Sample Quiz #6

MA 2071

Diagonalization

1. a. What is the definition of eigenvector of a linear transformation?

b. For the linear transformation L(y) = d²y/dx² which of the following are eigenvectors? For those that are, what is their eigenvalue??

i) y = e^3x ii) y= x³ iii) y= cos(5x) iv) y = ln(x)

2. For the given matrix, find its eigenvalues and an eigenvector for each

3. For a given eigenvalue of a matrix, why is the corresponding eigenvector not unique? Feel free to use your work from Problem #2 to illustrate the point.

4. The following matrix has eigenvalues computed to be 1,1 and 4(note correction from earlier versison which had a 1 in the 3,3 entry)

a) diagonalize it

b) use your result in part a) to find the 2^nd and 5^th and powers of it

c) what is the determinant of A based upon your Diagonalization?? (this is easy! use things you've learned about determinants)

d) What is the inverse of A based upon your Diagonalization ?

5. Prove that the eigenvalues of an upper triangular matrix are the entries on the diagonal of it (OK to use a

3x3 for purposes of a proof).

6. Prove: if A has no inverse then 0 is an eigenvalue of A