Eigenbases make powers easy

Mathematics · Linear Algebra · ages 19-20

Name ______________________ Date ____________

What is the largest eigenvalue of the diagonal matrix with diagonal entries 4 and 7?
Answer: ______________
What is the largest eigenvalue of the diagonal matrix with diagonal entries 4 and 7?
Answer: ______________
Lee says P holds the eigenvalues and D holds the eigenvectors. Is Lee right?
Circle one: True False
In the factorisation A = P D P inverse, what sits on the diagonal of D?
- The eigenvalues of A
- The eigenvectors of A
- All ones
A 2 by 2 matrix with two different eigenvalues cannot be diagonalised. True or false?
Circle one: True False
A = P D P^-1 with D = diag(2, 3). What is the trace of A^2?
Answer: ______________
Which matrix cannot be diagonalised?
- [[1, 0], [0, 3]]
- [[2, 0], [0, 2]]
- [[2, 1], [0, 2]]
A = P D P inverse with D = diag(2, 3). What is the trace of A squared?
Answer: ______________
A 2 by 2 matrix has eigenvalues 1 and 2. What is the trace of A to the 10th?
Answer: ______________
A 2 by 2 matrix has eigenvalues 1 and 2. What is the trace of A^10?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

7 · A diagonal matrix carries its eigenvalues on the diagonal: 4 and 7, largest 7.
7 · A diagonal matrix carries its eigenvalues on the diagonal: 4 and 7, largest 7.
False · Lee is wrong: D holds eigenvalues on its diagonal, P holds the eigenvectors.
The eigenvalues of A · D is diagonal with the eigenvalues; P holds the matching eigenvectors.
False · False: distinct eigenvalues give independent eigenvectors, which is exactly when diagonalisation works.
13 · A^2 has eigenvalues 4 and 9, and trace = 4 + 9 = 13.
[[2, 1], [0, 2]] · [[2, 1], [0, 2]] has eigenvalue 2 twice but only one eigenvector direction, so no eigenbasis exists.
13 · A squared has eigenvalues 4 and 9, and trace = 4 + 9 = 13.
1025 · Eigenvalues to the 10th: 1 and 1024; trace = 1025.
1025 · Eigenvalues to the 10th: 1 and 1024; trace = 1025.