operatios ad Properties a1,1 b1,1 a1, b1, AB am,1 bm,1 am, b m, a1,1 a1, A am,1 a m, A BBA Additio is commutative A BCABC Additio is associative A0A Neutral elemet of additio AA 0 Iverse elemet of additio A A Scalar multiplicatio is associative 1A A Neutral elemet of scalar multiplicatio AAA Scalar multiplicatio is distributive AB AB Scalar multiplicatio is distributive 7

8 Let A be a m matri ad B be a p matri The product C=AB is a m p matri with elemets row times colum Matri Multiplicatio (1) a1,1 a1, a1 b1,1 b 1, p b1 A a1 a B b1 b p am,1 a m, am b,1 b, p b Note: umber of colums of A has to equal umber of rows of B Equivalet formulatios of the matri multiplicatio: c a b a b i, i i, k k, k 1 a1, k bk,1 a1, k bk, p k1 k1 ab 1 1 ab 1 p ab 1 C Ab Ab a b 1 p k k k 1 m 1 m p m amk, bk,1 amk, b ab ab ab k, p k1 k1 8

9 Matri Multiplicatio () Special cases m = 1, > 1, p = 1 (row vector times colum vector) cab k 1 a b k k m = 1, > 1, p > 1 (row vector times matri) k 1 c ab a b k k m > 1, > 1, p = 1 (matri times colum vector) k 1 cab a k b k scalar row vector colum vector m > 1, = 1, p > 1 (colum vector times row vector) a1,1b1,1 a1,1b 1, p Cab matri am,1b1,1 am,1 b 1, p Ier or scalar product Matri-vector products Outer or dyadic product 9

10 Properties A B CACBC Matri multiplicatio is distributive AB CABAC Matri multiplicatio is distributive ABAB AB Mied scalar / matri multiplicatio is associative AB C A BC Matri multiplicatio is associative Note: matri multiplicatio is ot commutative i geeral Eample 6 A 1 7 B Matri Multiplicatio (3) C AB 11 6 BA 5 19 AB BA AC 146 CA 146 AC CA 10

11 Traspose of a matri Traspose ad ermitia Traspose T a1,1 a1, a1 a1,1 am,1 a 1 T T T A a1 a A a1 am T am,1 a m, am a1, a m, a Row vectors become colum vectors ad vice versa ermitia traspose of a comple matri * * * * a1,1 a 1, a1,1 a m,1 a 1 T * A A a1 a m * * * * am,1 a m, a1, a m, a Traspose of the comple cougate matri Properties T A T A A A T T T A B A B ad AB A B AB B A AB B A T T T T 11

12 Determiat of a matri det A a1,1 a1, A a a a a a a,1, Determiat of a 3 3 matri (Sarrus rule) det A a a a 1,1 1, 1,3 Determiat of a matri Determiats (1) 1,1,,1 1, a a 1,1 1, a a a a a a a a a a a a a a,1,,3,1, 1,1, 3,3 1,,3 3,1 1,3,1 3, a a a a a a a a a a a a a 3,1 3, 3,3 3,1 3, 3,1, 1,3 3,,3 1,1 3,3, 1 1, Let the (-1) (-1) matri A i, equal A without the i-th row ad -th colum Recursive defiitio of determiat by cofactor epasio colum epasio row epasio A i, : mior matri i i A ai, A i, det A( 1) ai, det Ai, i1 1 det ( 1) det a det A i, : mior 1

13 Determiats () Fudametal properties Liearity i colums (rows) Echagig two colums (rows) Determiat of idetity matri a a a a a a a a a1 a1 a det I 1 Some additioal properties Symmetry i colums ad rows Zero colum (row) Two equal colums (rows) Multiple of oe colum (row) Scalar multiplicatio Addig two colums (rows) Determiat of matri product det A 0 a det T det A 0 1 a1 0 a a a a a 1 1 a1 a a det A det AB det A det B All properties valid for arbitrary matrices A det A 13

14 Determiat of diagoal or triagular matri At least oe factor is zero for all det D d i1 ii, Determiats (3) det L ii, det U uii, i1 i1 Efficiet calculatio of determiat Determiat uaffected by addig multiples of rows (colums) to rows (colums) Trasform A ito triagular matri by elemetary row (colum) operatios Practical meaig of the determiat If det A = 0 the matri A is sigular det A equals volume of parallelepiped with edges give by rows (colums) of A Gives formulas for the pivots used for solvig liear equatio systems 14

