4 Linear algebra IV

1. New: 8.5.6 Old: 7.5.6 Use row operations ONLY to reduce to upper triangular form!

2. New: 8.7.8 Old: 7.7.8. Use minors to do so.

3. New: 9.1.4, 9.1.6 Old: 8.1.4, 8.1.6. Find a complete set of independent eigenvectors for each eigenvalue. Make sure to write the null space for any multiple eigenvalues. No Gerschgorin. State whether singular and/or defective.

4. New: 9.1.14 Old: 8.1.14 Find a complete set of independent eigenvectors for each eigenvalue. Make sure to write the null space for any multiple eigenvalues. No Gerschgorin. Explain whether singular and/or defective or not.

5. New: 9.1.4, 9.1.6 Old: 8.1.4, 8.1.6. Refer to your previous work on these matrices. Check that $E^{-1}AE$ is indeed $\Lambda$ for the eigenvalues and eigenvectors you found. If not, explain why not.

6. New: 9.2.11 Old: 8.2.13. First, ensure that the book knows what it is talking about by taking the matrix of 9.1.6/8.1.6,
   
   $$A = \left( \begin{array}{cc} 0 & 1 0 & 0 \end{array} \right),$$
   
   
   and showing that $A^2$ can be diagonalized. Then check that $A$ can indeed be diagonalized as the book says. If you find that the author knows what he is talking about, prove that the theorem is true for any arbitrary matrix $A$. If you find that the author has no clue, then prove, for every matrix $A$, that if $A$ is diagonalizable, $A^2$ is diagonalizable. Hint: relate the eigenvectors and eigenvalues of $A^2$ to those of $A$.

7. Use the theorem given in New: 9.2.12 / Old: 8.2.14 to find a square root of the matrix of 9.2.5/8.2.5. That means you should find a matrix $A$ so that $A^2$ is the matrix of question 9.2.5/8.2.5. Indicate $\sqrt{-1}=i$. Verify by multiplication that indeed $A^2$ gives the given matrix.

   Show that if you simply take the square root of each coefficient in the original matrix, the square matrix does not give the original matrix.

   Note: the eigenvalues and eigenvectors of the 9.2.5/8.2.5 matrix are:
   

   $$\lambda_1 = 0 \quad \vec e_1 = \left(\begin{array}{r} 0 ... ...ft(\begin{array}{r} 0 -3 2 \end{array}\right) \qquad$$