A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula aay of objects (calle enties) Those enties ae usually numbes, but they can also inclue functions, vectos, o even othe matices Each enty s position is aesse by the ow an column (in that oe) whee it is locate Fo example, a epesents the enty positione at the th ow an the n column of the matix A The size of a matix The size of a matix is specifie by numbes [numbe of ows] [numbe of columns] Theefoe, an m n matix is a matix that contains m ows an n columns A matix that has equal numbe of ows an columns is calle a squae matix A squae matix of size n n is usually efee to simply as a squae matix of size (o oe) n Notice that if the numbe of ows o columns is, the esult (espectively, a n, o an m matix) is just a vecto A n matix is calle a ow vecto, an an m matix is calle a column vecto Theefoe, vectos ae eally just special types of matices Hence, you will pobably notice the similaities between many of the matix opeations efine below an vecto opeations that you might be familia with 8, Zachay S Tseng D- - 6
Two special types of matices Ientity matices (squae matices only) The n n ientity matix is often enote by I n I, I, etc Popeties (assume A an I ae of the same size): AI IA A I n x x, x any n vecto Zeo matices matices that contain all-zeo enties Popeties: A A A A A Aithmetic opeations of matices (i) Aition / subtaction a c b ± e g f h a± e c± g b± f ± h 8, Zachay S Tseng D- - 7
(ii) Scala Multiplication a b ka kb k, fo any scala k c kc k (iii) Matix multiplication a c b e g f h ae bg ce g af cf bh h The matix multiplication AB C is efine only if thee ae as many ows in B as thee ae columns in A Fo example, when A is m k an B is k n The pouct matix C is going to be of size m n, an whose ij-th enty, c ij, is equal to the vecto ot pouct between the i- th ow of A an the j-th column of B Since vectos ae matices, we can also multiply togethe a matix an a vecto, assuming the above estiction on thei sizes is met The pouct of a matix an a - enty column vecto is a c b x y ax by cx y Note : Two squae matices of the same size can always be multiplie togethe Because, obviously, having the same numbe of ows an columns, they satisfy the size equiement outline above Note : In geneal, AB BA Inee, epening on the sizes of A an B, one pouct might not even be efine while the othe pouct is 8, Zachay S Tseng D- - 8
Deteminant (squae matices only) Fo a matix, its eteminant is given by the fomula et a c b a bc Note: The eteminant is a function whose omain is the set of all squae matices of a cetain size, an whose ange is the set of all eal (o complex) numbes 6 Invese matix (of a squae matix) Given an n n squae matix A, if thee exists a matix B (necessaily of the same size) such that AB BA I n, then the matix B is calle the invese matix of A, enote A The invese matix, if it exists, is unique fo each A A matix is calle invetible if it has an invese matix Theoem: Fo any matix A its invese, if exists, is given by a c b, A a bc c b a Theoem: A squae matix is invetible if an only if its eteminant is nonzeo 8, Zachay S Tseng D- - 9
8, Zachay S Tseng D- - Examples: Let A an B (i) A B ) ( ) ( (ii) AB 7 8 8 8 On the othe han: BA 9 8 6 (iii) et(a) (), et(b) 8 Since neithe is zeo, as a esult, they ae both invetible matices (iv) A / / 6 / 6 / ) (
7 Systems of linea equations (also known as linea systems) A system of linea (algebaic) equations, Ax b, coul have zeo, exactly one, o infinitely many solutions (Recall that each linea equation has a line as its gaph A solution of a linea system is a common intesection point of all the equations gaphs an thee ae only ways a set of lines coul intesect) If the vecto b on the ight-han sie is the zeo vecto, then the system is calle homogeneous A homogeneous linea system always has a solution, namely the all-zeo solution (that is, the oigin) This solution is calle the tivial solution of the system Theefoe, a homogeneous linea system Ax coul have eithe exactly one solution, o infinitely many solutions Thee is no othe possibility, since it always has, at least, the tivial solution If such a system has n equations an exactly the same numbe of unknowns, then the numbe of solution(s) the system has can be etemine, without having to solve the system, by the eteminant of its coefficient matix: Theoem: If A is an n n matix, then the homogeneous linea system Ax has exactly one solution (the tivial solution) if an only if A is invetible (that is, it has a nonzeo eteminant) It will have infinitely many solutions (the tivial solution, plus infinitely many nonzeo solutions) if A is not invetible (equivalently, has zeo eteminant) 8, Zachay S Tseng D- -