15 Trace ad diag operatio tr A a, i1 Vector ad Matri Norm Vector orm ( -orm, Euclidia legth) Matri orm ( -orm, spectral orm) Frobeius orm ii diag A a a 1,1, diag * i i i i1 i1 tr A A sup A A A F 0 m m tr AA a i, i1 1 i1 i 1 sup ma A 1 mi A A A i : sigular value of A ma (A) : largest sigular value of A mi (A) : smallest sigular value of A 15

16 Liear Combiatio Matri A m describes liear mappig of vector oto vector y m AA A m A: A Vector y is give by liear combiatio of the colum vectors a i y Aa11a am aii Importat subspaces Liearity: i1 Rage (spa, image): Subspace cosistig of all liear combiatios of a 1,, a is called the subspace spaed by A A spa A y y A, If the colums of A are liear idepedet, they form a basis of the spaed space Null space (kerel): The ull space cosists of all vectors such that A = 0 Aker A A0, a a 1 a 1a 1 y 16

17 Liear Combiatio Eample 1 3 y1 a1,1 y y a a y a, ,1 a 1 y y Lie i 3 Eample 3 a1,1 a1, 1 a,1 a, 11 a3,1 a 3, y a a a a 1 y a 1 a Plae i 3 Eample 3 3 a1,1 a1, a1,3 1 y a a a a a a a a a,1,, ,1 3, 3,3 3 a 3 a 1 a 17

18 System of m liear equatios i ukows Matri-vector otatio Geometric iterpretatios Liear Equatio Systems (1) a a a b 1,1 1 1, 1, 1 a a a b,1 1,, a a a b m,1 1 m, m, m a1,1 a1, a1, 1 b1 a,1 a, a, b am,1 am, a m, bm is the itersectio of m hyperplaes A b a b b is a liear combiatio of the colum vectors i Eteded coefficiet matri [A b] i a a a b a a a b a a a b 1,1 1, 1, 1,1,, m,1 m, m, m iai i1 b 18

19 Liear Equatio Systems () Illustratio for system (hyperplaes straight lies) itersectig straight lies parallel straight lies idetical straight lies a 1 b1 a b a 1 b1 a b a 1 b1 a b a 1, a liearly idepedet a 1, a parallel a 1, a, b parallel a a b b b a a 1 a1 1 a 1a 1 a uique solutio o solutio ifiite umber of solutios 19

20 Liear Equatio Systems (3) Elemetary operatios that result i equivalet liear equatio systems Iterchage two colums Multiply a equatio by a ozero scalar Add a costat multiple of oe equatio to aother As equatios correspod to rows of the eteded coefficiet matri [A b], the elemetary operatios are performed o the rows of this matri Apply elemetary operatios to solve task Apply operatios to the rows of the eteded coefficiet matri [A b] to simplify the calculatio of the solutio Calculatio of the iverse by Gauss-Jorda method Cholesky ad QR decompositio of matrices 0

21 Square liear equatio system A = b with equatios i ukows Cramer s rule Eample for =5 ad =3 det A Let A equal A with the -th colum 3 a1 a b a4 a5 5 replaced by b a a 1 iai a a i A a1 a 1 b a 1 a a1 a 3a3 a4 a5 The the -th elemet of is 3 a1 a a3 a4 a5 det A 3 deta det A Proof: substitute ito A ad use liearity i colums Three possibilities Liear Equatio Systems (4) b iai i1 det A 0 uique solutio det A 0 ad det A 0 for some o solutio det A 0 ad det A 0 for all ifiite umber of solutios

22 Gaussia Elimiatio (1) Eample: 33 system (1) Elimiatio Subtractig multiples of rows to create zeros Trasform system ito upper triagular form () Back-substitutio Solve for ukows Computatio i reverse order l a,1,1 / a1,1 3,1 3,1 / a1,1 l a l a / a (1) (1) 3, 3,, Pivot elemets Reduced systems a a a 1,1,1 3,1 1, 1,3 1,,3 3, 3,3 3 1,1 1, 1,3 1 (1) (1) (1) a, a,3 b (1) (1) (1) a3, a3,3 b , 1 1, 1,3 1 (1) (1) (1) a, a,3 b () () a3, 3 b a a b a a b a a b a a a b a a a b b / a () () 3 3 3,3 b a / a (1) (1) (1),3 3, b a a / a 1 1 1, 1,3 3 1,1 Etesio to (1): If ( 1) If a k, 0 for some k > echage rows ( 1) If a k, 0 for all k > move to et colum ( 1) a, 0

