EE448/528 Version 1.0 John Stensby. This matrix has m rows and n columns. Often, we use the notation {a ij } to denote a matrix.

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1 EE448/528 Vesion.0 John Stensby Chapte 3: atices - Elementay Theoy A matix is a ectangula aay of scalas. A = a a2 an a2 a22 a2n am am2 amn (3-) This matix has m ows and n columns. ften, we use the notation {a ij } to denote a matix. Repesenting a inea Tansfomation by a atix As given in Chapte 2, the definition of a linea tansfomation T : U V is somewhat abstact. nce bases fo spaces U and V ae specified, a matix can be used to povide a definitive epesentation fo a linea tansfomation. et U and V be vecto spaces, ove the same field F, of dimension n and m, espectively. et α, α 2,..., α n be a basis of U, and β, β 2,..., β m be a basis of V. et T : U V be a linea tansfomation. In tems of these bases, we can wite T( m α j) = aij βi, j n. (3-2) i= This equation defines an m n matix A that descibes T : U V with espect to the given bases. As shown below, this matix can be used as a epesentation fo tansfomation T. ften, we use the symbolic equation T α α α β β β 2 n = 2 m A (3-3) instead of the algebaic equation (3-2). The quantity α α α 2 n with columns made fom the basis vectos α, α 2,..., α n. is itself an n n matix CH3.DC age 3-

2 EE448/528 Vesion.0 John Stensby Given any X U, the matix A can be used to compute Y = T(X ) by using the coodinate vectos that epesent X and Y with espect the bases. Fo X U, we can wite n X = x j α j, (3-4) j= whee x i, i n, ae the coodinates fo vecto X. ow, use (3-4) to compute n n Y T( X) T xj α jj = xj T( α j) (3-5) j= j= F = = H G I K Use (3-2) in (3-5) to obtain F HG I KJ = F H I K n m m n m Y = x j aij βi G aij x jj βi = y i βi, (3-6) j= i= i= j= i= whee n yi = a ij xj, i m, (3-7) j= ae the coodinates of Y. The algebaic deivation of (3-7) has a somewhat symbolic countepat. Fist, ecall that X can be epesented symbolically as CH3.DC age 3-2

3 EE448/528 Vesion.0 John Stensby x X = α α α x2 2 n, (3-8) whee [x x 2... x n ] T is a coodinate vecto fo the vecto X (emembe: coodinates ae in italics and vecto components ae not italicized). ote that the x k ae just scalas, and apply T to (3-8) to obtain x TdXi = T x α α α 2 2 n (3-9) ow, use (3-3) in (3-9) to obtain x y T x dxi = A 2 [ β β β m ] m = y2 2 [ β β2 β ] yn, (3-0) whee y x a a2 an x y2 x2 a2 a22 a2 x n 2 A = yn am am2 amn = (3-) CH3.DC age 3-3

4 EE448/528 Vesion.0 John Stensby is the desied elationship between the coodinate vectos used to epesent X and Y. Equations (3-7) and (3-) state equivalent esults: the ows of m n matix A ae used to expess the coodinate vecto fo Y in tems of the coodinate vecto fo X. Example et U = V = R 2. et α = [ 0] T and α 2 = [0 ] T be a T( X) X common basis fo U, V. Define T as the tansfomation that otates vectos counteclockwise by π/2 adians. π/2 ote that T does not change the magnitude of the vecto. bseve that T F HG I T 0 KJ = 0 = HG KJ = and 0 F I = so that A = 0 0 is a matix epesentation fo the otation tansfomation. As shown above, thee is a well defined elationship between a linea tansfomation and the matix used to epesent the tansfomation. Hence, almost all of the definitions, given in Chapte 2, dealing with linea tansfomations have countepats when dealing with matices. Fo example, let T : U V be epesented by matix A. Then the vectos in R(T) ae epesented by coodinate vectos in the subspace CH3.DC age 3-4

