Applications in Linear Algebra and Uses of Technology

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1 1 TI-89: Let A Applicatios i Liear Algebra ad Uses of Techology,adB type i: [1,,;4,5,6;7,8,10] press: STO type i: A type i: [4,-1;-1,4] press: STO (1) Row Echelo Form: MATH/matrix (4)/ref (A) Row Reduced Echelo Form: MATH/matrix (4)/rref (A) MATH/matrix (4)/ref (A): MATH/matrix (4)/ref (A): rak(a) () Determiat: MATH/matrix (4)/det (A) press: MATH/matrix/det( type i: det(a) Aswer is () Iverse: A^ 1 type i: A^(-1) Aswer is (4) Eigevalues ad Eigevectors: MATH/matrix (4)/eigV1(B), MATH/matrix (4)/eigVc(B) MATH/matrix (4)/eigV1(B):, MATH/matrix (4)/eigVc(B): Scietific Notebook: The software is istalled o the PCs i Labs i Grimsley Hall ad Lab 15 i Thompso Hall a Typig i Scietific Notebook: Click o T/M for text/math, click o N x for subscript ad N x for superscript Iitialize 4 matrix: click o T/M if it is ot i M, type i A, click o the ico ()[] ad the ext oe (for matrix), type i elemets of A

2 A b Fuctios i Matrix Computatio: Click o Compute adtheclickomatrices All built-i fuctios i matrix computatio are listedthere Sofar,wehavestudied: determiat, Gaussia Elimiatio, iverse, rak, traspose, eigevalues ad eigevectors The computatios of these fuctios ca be doe by clickig o these fuctios The computatios of some of these fuctios ca also doe bu typig i: deta, A 1, A T Example Let A Use the Scietific Notebook to fid A T, deta, rak of A, A 1, Gaussia Elimiatio ad eigevalues ad eigevectors To defie A : highlight o A , ad click o the ico fx " To compute the determiat of A; type deta ad highlight deta adtheclicko?", we have: To compute the rak of A : highlight o A, click o Compute, Matrix, Rak, we have: rak: To compute the iverse of A : type A 1 ad highlight it ad the click o?", we have 4 1 A Gaussia Elimiatio: highlight o A,, click o Compute, Matrix, ad Gaussia Elimiatio, we have Gaussia elimiatio: MatLab: The software is also istalled o the PCs i Labs i Grimsley Hall ad Lab 15 i Thompso Hall a Typig i Matrices i MatLab: 1 4 Type i the matrix A A[14;8765;910111] after :

3 If the ed with ;, the A is ot displayed o the scree, otherwise, A is displayed A ca also be iitialized as follows A[ ] b Fuctios i Matrix Computatio: MatLab has may built-i fuctios Type help matlab\matfu to see a list of fuctios Here is a list of MatLab fuctios for determiat, Gaussia Elimiatio, iverse, rak, traspose, eigevalues ad eigevectors: determiat det(a) Gaussia Elimiatio rref(a) iverse iv(a) ora^(-1) rak rak(a) traspose A eigevalues oly eig(a) eigevalues ad eigevectors: [VD]eig(A) Example Let A Elimiatio ad eigevalues ad eigevectors A[1;456;7810]; A as det(a) as - rak(a) as iv(a) as eig(a) as [V D]eig(A) V Use MatLab to fid A, deta, raka, A 1, Gaussia