23 Special cases All diagoal elemets ozero 0 uique solutio 0 0 Gaussia Elimiatio () Zero row i coefficiet matri, correspodig right had side ozero o solutio Zero rows i coefficiet matri, correspodig right had sides zero ifiite umber of solutios Free parameters 3

24 Gaussia Elimiatio (3) Geeral formulatio of the algorithm (1) Iitializatio ad elimiatio (0) (0) A : A, b : b for : 1to m1do fid pivot elemet a for i: 1to m do ed ed l a / a ( 1) ( 1) i, i,, ( 1), for k: 1to do ed a a l a ( ) ( 1) ( 1) ik, ik, i, k, b b l b ( ) ( 1) ( 1) i i i, Pivot search : ide of first ozero colum if o the r: 1, break echage rows, so that Back-substitutio choose values for free parameters for : r dowto 1do b a ed ( 1) a, ( 1) ( 1), k k ( 1) k 1 a, 0 1 4

25 Result after Elimiatio Step Number of ozero rows o left had side: Rak of Matri A (umber of liear idepedet equatios) ( r) ( r) r m r m ad b b Solutio eists oly if or Gaussia Elimiatio (4) a * * * * 0 0 a * * (0) (0) 1, (1) (1) a, * * * b ( r1) ( r1) r, r r ( r) br 1 Uique solutio if r o free parameters Ifiite umber of solutios if r r free parameters b b b ( r ) m r m r r1 m 0 rak A r 5

26 Liear equatio system A b Basic idea of iterative algorithms (0) Start with iitial estimate of the solutio vector ( 1) Fid improved approimatio k from previous approimatio Stop after covergece Jacobi Iterative Solutio of Liear Equatio Systems Solve row i for ukow i Parallel implemetatio possible Gauss-Seidel Use already updated values Better covergece behavior tha Jacobi No parallel implemetatio possible Cougate Gradiet a b for 1i i, i More complicated implemetatio, but usually fast covergece 1 i1 ( k 1) ( k ) ( k ) 1 i bi ai, ai, 1 i1 ai, i i1 ( k 1) ( k 1) ( k ) 1 i bi ai, ai, 1 i1 ai, i ( k ) 6

27 Iverse Matri (1) Iverse A -1 of a square matri A 1 1 A A AA I Relatio of iverse to liear equatio systems 1 A b A b Calculatio of the iverse by Gauss-Jorda method simultaeous liear equatio systems A1 A AX I A I 1 Forward elimiatio A I U L 1 1 Backward elimiatio U L I A Iverse eists oly if AX = I has a uique solutio ( A osigular) Coditio: Properties 1 A 1 AB 1 A B A A A rak 1 A det A 0 7

28 Iverse Matri () Matri Iversio Lemma (A m m, B m, C, D m ) A BCD A A B C DA B DA A A B I CDA B CDA Iverse of block matri E: A B E with A m m, B m, C m, D C D E E F F BD DCF D D CF BD A A BG CA A BG G CA G Schur complemet of A w.r.t E 1 F A BD C Schur complemet of D w.r.t E 1 G D CA B 8

29 LU Decompositio Every ivertible matri A ca be writte as the product of a lower triagular matri L ad a upper triagular matri U Applicatio: Solutio of liear equatio system A LU b with costat coefficiet matri A for differet right had sides Iversio of triagular matrices easy solve Ly b ad the U y Calculatio of LU decompositio by Gaussia elimiatio 1 Forward elimiatio: ALU I U L ( 1) ( 1) L cotais factors l a / a from the elimiatio steps i, i,, Direct calculatio of LU decompositio (eample: 33 matri) a1,1 a1, a1, r1,1 r1, r1,3 r1,1 r1, r1,3 a,1 a, a,3 l, r, r,3 l,1r1,1 l,1r1, r, l,1r1,3 r,3 a3,1 a3, a 3,3 l3,1 l3, r 3,3 l3,1r1,1 l3,1r1, l3,r, l3,1r1,3 l3,r,3 r 3,3 Calculatio order: r r r l l r r l r 1,1 1, 1,3,1 3,1,,3 3, 3,3 A LU 9

30 Cholesky Decompositio Let A be hermitia A A positive defiite A0 the A is fully characterized by a lower triagular matri L Cholesky decompositio A LL Similar to LU decompositio But computatioal compleity reduced by factor Eample: 3 3 matri * * l1,1 l1,1l,1 l1,1l 3,1 a1,1 a1, a1,3 * * * a,1 a, a,3 l,1l1,1 l,1 l, l,1l3,1 l,l 3, a * * * 3,1 a3, a 3,3 l3,1l1,1 l3,1l,1 l3,l, l3,1 l3, l3,3 Calculatio order: l1,1 l,1 l3,1 l, l3, l3,3 Algorithm (0) A : A for k: 1to do l a ( k 1) kk, kk, for i: k 1to do ed ed ed l a / l ( k 1) * ik, ik, kk, for : k 1to i do a a l l ( k) ( k1) * i, i, i, k, k 30

