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 682345163412, find the eigenvalues and eigenvectors and determine whether it is defective.

  • A 5 4 3 1 , 7 8 3 1 , defective
  • B 1 1 3 1 , 1 2 1 1 , defective
  • C 4 1 1 1 , 4 3 2 1 , defective
  • D 3 1 1 1 , 1 2 1 1 , defective
  • E 1 1 4 1 , 4 3 2 1 , defective

Q2:

Find the eigenvalues and eigenvectors of the matrix 37192182310, and hence determine whether the matrix is defective.

  • AThe eigenvalues are 1 with corresponding eigenvector 322, 2 with corresponding eigenvector 713, and 3 with corresponding eigenvector 19810. The matrix is not defective.
  • BThe eigenvalues are 1 with corresponding eigenvector 311, 2 with corresponding eigenvector 121, and 3 with corresponding eigenvector 211. The matrix is not defective.
  • CThe eigenvalues are 10 with corresponding eigenvector 821, and 1 with corresponding eigenvector 311. The matrix is defective.
  • DThe eigenvalues are 1 with corresponding eigenvector 422, 2 with corresponding eigenvector 733, and 3 with corresponding eigenvector 1987. The matrix is not defective.
  • EThe eigenvalues are 10 with corresponding eigenvector 1322, and 1 with corresponding eigenvector 723. The matrix is defective.

Q3:

Find the eigenvalues and eigenvectors of the matrix 20918656301427, and hence determine whether the matrix is defective.

  • AThe eigenvalues are 1 with corresponding eigenvector 34141, 3 with corresponding eigenvector 9133131, and 2 with corresponding eigenvector 1211. The matrix is defective.
  • BThe eigenvalues are 1 with corresponding eigenvector 13791, 3 with corresponding eigenvector 32321, and 2 with corresponding eigenvector 3211. The matrix is not defective.
  • CThe eigenvalues are 1 with corresponding eigenvector 13791, 3 with corresponding eigenvector 32321, and 2 with corresponding eigenvector 3211. The matrix is not defective.
  • DThe eigenvalues are 1 with corresponding eigenvector 78181, 3 with corresponding eigenvector 6116111, and 2 with corresponding eigenvector 17671. The matrix is defective.
  • EThe eigenvalues are 1 with corresponding eigenvector 34141, 3 with corresponding eigenvector 9133131, and 2 with corresponding eigenvector 1211. The matrix is not defective.

Q4:

Find the eigenvalues and eigenvectors of the matrix 720810246, and hence determine whether the matrix is defective.

  • AThe eigenvalues are 3 with corresponding eigenvector 126 and 6 with corresponding eigenvector 002.
  • BThe eigenvalues are 3 with corresponding eigenvector 122 and 6 with corresponding eigenvector 001.
  • CThe eigenvalues are 3 with corresponding eigenvector 122 and 6 with corresponding eigenvector 210.
  • DThe eigenvalues are 6 with corresponding eigenvector 254 and 3 with corresponding eigenvector 541.
  • EThe eigenvalues are 3 with corresponding eigenvector 241 and 6 with corresponding eigenvector 182.

Q5:

Find the eigenvalues and eigenvectors of the matrix 117104115, and hence determine whether the matrix is defective.

  • AThe eigenvalues are 2 with corresponding eigenvector 211 and 1 with corresponding eigenvector 311.
  • BThe eigenvalues are 2 with corresponding eigenvector 121 and 1 with corresponding eigenvector 111.
  • CThe eigenvalues are 2 with corresponding eigenvector 121 and 1 with corresponding eigenvector 311.
  • DThe eigenvalues are 2 with corresponding eigenvector 121 and 1 with corresponding eigenvector 111.
  • EThe eigenvalues are 2 with corresponding eigenvector 211 and 1 with corresponding eigenvector 311.

Q6:

Find a basis for the eigenspace of each eigenvalue of the matrix 3121139806, and hence determine whether the matrix is defective.

  • AA basis for the eigenspace associated with the eigenvalue 2 is 111, a basis for the eigenspace associated with the eigenvalue 3 is 9821, and a basis for the eigenspace associated with the eigenvalue 6 is 011. The matrix is therefore not defective.
  • BA basis for the eigenspace associated with the lone eigenvalue 0 is 34321. The matrix is therefore defective.
  • CA basis for the eigenspace associated with the lone eigenvalue 1 is 78141. The matrix is therefore defective.
  • DA basis for the eigenspace associated with the eigenvalue 2 is 111, a basis for the eigenspace associated with the eigenvalue 3 is 9821, and a basis for the eigenspace associated with the eigenvalue 6 is 011. The matrix is therefore not defective.
  • EA basis for the eigenspace associated with the lone eigenvalue 0 is 34141. The matrix is therefore defective.

Q7:

Find a basis for the eigenspace of each eigenvalue of the matrix 116756172, and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 6, 2+6𝑖, and 26𝑖 are 111, 𝑖𝑖1, and 𝑖𝑖1 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 6, 2+6𝑖, and 26𝑖 are 71719171, 𝑖𝑖1, and 𝑖𝑖1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 6, 2+6𝑖, and 26𝑖 are 111, 𝑖𝑖1, and 𝑖𝑖1 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 6, 2+6𝑖, and 26𝑖 are 111, 116535𝑖, and 116535𝑖 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 6, 2+6𝑖, and 26𝑖 are 111, 116535𝑖, and 116535𝑖 respectively. The matrix is therefore not defective.

Q8:

Find a basis for the eigenspace of each eigenvalue of the matrix 2121129807, and hence determine whether the matrix is defective.

