1. Eigenvectors and Linear Transformations Recall the definition of similar matrices: Let A and C be n  n 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 are essentially the same when viewed from the proper perspective. Of course, this is a huge breakthrough since the mapping 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.

2. 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.

3. The Matrix of a Linear Transformation wrt Given Bases Let V and W be n-dimensional and m-dimensional vector spaces, respectively. Let T:V  W be a linear transformation. Let B = {b 1, b 2,..., b n } and B' = {c 1, c 2,..., c m } be ordered bases for V and W, respectively. Then M is the matrix representation of T relative to these bases where Example. Let B be the standard basis for R 2, and let B' be the basis for R 2 given by If T is rotation by 45º counterclockwise, what is M?

4. 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 Example. Let T: be defined by This is the _____________ operator. Let B = B' = {1, t, t 2, t 3 }.

5. Similarity of two matrix representations: Here, the basis B of is formed from the columns of P. Multiplication by A Multiplication by C Multiplication by P –1 Multiplication by P

6. A linear operator: geometric description Let T: be defined as follows: T(x) is the reflection of x in the line y = x. x y x T(x)

7. Standard matrix representation of T and its eigenvalues Since T(e 1 ) = e 2 and T(e 2 ) = e 1, the standard matrix representation A of T is given by: The eigenvalues of A are solutions of: We have The eigenvalues of A are: +1 and –1.

8. A basis of eigenvectors of A Let Since Au = u and Av = –v, it follows that B ={u, v} is a basis for consisting of eigenvectors of A. The matrix representation of T with respect to basis B:

9. Similarity of two matrix representations The change-of-coordinates matrix from B to the standard basis is P where Note that P -1 = P T and that the columns of P are u and v. Next, That is,

10. A particular choice of input vector w Let w be the vector with E coordinates given by x y u T(w) v w x y

11. Transforming the chosen vector w by T Let w be the vector chosen on the previous slide. We have The transformation w T(w) can be written as Note that

12. 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