31 Every m matri A ca be writte as A QR where Q is a m matri with orthoormal colums q q i 1 0 for i for i R is a upper triagular matri Colums of A are represeted i the orthoormal base defied by Q a k r k i, k i i1 Illustratio for the m case q a a q q 1 1 QR Decompositio (1) r1,1 r1, 0 r, r q r q r q 1,1 1 1, 1, Q Q q r,q I q 1 a r q r q 1, 1, r1,q1 a r q 1 1,1 1 31

32 Calculatio of QR decompositio by modified Gram-Schmidt algorithm Calculate legth (Euclidea orm) of a 1 r 1,1 Normalize a 1 to have uit legth q 1 Proectio of a,...,a oto q 1 r 1, (1) Subtract compoets of a,...,a parallel to q 1 q Cotiue with et colum Q is computed colum by colum from left to right R is computed row by row from top to bottom Illustratio for the m case r1,1 r1, 0 r, r q r q r q a a q q 1 1 QR Decompositio () 1,1 1 1, 1, q r,q for k: 1to do r a kk, q / r k k k, k for i: k 1to do r ed ed q1 r1,q1 a k q a ki, k i a a r q i i k, i k a r q r q 1, 1, a r q 1 1,1 1 3

33 ouseholder reflectio for real valued sigals Reflectio of vector across the plai surface whose ormal vector is u ( u =1) is achieved by orthoormal matri u Θ Iuu Reflected vector with y = ouseholder matri is symmetric (Q = Q T ) ad orthogoal (Q -1 = Q T ) Reflectio ito specific directio T y Θuu T u Reflected vector should cotai oly oe o-dimiishig elemet reflectio creates -1 elemets equal to zero T Θ Iuu with Applicatio with respect to matri A QR Decompositio (3) u u y y T u u 1 y Θ y * * * a1 * * ΘA Θ * * * 0 * * * * * 0 * * u u 0 33

34 QR Decompositio (4) ouseholder reflectios for comple valued sigals Θ I(1 w) uu with u y ad w u y u y 0 Special case: create zeros i a vector Applicatio to QR decompositio of m matri A R: A, Q: Im for k: 1to do R( k: m, k) T y [ 0] calculate u, w, Θ R( k: m, k: ) Θ R( k: m, k: ) T Iitializatio Loop through all colums Create zeros below the mai diagoal i k-th colum of R u y uu uu y ed Q(:, k: m) Q(:, k: m) Θ Update uitary matri Q 34

35 (0) (0) (0) a1,1 a1, a 1,3 (0) (0) (0) A a,1 a, a1,3 Q0 R 0 (0) (0) (0) a3,1 a3, a1,3 * * * a a a QR Decompositio (5) (1) (1) (1) 1,1 1, 1,3 (1) (1) * * * 0 a, a1,3 1 1 (1) (1) * * * 0 a3, a1,3 Q Θ Θ R Q R * * * a a a QR decompositio of 3 3 matri A (1) () () 1,1 1, 1, () () 1 1 * * * 0 a, a 1,3 0 0 Θ Θ () * * * 0 0 a 1,3 Q R Q R * * * (1) () (3) 1,1 1, 1,3 I 0 I 0 ( ) (3) * * * 0 a, a 1, (3) * * * 0 0 a 13, Q R Q R a a Step 0: Iitializatio of Q ad R a Step 1: Create zeros i first colum of R Step : Create zeros i secod colum of R Step 3: Create real-valued lower right elemet i R 35

36 Gives rotatios G(, ik, ) G(, ik, ) Let equal a idetity matri ecept for is uitary ad describes a rotatio Special choices for c ad s: c Liear trasformatio y G(, ik, ) QR Decompositio (6) g g cosc * ii, kk, g g si s * ik, ki, i k ad s i k i k Gives rotatio ca create zero while chagig oly oe other elemet Eample * * * * * * * * * * * * (,3, 1) (1,, ) (,3, 3 * * * G * * * G 0 * * G A ) 0 * * R * * * 0 * * 0 * * 0 0 * y, y 0, y i, k i i k k R G(,3, 3) G(1,, ) G(,3, 1) A QG(,3, 1) G(1,, ) G(,3, 3) 36

