# 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.

Q1:

For the matrix find the eigenvalues and eigenvectors and determine whether it is defective.

• A, defective
• B, defective
• C, defective
• D, defective
• E, defective

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:

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 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:

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 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:

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 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:

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 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:

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 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:

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 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:

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 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.

Q16:

Let Determine the Jordan normal form for .

• A
• B
• C
• D

Q17:

Given that the eigenvalues of the nondefective matrix are 1 and , find .

• A
• B
• C
• D
• E

Q18:

Find and .

• A,
• B,
• C,
• D,
• E,

Q19:

Can a real matrix which has a nonreal eigenvalue be defective?

• Ayes
• Bno

Q20:

Suppose an matrix is diagonalizable. Then, which of the following does always have?

• A linearly independent columns
• BExactly as many linearly independent eigenvectors as eigenvalues
• C linearly independent eigenvectors
• D nonzero eigenvalues
• E distinct eigenvalues