4 D rref(a) as B[1;456;789];b[1;;]; rref([b b]) as Solvig Systems of Liear Equatio: Cosider solvig systems of liear equatios: Ax b If A is a osigular (square) matrix, the we kow x A 1 b If A is a square matrix ad the system Ax b is cosistet, the the geeral solutio ca be solved from the reduced augmeted matrix A b Scietific Notebook solves both types of systems ad provides either a uique solutio or the geeral solutio Whe A is osigular, MatLab computes the uique solutio x A 1 b Whe A is rectagular ad the system is cosistet, we ca have a reduced form for the augmeted matrix A b from MatLab ad the compute the geeral solutio ourselves Example Fid the solutio of the system liear equatios: x 1 x x 1 4x 1 x x 1 x 1 x x Scietific Notebook: Highlight o the equatiosclick o Compute, Solve, Exact Solutio is: x 18 1,x 1,x 1 1 So, the uique solutio is x MatLab: A[-1;4-1;1-];b[1;-1;] xiv(a)*b x Traffic Flow Problems: Compute traffic flows: x 1, x, x, x 4,adx 5

5 5 ukows: x 1,,x 5 ad the relatio of these ukows is: Itersectio Flow I Flow Out A x 1 x B x x 4 00 x C x 4 x 5 D x 1 x The system for solvig the flow patter for the etwork: x 1 x 800 x x x 4 00 x 4 x x 1 x Scietific Notebook: Highlight o the equatiosclick o Compute, Solve, Exact Because it is a system of 4 equatios i 5 ukows, it is cosistet (by atural) with ifiitely may solutios We will be asked to iput ukows that we wat to solve for Iput x 1, x, x, x 4 (use the ico N x to iput a subscript) We have the geeral solutio of the system: Solutios : x 4 x 5 500,x 400,x x 5 00,x 1 x That is, t 600 x t t 500 t, t is ay real umber

6 MatLab: x 1 x 800 x x x 4 00 x 4 x x 1 x A[11000;01-110;00011;10001];b[800;00;500;600]; rref([a b]) as t 00 t Solve the geeral solutio: x t t 6 Codig ad Decodig: A commo way of sedig a coded message is to assig a iteger value to each letter of the alphabet ad to sed the message as a strig of itegers Let us assig 0 to a space, 1-6 to a-z, 7 to, 8 to ad 9 to! space a b c d e f g h i j k l m o p q r s t u v w x y z,! Example message: solutios of hw six have bee posted o the course web Covert each letter to a assiged umber ad list all umbers i the order of words as a matrix: the coded message Read umbers from the top to the bottom ad from the left to the right Disguise the message further (the message to be set): further coded message A the coded message, wherea is a matrix ad is the umbers i each colum For this example, the matrix A is 5 5 Coditios o the matrix A : a I order to decode the coded message from the further coded message, the matrix A is ivertible The

7 the coded message A 1 the further coded message b The matrix A is desiged so that elemets of A 1 are itegers How ca we desig a matrix A that satisfies both coditio? Recall: A 1 AdjA deta Observe that if all elemets of A are itegers ad deta 1ordetA 1, the A 1 AdjA or A 1 AdjA whose elemets are itegers Now the questio is how to geerate a matrix whose elemets are all iteger ad whose determiat is 1 or -1 Observe also that: i If the matrix C is obtaied from B by a elemetary row operatio Type III, the detb detc ii If the matrix C is obtaied from B by a elemetary row operatio Type I, the detb detc So if we start with a matrix B whose determiat is 1 the a matrix A ca be obtaied by carryig out several elemetary operatios Type I ad III For example, Scietific Notebook: R 1 R R R R 1 R R 1 R R R R R B R 1 R 4 R R R 4 R R 1 R 5 R R R 5 R R R 1 R R 4 R 1 R R R R R 4 R R R R 4 R R 4 R 4 R R R 5 R R 4 R 5 R R 5 R 1 R R 5 R R R 5 R R R 5 R 4 R A , A MatLab:

8 Beye(5); for i:5, B(i,:)i*B(1,:)B(i,:); ed for i:5, for j1:5, if j~i, B(j,:)B(i,:)B(j,:); ed if ed ed AB; The further coded message A the coded message Check: A 1 C C Least Squares Problems (Liear Regressio): k 1adP 1 x a 0 a 1 x Ea 0,a 1 y i a 0 a 1 x i i0