37 Eample for Gives Rotatio Applicatio of ratio matri to vector 4 y G(,4, ) c 0 s c s 4 c s s 0 c 4 s c4 s c4 0 with c s s 4 4 c

38 Special eigevalue problem for arbitrary matrices A A I0 Coditio for eistece of otrivial solutios 0 Characteristic polyomial of degree has to be zero Zeros i of polyomial are the eigevalues of A with algebraic multiplicity k i Eigevectors Solve liear equatio systems i i for all eigevalues i Dimesio of solutio space is called geometric multiplicity g i (1 g i k i ) Eigevectors belogig to differet eigevalues are liearly idepedet Diagoalizatio of a matri A Eigevalues ad Eigevectors (1) 1 k k l A A I 1 l p ( ) det 0 Defie the matri X = [ 1 ] ad the diagoal matri = diag( 1,, ) 1 AX XΛ X AXΛ Oly possible for liearly idepedet eigevectors A I 0 38

39 Eigevalues ad Eigevectors () Some useful geeral properties A A T i * i A, i i A ivertible all A positive defiite all i 0 0 Properties for hermitia matrices, i.e. A = A A All eigevalues are real Eigevectors belogig to differet eigevalues are orthogoal Algebraic ad geometric multiplicities are idetical Cosequece: all eigevectors ca be chose to be mutually orthogoal A hermitia matri A ca be diagoalized by a uitary matri V V AV Λ AVΛV m 1 X AX i m i A I, i i,, i i 1 X i det A trace A i1 i1 Eigevalue decompositio i i 39

40 Sigular Value Decompositio (SVD) (1) Every m matri A of rak r ca be writte as Σ0 0 AUΣV U V 0 0 Sigular values i of A = square roots of ozero eigevalues of A A or AA Uitary m m matri U cotais left sigular vectors of A = eigevectors of AA Uitary matri V cotais right sigular vectors of A = eigevectors of A A Verificatio with eigevalue decompositio Σ0 0 Σ0 0 A AVΣ U UΣV V V AA UΣV VΣU U U Four fudametal subspaces: the vectors u 1,...,u r spa the colum space of A u r+1,...,u m spa the left ullspace of A v 1,...,v r spa the row space of A v r+1,...,v spa the right ullspace of A with the matri of sigular values orthogoal orthogoal Σ diag(,, r ) diag( S,, S ) r 40

41 Sigular Value Decompositio (SVD) () Illustratio of the fudametal subspaces Cosider liear mappig A with orthogoal decompositio r rowspace r columspace A A r 0 A 0 right ullspace left ullspace 41

42 Pseudo Iverse ad Least Squares Solutio (1) Iverse A -1 eists oly for square matrices with full rak Geeralizatio: (Moore-Perose) pseudo iverse A + 1 Σ0 0 Σ0 0 AUΣV U V A VΣ U V U Special cases for full rak matrices 1 for rak m A AA A A spa{a} 1 A A A for raka Applicatio: Least squares solutio of a liear equatio system Problem: fid vector that miimizes the euclidea distace betwee A ad b Solutio: proect b oto the colum space of A ad solve A=b c If o uique solutio eists take solutio vector with shortest legth mi A b A b b c b e 4

43 Pseudo Iverse ad Least Squares Solutio () Illustratio of the least squares solutio of a liear equatio system rowspace A b columspace b c A Ab 0 b 0 right ullspace b left ullspace 43

44 Coditio Number Coditio umber is a idicator for the orthogoality of a matri A cod Solutio of the liear equatio system b=a is give by =A -1 b The coditio umber cod(a) describes the impact of a error b i the observatio data b (e.g. measuremet errors, oise, ) to the solutio Usig two estimatios the followig relatio is achieved 1 ma A mi A A A A 1 cod A 1 A b b 1 error A b 1 1 A b A b b / mi A b A b A ma A Relative error A b b cod A b b ma A mi cod (A) =1 for uitary matri b b ma A A mi Eample: cod(a)=100 ad b/b= 0.1% / = =10% 44

45 Olie: Selected Literature Gutkecht: Lieare Algebra effero: Elemetary Liear Algebra Matthews: Elemetary Liear Algebra Wedderbur: Lectures o matrices The Matri Cookbook Prited: B. Bradie: A Friedly Itroductio to Numerical Aalysis, Pearso 006 G. Strag: Liear Algebra ad its Applicatios, ardcourt 1988 Johso, Riess, Arold: Itroductio to Liear Algebra, Addiso Wesley 00 K. ardy: Liear Algebra for Egieers ad Scietists usig Matlab, Pearso