  • AA basis for the eigenspace associated with the lone eigenvalue 1 is 34141. The matrix is therefore defective.
  • BA basis for the eigenspace associated with the lone eigenvalue 0 is 78141. The matrix is therefore defective.
  • CA basis for the eigenspace associated with the lone eigenvalue 1 is 34321. The matrix is therefore defective.
  • DA basis for the eigenspace associated with the eigenvalue 7 is 011, and a basis for the eigenspace associated with the eigenvalue 2 is 9821. The matrix is therefore not defective.
  • EA basis for the eigenspace associated with the eigenvalue 7 is 011, and a basis for the eigenspace associated with the eigenvalue 2 is 9821. The matrix is therefore not defective.

Q9:

Find a basis for the eigenspace of each eigenvalue of the matrix 420240222 and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 2 and 6 are 110,001, and 111 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 2 and 4 are 110,001, and 001 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 2 and 4 are 110,001, and 14114 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 2 and 6 are 110,001, and 15115 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 2 , 6, and 4 are 110, 111, and 001 respectively. The matrix is therefore not defective.

Q10:

Find a basis for the eigenspace of each eigenvalue of the matrix 963060369 and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 6 and 12 are 210,101, and 101 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 6 and 9 are 1501, and 1601 respectively. The matrix is therefore defective.
  • CBases for the eigenspaces associated with the eigenvalues 6 and 12 are 1501, and 1701 respectively. The matrix is therefore defective.
  • DBases for the eigenspaces associated with the eigenvalues 6 and 9 are 210,101, and 000 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 6 and 12 are 210,101, and 1701 respectively. The matrix is therefore not defective.

Q11:

Find a basis for the eigenspace of each eigenvalue of the matrix 211232221 and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 1 and 2 are 112,101 and 1211 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 1 and 2 are 111 and 1211 respectively. The matrix is therefore defective.
  • CBases for the eigenspaces associated with the eigenvalues 1 and 2 are 110,101 and 1211 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 1 and 2 are 110,101 and 1201 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 1 and 2 are 112 and 1211 respectively. The matrix is therefore defective.

Q12:

Find a basis for the eigenspace of each eigenvalue of the matrix 422022202 and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 4, 1+𝑖, and 1𝑖 are 111, 12𝑖1𝑖, and 12𝑖1+𝑖 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 4, 2+2𝑖, and 22𝑖 are 2131, 𝑖𝑖1, and 𝑖𝑖1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 4, 2+2𝑖, and 22𝑖 are 111, 𝑖𝑖1, and 𝑖𝑖1 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 4, 2+2𝑖, and 22𝑖 are 2131, 𝑖𝑖1, and 𝑖𝑖1 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 4, 1+𝑖, and 1𝑖 are 2131, 12𝑖1𝑖, and 12𝑖1+𝑖 respectively. The matrix is therefore not defective.

Q13:

Find a basis for the eigenspace of each eigenvalue of the matrix 126174449189 and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 0, 13, and 17 are 13231, 71454, and 5214131 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 0, 13, and 17 are 13231, 71454, and 5214131 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 0, 18, and 12 are 13231, 101, and 210 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 0 and 4 are 13791, 11191 respectively. The matrix is therefore defective.
  • EBases for the eigenspaces associated with the eigenvalues 0, 18, and 12 are 13231, 101, and 210 respectively. The matrix is therefore not defective.

Q14:

Find a basis for the eigenspace of each eigenvalue of the matrix 420210226, and hence determine whether the matrix is defective.

  • ABases for the eigenspaces associated with the eigenvalues 6, 4+2𝑖, and 42𝑖 are 152321, 1𝑖1, and 𝑖11 respectively. The matrix is therefore not defective.
  • BBases for the eigenspaces associated with the eigenvalues 6, 4+2𝑖, and 42𝑖 are 152321, 1𝑖1, and 1𝑖1 respectively. The matrix is therefore not defective.
  • CBases for the eigenspaces associated with the eigenvalues 6, 4+2𝑖, and 42𝑖 are 110, 1𝑖1, and 1𝑖1 respectively. The matrix is therefore not defective.
  • DBases for the eigenspaces associated with the eigenvalues 6, 4+2𝑖, and 42𝑖 are 001, 1𝑖1, and 1𝑖1 respectively. The matrix is therefore not defective.
  • EBases for the eigenspaces associated with the eigenvalues 6, 4+2𝑖, and 42𝑖 are 001, 1𝑖1, and 1𝑖1 respectively. The matrix is therefore not defective.

Q15:

Suppose a matrix 𝐴 is an 𝑚×𝑚 matrix which has characteristic polynomial 𝑝(𝑥)=𝑥(2𝑥)(4𝑥)(7𝑥). Is 𝐴 diagonalizable? Why, or why not?

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

Q16:

Let 𝐴=012003000. Determine the Jordan normal form for 𝐴.

  • A 0 1 0 0 0 1 0 0 0
  • B 0 0 3 0 0 0 0 0 0
  • C 9 8 1 0 9 2 0 0 3
  • D 0 1 2 0 0 3 0 0 0

Q17:

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

  • A 𝐴 = 1 2 𝐼
  • B 𝐴 = 𝐼 1 2
  • C 𝐴 = 𝐼
  • D 𝐴 = 1 2 𝐼
  • E 𝐴 = 𝐼

Q18:

Find 321120 and lim321120.

  • A 2 + 1 2 2 + 2 2 1 2 + 1 2 2 + 1 , 2 2 1 1
  • B 2 1 2 2 2 2 1 2 1 2 2 1 , 2 2 1 1
  • C 1 0 0 1 2 , 1 0 0 0
  • D 2 1 2 1 1 1 + 1 2 , 2 1 1 1
  • E 1 0 0 1 2 , 1 0 0 1

Q19:

Can a real 3×3 matrix which has a nonreal eigenvalue be defective?

  • Ayes
  • Bno

Q20:

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

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

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