5 EE448/528 Vesion.0 John Stensby y x R( A) = y 2 x 2 = A whee [ x x x ] 2 n T is a coodinate vecto in U (3-2) ym xm Clealy, R(A) is the span of the columns of A. ikewise, the vectos in K(T) ae epesented by coodinate vectos in the subspace K( A ) = 0 0 x x 0 x2 x2 : A = (3-3) atices: Some Special Kinds In ou wok (and most applications), the elements of a matix belong to the field of eal numbes, denoted as R, o the field of complex numbes, denoted as C. If A is an m n matix ove the eal numbes we say A R m n. ikewise, if A is an m n matix ove the complex numbes we say A C m n. An n n matix R n n is said to be othogonal if T = - so that T = - = I, the n n identity matix. The columns (and ows) of ae othogonal. et = q q q 2 n, (3-4) then is othogonal if and only if qi, q j = q j T qi = i = j = 0 i j (3-5) CH3.DC age 3-5

6 EE448/528 Vesion.0 John Stensby An n n matix U C n n is said to be unitay if U * = U - so that U * U = U - U = I, the n n identity matix (hee, the * denotes complex conjugate tanspose). The columns (and ows) of U ae othogonal. et U = u u u 2 n, (3-6) then U is unitay if and only if ui, u j = u j ui = i = j = 0 i j (3-7) A squae matix A R n n is said to be symmetic if A T = A. Symmetic matices show up in many applications. An impotant attibute of symmetic matices is that they always have ealvalued eigenvalues. Also, they ae othogonal simila to a diagonal matix containing the eigenvalues of A. That is, thee always exists an othogonal matix R n n (i.e., T = o T = I) such that - A = T A = D, whee D R n n is a diagonal matix containing the eigenvalues of A. The columns of ae the eigenvectos of A, and they ae othonomal. nly a few applications deal with complex-valued, symmetic matices (As a geneal obsevation, linea algeba books do not conside complex-valued symmetic matices). Howeve, fo the complex numbe field, the countepat of the symmetic matix is the Hemitian matix. A squae matix A C n n is said to be Hemitian if A * = A. Hemitian matices show up in many applications. An impotant attibute of Hemitian matices is that they always have ealvalued eigenvalues. Also, they ae unitay simila to a diagonal matix containing the eigenvalues of A. That is, thee always exists an unitay matix U C n n (U * = U o U T U = I) such that U - AU = U * AU = D, whee D C n n is a diagonal matix containing the eigenvalues of A. The columns of U ae the eigenvectos of A, and they ae othonomal. CH3.DC age 3-6

7 EE448/528 Vesion.0 John Stensby Vecto/Tansfomation Repesentation With Respect to thonomal Basis et α i, i n, be an othonomal basis of vecto space U. Then αi, α j = α j αi = i = j = 0 i j (3-8) Theoem 3. et X U be abitay. Then we can make the epesentation n X = X, α α + X, α2 α2 + + X, αn αn = X, αk αk k= n oof: X X, αk αk, α j = X, α j X, α j = 0 fo j n. k= Since X n and X,α k α k have equal coodinates, they must be the same vecto. k= This theoem gives us a convenient and useful epesentation fo an abitay vecto. Also, the poof illustates an impotant technique fo showing that two vectos ae equal. That is, to show that two vectos ae equal, it is sufficient to show that they have the same coodinates (given an undelying basis). Theoem 3.2 et T : U V. et α j, j n, and β i, i m, be othonomal bases on U and V, espectively. et m n matix A = {a ij } epesent T with espect to these bases. Then aij = T( α j), βi i m, j n, (3-9) oof: By Theoem 3. we have CH3.DC age 3-7