8 Eigenvalues an Eigenvectos Given a squae matix A, suppose thee ae a constant an a nonzeo vecto x such that Ax x, then is calle an Eigenvalue of A, an x is an Eigenvecto of A coesponing to Do eigenvalues/vectos always exist fo any given squae matix? The answe is yes How o we fin them, then? Rewite the above equation, we get Ax x The next step woul be to facto out x But oing so woul give the expession (A ) x Notice that it equies us to subtact a numbe fom an n n matix That s an unefine opeation Hence, we nee to futhe efine it by ewiting the tem x I x, an then factoing out x, obtaining (A I) x This is an n n system of homogeneous linea (algebaic) equations, whee the coefficient matix is (A I) We ae looking fo a nonzeo solution x of this system Hence, by the theoem we have just seen, the necessay an sufficient conition fo the existence of such a nonzeo solution, which will become an eigenvecto of A, is that the coefficient matix (A I) must have zeo eteminant Set its eteminant to zeo an what we get is a egee n polynomial equation in tems of The case of a matix is as follow: A I a c b a c b 8, Zachay S Tseng D- -
Its eteminant, set to, yiels the equation et a c b ( a )( ) bc ( a ) ( a bc) It is a egee polynomial equation of, as you can see This polynomial on the left is calle the chaacteistic polynomial of the (oiginal) matix A, an the equation is the chaacteistic equation of A The oot(s) of the chaacteistic polynomial ae the eigenvalues of A Since any egee n polynomial always has n oots (eal an/o complex; not necessaily istinct), any n n matix always has at least one, an up to n iffeent eigenvalues Once we have foun the eigenvalue(s) of the given matix, we put each specific eigenvalue back into the linea system (A I) x to fin the coesponing eigenvectos 8, Zachay S Tseng D- -
8, Zachay S Tseng D- - Examples: A A I Its chaacteistic equation is 6) )( ( 6 ) )( ( et The eigenvalues ae, theefoe, an 6 Next, we will substitute each of the eigenvalues into the matix equation (A I) x Fo, the system of linea equations is (A I) x (A I) x x x Notice that the matix equation epesents a egeneate system of linea equations Both equations ae constant multiples of the equation x x Thee is now only equation fo the unknowns, theefoe, thee ae infinitely many possible solutions This is always the case when solving fo eigenvectos Necessaily, thee ae infinitely many eigenvectos coesponing to each eigenvalue
Solving the equation x x, we get the elation x x Hence, the eigenvectos coesponing to ae all nonzeo multiples of k Similaly, fo 6, the system of equations is 6 (A I) x (A 6 I) x x x 6 Both equations in this secon linea system ae equivalent to x x Its solutions ae given by the elation x x Hence, the eigenvectos coesponing to 6 ae all nonzeo multiples of k Note: Evey nonzeo multiple of an eigenvecto is also an eigenvecto 8, Zachay S Tseng D- -
Two shot-cuts to fin eigenvalues: If A is a iagonal o tiangula matix, that is, if it has the fom a a b a, o, o c Then the eigenvalues ae just the main iagonal enties, a an in all examples above If A is any matix, then its chaacteistic equation is et a c b ( a ) ( a bc) If you ae familia with teminology of linea algeba, the chaacteistic equation can be memoize athe easily as Tace(A) et(a) Note: Fo any squae matix A, Tace(A) [sum of all enties on the main iagonal (unning fom top-left to bottom-ight)] Fo a matix A, Tace(A) a 8, Zachay S Tseng D- - 6
A shot-cut to fin eigenvectos (of a matix): Similaly, thee is a tick that enables us to fin the eigenvectos of any matix without having to go though the whole pocess of solving systems of linea equations This shot-cut is especially hany when the eigenvalues ae complex numbes, since it avois the nee to solve the linea equations which will have complex numbe coefficients (Waning: This metho oes not wok fo any matix of size lage than ) We fist fin the eigenvalue(s) an then wite own, fo each eigenvalue, the matix (A I) as usual Then we take any ow of (A I) that is not consiste of entiely zeo enties, say it is the ow vecto (α, β) We put a minus sign in font of one of the enties, fo example, (α, β) Then an eigenvecto of the matix A is foun by switching the two enties in the above vecto, that is, k (β, α) Example: Peviously, we have seen A The chaacteistic equation is Tace(A) et(a) 6 ( )( 6), which has oots an 6 Fo, the matix (A I) is Take the fist ow, (, ), which is a non-zeo vecto; put a minus sign to the fist enty to get (, ); then switch the enty, we now have k (, ) It is inee an eigenvecto, since it is a nonzeo constant multiple of the vecto we foun ealie On vey ae occasions, both ows of the matix (A I) have all zeo enties If so, the above algoithm will not be able to fin an eigenvecto Instea, une this cicumstance any non-zeo vecto will be an eigenvecto 8, Zachay S Tseng D- - 7
Execises D-: Let C 7 an D Compute: (i) C D an (ii) C D Compute: (i) CD an (ii) DC Compute: (i) et(c), (ii) et(d), (iii) et(cd), (iv) et(dc) Compute: (i) C, (ii) D, (iii) (CD), (iv) show (CD) D C Fin the eigenvalues an thei coesponing eigenvectos of C an D Answes D-: (i), (ii) 8 (i), (ii) 8 (i) 8, (ii), (iii) 6, (iv) 6 /8 /8 / /6 /6 (i), (ii) 7 /8 /8, (iii), (iv) The / / equality is not a coincience In geneal, fo any pai of invetible matices C an D, (CD) D C s s (i), k ;, k 7s ; s any nonzeo numbe s s (ii), k ;, k s / ; s any nonzeo numbe s 8, Zachay S Tseng D- - 8