9 Note that Ea 0,a 1 is a quadratic fuctio i a 0 ad a 1 ad is cocave up So Ea 0,a 1 attais its miimum value at the critical poit a 0,a 1 which is the solutio of the followig system of two equatios: Rewrite the equatios as follows: E a 0 E a 1 i0 y i a 0 a 1 x i 1 0 i0 y i a 0 a 1 x i x i 0 y 0 y 1 y 1a 0 a 1 x 0 x 1 x 0 y 0 x 0 y 1 x 1 y x a 0 x 0 x 1 x a 1 x 0 x 1 x 0 which are equivalet to the followig equatios: y 0 y 1 y 1a 0 a 1 x 0 x 1 x y 0 x 0 y 1 x 1 y x a 0 x 0 x 1 x a 1 x 0 x 1 x Express above system i matrix-vector otatio: 1 x 0 x 1 x a 0 y 0 y 1 y (1) x 0 x 1 x x 0 x 1 x a 1 y 0 x 0 y 1 x 1 y x ad the solutio is of the form: () a 0 a 1 1 x 0 x 1 x x 0 x 1 x x 0 x 1 x 1 y 0 y 1 y y 0 x 0 y 1 x 1 y x 1 i0 x i i0 x i i0 x i 1 i0 y i i0 x i y i Note that the a c a 0 b d 1 1 ad bc () 1 a 1 1 i0 x i Observe that d c b a 1 x i i0 i0 x i if ad bc 0 Hece, if x i i0 i0 0, x i i0 x i i0 x i 1 i0 y i i0 x i y i 1 x 0 1 x 0 x 1 x x 0 x 1 x x 0 x 1 x x 0 x 1 x 1 x 1 ad 1 x

10 y 0 y 0 y 1 y y 0 x 0 y 1 x 1 y x x 0 x 1 x y 1 y Defie 1 x 0 y 0 A 1 x 1, y y 1 ad a a 0 a 1 1 x The the system of equatios i (1) is equivalet to the form: A T Aa A T y ad the solutio i () is equivalet to the form: (4) a A T A 1 A T y The miimum approximatio error for the liear regressio: E mi a 0,a 1 y i a 0 a 1 x i i0 Steps to compute the vector a (1) Compute c 1 i0 x i, c i0 a 0 a 1 i (): y x i, b 1 i0 y i ad b i0 x i y i () Compute a 0 a 1 A T A 1 A T y 1 1c c 1 c c 1 c 1 1 b 1 b Example Let fx x 1 Give 0,1,,,8, Fid P 1 x a 0 a 1 x Solutio: (1) c 1 i0 y i 1 6, i0 a 0 a 1 x i , c x i , i x i y i i , P 1 x x E mi a 0,a

11 y y x 1, y P 1 x, x i,y i 8 Cotrollability: (EE Systems II (Digital Cotrol Theory)) Let A be a matrix, B be a 1 vector Cosider the system which geerates the k 1th state vector xk 1 xk 1 Axk Buk where the sequece of costats: u0, u1,, ad the iitial vector x0 are give Observe that x1 Ax0 Bu0 x Ax1 Bu1 AAx0 Bu0 Bu1 A x0 ABu0 Bu1 x A x0 A Bu0 ABu1 Bu Asystemissaidtobecotrollable if it is possible to force the system from ay iitial state, x0, to a arbitrary fial state, xf i a fiite umber of steps Let C B, AB, A B,, A 1 B C is called a cotrollability matrix The system is cotrollable if rakc x Example Let A, B, u0, u1 1 ad x0 0 Cosider the system: xk 1 Axk Buk a Compute x1 ad x b Determie if the system is cotrollable 1 Scietific Notebook: a i

12 x x ii Defie C B, AB Sice rakc, the system is cotrollable MatLab: A[0 1;0 ]; B[0;]; x[-1;]; u[;-1] xa*xb*u(1) xa*xb*u() rak(c[b A*B]);