8 EE448/528 Vesion.0 John Stensby T( α) = T( α), β β + T( α), β2 β2 + + T( α), βm βm T( α2) = T( α2), β β + T( α2), β2 β2 + + T( α2), βm βm (3-20) T( αn ) = T( αn ), β β + T( αn ), β2 β2 + + T( αn ), βm βm which can be ewitten as T α α α β β β 2 n = 2 m T( α), β T( α2), β T( αn ), β T( α), β2 T( α2), β2 T( αn ), β2 T( α), βm T( α2), βm T( αn ), βm (3-2) Hence, Equation (3-9) epesents T as claimed. Rank and ullity of a atix The ank and nullity of matix {a ij } ae the ank and nullity of the linea tansfomation that the matix epesents. The ank of T : U V is the dimension of subspace T(U). ow, let α, α 2,..., α n be a basis fo U. Then subspace T(U) is spanned by the n vectos T(α ), T(α 2),..., T(α n). Hence, Rank(T) is the maximum numbe of linealy independent elements in T(α), T(α2),..., T(α n). But note that T α α α β β β 2 n = 2 m A (3-22) implies CH3.DC age 3-8

9 EE448/528 Vesion.0 John Stensby T( α β β β ) = 2 m T( α ) β β β 2 = 2 m T( α ) β β β 3 = 2 m a a2 am a2 a22 am2 a3 a23 am3 an T( α ) β β β a n n = 2 2 m amn. (3-23) bseve that T(α 2) is independent of T(α ) if and only if [a 2... a m2 ] T is independent of [a... a m ] T. And, T(α 3) is independent of T(α ) and T(α 2) if and only if [a 3... a m3 ] T is independent of [a... a m ] T and [a 2... a m2 ] T. Continue this agument to its conclusion: T(αn) is independent of T(α ), T(α 2),..., T(α n-) if and only if [a n... a mn ] T is independent of [a... a m ] T, [a 2... a m2 ] T,..., [a n-... a mn- ] T, the fist n- columns of m n matix A. Hence, Rank(T) is equal to the maximum numbe of linealy independent columns of matix A (also, we know that column ank is equal to ow ank so that Rank(T) is equal to the maximum numbe of linealy independent ows of matix A). Finally, we say that m n matix A has full ank if ank[a] = min(m, n). CH3.DC age 3-9

10 EE448/528 Vesion.0 John Stensby on-singula atices Conside the impotant case whee n-dimensional U = V so that T : U U. Simple, intuitive aguments can be given fo ) T : U U is one-to-one and onto if and only if Rank(T) = n, which is equivalent to 2) T : U U is one-to-one and onto if and only if ullity(t) = 0. If Rank(T) = n, then the n columns of A ae independent, Rank(A) = n, and A is said to be nonsingula. In this case, the matix A - exists, and it epesents the linea opeato T -. Change of Basis - Impact on Vecto Repesentation With espect to an abitay but fixed basis, we have epesented both vectos and linea tansfomations as n-tuples and matixes, espectively. Howeve, the epesentations ae entiely dependent on the chosen basis. The vectos and tansfomations have meaning independent of any paticula choice of basis. et α, α 2,..., α n be a basis of n-dimensional U. With espect to this basis, an X U can be epesented as n X = xi αi, (3-24) i= o x X = α α α x2 2 n, (3-25) wee coodinates x i, i n, depend both on the vecto X and the basis. ow, let us change the basis fo U, let α, α 2,..., α n be the "new" basis fo vecto space U. Futhemoe the "old" basis and the "new" basis ae elated by CH3.DC age 3-0

11 EE448/528 Vesion.0 John Stensby n α j = p ijαi, j n, (3-26) i= which is given symbolically as α α α = α α α 2 n 2 n, (3-27) whee = {p ij } is an n n matix called the matix of tansition fom the "old" basis α, α 2,..., α n to the "new" basis α, α 2,..., α n. The columns of ae n-tuples epesenting the "new" basis in tems of the "old" basis. Hence the columns of ae independent, and is non-singula. ow, let x i, i n, be the "new" coodinates of X. We use (3-27) to wite x x x X = 2 n n = x α α α α α α (3-28) ow, compae the last poduct on the ight hand side of (3-28) with (3-25) to obtain x x2 = x x2. (3-29) Hence, fo k n, the k th ow of elates the "new" and old coodinates x k and x k, espectively. Change of Basis on Space U - Impact on atix Repesentation fo T : U V et α, α 2,..., α n and β, β 2,..., β m be bases of n-dimensional U and m-dimensional V, espectively. With espect to these bases, let T : U V be epesented by m n matix A so that CH3.DC age 3-

