4 10/22

1. New: 8.7.8 Old: 7.7.8. Using minors, unless you used minors in 7.8.6/6.9.6, then using elimination.

2. New: 9.1.4, 9.1.6 Old: 8.1.4, 8.1.6. Find a complete set of eigenvectors. No Gerschgorin. State whether singular and/or defective.

3. New: 9.1.14 Old: 8.1.14 Find a complete set of eigenvectors. No Gerschgorin. State whether singular and/or defective.

4. New: 9.1.4, 9.1.6 Old: 8.1.4, 8.1.6. Redux. Check that $E^{-1}AE$ is indeed $\Lambda$ for the eigenvalues and eigenvectors you found. If not, explain why not.

5. New: 9.2.5 Old: 8.2.5 Verify by multiplication that $E^{-1}AE$ is indeed $\Lambda$.

6. New: 9.2.11 Old: 8.2.13 This is another of these questions the students can easily answer. First, for the matrix $A$ of 9.1.6/8.1.6, find $A^2$ and comment on the problem. Then prove, for any matrix $A$, that if $A$ is diagonalizable, $A^2$ is diagonalizable. Hint: find eigenvectors and eigenvalues of $A^2$ in terms of those of $A$.

7. New: 9.2.12 Old: 8.2.14. Note that if $E^{-1}AE=\Lambda$, then $A=E\Lambda E^{-1}$. Then show that if the theorem is true for any value $k$, including $k=1$, it is true for the next larger value of $k$, implying the theorem by recursion. Use the theorem to find a square root of the matrix of 9.2.5/8.2.5, i.e. a matrix $A$ whose square is the matrix of question 9.2.5/8.2.5. Indicate $\sqrt{-1}=i$.