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Eigenvectors and Linear Transformations • Recall the definition of similar matrices: Let A and C be nn matrices. We say that A is similar to C in case A = PCP-1 for some invertible matrix P. • A square matrix A is diagonalizable if A is similar to a diagonal matrix D. • An important idea of this section is to see that the mappings x Ax and w Dw are essentially the same when viewed from the proper perspective. Of course, this is a huge breakthrough since the mapping w Dw is quite simple and easy to understand. In some cases, we may have to settle for a matrix C which is simple, but not diagonal. Similarity Invariants for Similar Matrices A and C Property Description Determinant A and C have the same determinant Invertibility A is invertible <=> C is invertible Rank A and C have the same rank Nullity A and C have the same nullity Trace A and C have the same trace Characteristic Polynomial A and C have the same char. polynomial Eigenvalues A and C have the same eigenvalues Eigenspace dimension If is an eigenvalue of A and C, then the eigenspace of A corresponding to and the eigenspace of C corresponding to have the same dimension. The Matrix of a Linear Transformation wrt Given Bases • Let V and W be n-dimensional and m-dimensional vector spaces, respectively. Let T:VW be a linear transformation. Let B = {b1, b2, ..., bn} and B' = {c1, c2, ..., cm} be ordered bases for V and W, respectively. Then M is the matrix representation of T relative to these bases where T( x)B' MxB , and M T(b1 )B' T(b2 )B' T( bn )B' . • Example. Let B be the standard basis for R2, and let B' be the basis for R2 given by c1 2 2 2 2 22 , c 2 2 . 2 If T is rotation by 45º counterclockwise, what is M? Linear Transformations from V into V • In the case which often happens when W is the same as V and B' is the same as B, the matrix M is called the matrix for T relative to B or simply, the B-matrix for T and this matrix is denoted by [T]B. Thus, we have T(x)B TB xB . • Example. Let T: P3 P3 be defined by T(a 0 a 1 t a 2 t a 3 t ) a 1 2a 2 t 3a 3 t . 2 3 1 2 This is the _____________ operator. Let B = B' = {1, t, t2, t3}. ?? ?? [T] B ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? . ?? ?? Similarity of two matrix representations: A PCP 1 x Multiplication by A Multiplication by P–1 xB Ax Multiplication by P Multiplication by C AxB Here, the basis B of R n is formed from the columns of P. A linear operator: geometric description • Let T: R 2 R 2 be defined as follows: T(x) is the reflection of x in the line y = x. y T(x) x x Standard matrix representation of T and its eigenvalues • Since T(e1) = e2 and T(e2) = e1, the standard matrix representation A of T is given by: 0 1 A [T] E . 1 0 • The eigenvalues of A are solutions of: det(A I) 0. • We have det 1 1 2 1 0. • The eigenvalues of A are: +1 and –1. A basis of eigenvectors of A • Let 2 2 2 2 u , v . 2 2 2 2 • Since Au = u and Av = –v, it follows that B ={u, v} is a 2 basis for R consisting of eigenvectors of A. • The matrix representation of T with respect to basis B: 1 0 D [T] B . 0 1 Similarity of two matrix representations • The change-of-coordinates matrix from B to the standard basis is P where 2 2 2 2 P . 2 2 2 2 • Note that P-1= PT and that the columns of P are u and v. • Next, 2 2 2 2 1 0 2 2 PDP 2 2 0 1 2 2 2 2 -1 • That is, P[T] B P 1 TE . 2 2 0 1 . 2 2 1 0 A particular choice of input vector w • Let w be the vector with E coordinates given by 0 w [w ]E . 2 y x y w v u T(w) x Transforming the chosen vector w by T • Let w be the vector chosen on the previous slide. We have 1 [w ] B . 1 • The transformation w T(w) can be written as P 1 D P 2 0 1 1 w T( w ) 2 alias 1 alibi 1 alias 0 • Note that 1 [T( w )]B . 1 What can we do if a given matrix A is not diagonalizable? • Instead of looking for a diagonal matrix which is similar to A, we can look for some other simple type of matrix which is similar to A. • For example, we can consider a type of upper triangular matrix known as a Jordan form (see other textbooks for more information about Jordan forms). • If Section 5.5 were being covered, we would look for a matrix of the form a b b a .