# Worksheet: Matrix Diagonalization

In this worksheet, we will practice determining whether a given matrix is diagonalizable or not, how to diagonalize it, and how to use that to find high powers of the matrix.

**Q2: **

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

- AThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.
- BThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.
- CThe eigenvalues are 10 with corresponding eigenvector , and 1 with corresponding eigenvector . The matrix is defective.
- DThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.
- EThe eigenvalues are 10 with corresponding eigenvector , and 1 with corresponding eigenvector . The matrix is defective.

**Q3: **

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

- AThe eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and with corresponding eigenvector . The matrix is defective.
- BThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is not defective.
- CThe eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and with corresponding eigenvector . The matrix is not defective.
- DThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is defective.
- EThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is not defective.

**Q4: **

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

- AThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
- BThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
- CThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
- DThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
- EThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .

**Q5: **

- AThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
- BThe eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
- CThe eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
- DThe eigenvalues are 2 with corresponding eigenvector and 1 with corresponding eigenvector .
- EThe eigenvalues are 2 with corresponding eigenvector and 1 with corresponding eigenvector .

**Q6: **

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

- AA basis for the eigenspace associated with the eigenvalue 2 is , a basis for the eigenspace associated with the eigenvalue 3 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
- BA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.
- CA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.
- DA basis for the eigenspace associated with the eigenvalue 2 is , a basis for the eigenspace associated with the eigenvalue 3 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
- EA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.

**Q7: **

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

- ABases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

**Q8: **

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

- AA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.
- BA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.
- CA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.
- DA basis for the eigenspace associated with the eigenvalue 7 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
- EA basis for the eigenspace associated with the eigenvalue 7 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.

**Q9: **

- ABases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues 2 and 4 are , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues 2 and 6 are , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

**Q10: **

- ABases for the eigenspaces associated with the eigenvalues 6 and 12 are , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore defective.
- CBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore defective.
- DBases for the eigenspaces associated with the eigenvalues 6 and 9 are , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues 6 and are , and respectively. The matrix is therefore not defective.

**Q11: **

- ABases for the eigenspaces associated with the eigenvalues and 2 are and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues 1 and 2 are and respectively. The matrix is therefore defective.
- CBases for the eigenspaces associated with the eigenvalues 1 and 2 are and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues 1 and are and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues and 2 are and respectively. The matrix is therefore defective.

**Q12: **

- ABases for the eigenspaces associated with the eigenvalues 4, , and are , , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues 4, , and are , , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

**Q13: **

- ABases for the eigenspaces associated with the eigenvalues 0, 13, and are , , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues 0, 13, and are , , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues 0, 18, and are , , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues 0 and are , respectively. The matrix is therefore defective.
- EBases for the eigenspaces associated with the eigenvalues 0, 18, and are , , and respectively. The matrix is therefore not defective.

**Q14: **

- ABases for the eigenspaces associated with the eigenvalues 6, , and are , , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues 6, , and are , , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues 6, , and are , , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues 6, , and are , , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues 6, , and are , , and respectively. The matrix is therefore not defective.

**Q15: **

Suppose a matrix is an matrix which has characteristic polynomial . Is diagonalizable? Why, or why not?

- ANo, because is singular.
- BYes, because all the roots of are real.
- CMaybe, it depends on whether has a degree equal to the dimension of the space.
- DMaybe, it depends on the dimensions of the eigenspaces.
- ENo, because has repeated roots.