12 EE448/528 Vesion.0 John Stensby T α α2 αn = β β2 β m A. (3-30) ow, suppose we want to change the basis of U to α, α 2,..., α n, the "new" basis fo U (the basis fo V emains unchanged). The "new" basis fo U is elated to the "old" basis fo U by (3-26) and (3-27). et m n A epesent T with espect to the bases α, α 2,..., α n and β, β 2,..., β m. Then, T α = α2 αn β β2 βm A. (3-3) ow, use (3-27) in (3-3) to obtain T α α α β β β 2 n = 2 m A. (3-32) n the ight, multiply this esult by - to obtain T α α α β β β 2 n = 2 m A. (3-33) Compae (3-30) and (3-33) and conclude that A = A - A = A. (3-34) Change of Basis on Space V - Impact on atix Repesentation fo T : U V Conside changing the basis fo the co-domain vecto space V. Again, m n matix A epesents T with espect to the "old" bases α, α 2,..., α n and β, β 2,..., β m. et β, β 2,..., β m be a "new" basis fo space V, and let CH3.DC age 3-2

13 EE448/528 Vesion.0 John Stensby m β j = q ijβi, j m, (3-35) i= β β 2 β m = β β2 β m, (3-36) whee m m, nonsingula matix = {q ij } is the matix of tansition that elates the "old" and "new" basis fo V. et m n matix A epesent T with espect to "old" α, α 2,..., α n and "new" β, β 2,..., β m so that T α α α β β β 2 n = 2 m A. (3-37) ow, use (3-36) in (3-37) to obtain T α α α β β β 2 n = 2 m A. (3-38) ow, compae (3-38) and (3-30) to obtain A = A A = - A (3-39) Simultaneous Change of Basis on U and V - Impact on Repesentation fo T : U V Again, m n matix A epesents T with espect to the "old" bases α, α 2,..., α n and β, β 2,..., β m. et α, α 2,..., α n and β, β 2,..., β m be "new" bases fo spaces U and V, espectively. Also, let and be matices of tansition as descibed by (3-26)/(3-27) and (3-35) /(3-36), espectively. et A be the matix epesenting T with espect to the new bases on U and V. Combine the esults leading to (3-34) and (3-39) to conclude that CH3.DC age 3-3

14 EE448/528 Vesion.0 John Stensby A = - A. (3-40) The case U = V and T : U U Finally, we conside the special case whee n-dimensional U = V and T : U U. et n n matix A epesent T with espect to "old" basis α, α 2,..., α n. Then the matix epesenting T with espect to the new basis α, α 2,..., α n is A = - A A = A. (3-4) We say that - A is a similaity tansfomation applied to A and that A and A ae simila. Change of bases and similaity tansfomations have many applications in engineeing and the physical sciences. et the n n matix of tansition be epesented in column fom as = p p p 2 n, (3-42) whee p k is the k th column of. Then A = A can be witten as p p p A A p p p 2 n = 2 n, (3-43) which implies that the i th column of A is just the epesentation of Ap i with espect to the columns p, p 2,..., p n. et a ij denote the elements of A. Then we have a j p + a 2j p2 + + a nj pn = Ap j, j n, (3-44) which is equivalent to CH3.DC age 3-4

15 EE448/528 Vesion.0 John Stensby n a kj pk = A p j, j n. (3-45) k= Example et A = , = We want to compute A = - A. Computing the invese - is too much wok - don't do it!. Instead, use A = A to wite p p2 p3 A = A p p2 p A = which leads to A = by inspection! In most cases, the method illustated by the pevious example is much easie than computing the invese - and then the poduct - A. CH3.DC age 3-5