Eigenvectors and Linear Transformations

1. Eigenvectors and Linear Transformations • Recall the definition of similar matrices: Let A and C be nn matrices. We say that Ais 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 PropertyDescription 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 = {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 • Example. Let B be the standard basis for R2, and let B' be the basis for R2 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, t2, t3}.

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

7. 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: • 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= PT 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 y x y w v u x T(w)

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