- If A is the matrix of a linear transformation which rotates all vectors in ℝ
^{2}through 30^{∘}, explain why A cannot have any real eigenvalues. - If A is an n×n matrix and c is a nonzero constant, compare the eigenvalues of A and cA.
- If A is an invertible n × n matrix, compare the eigenvalues of A and A
^{−1}. More generally, for m an arbitrary integer, compare the eigenvalues of A and A^{m}. - Let A,B be invertible n×n matrices which commute. That is, AB = BA. Suppose x is an eigenvector of B. Show that then Ax must also be an eigenvector for B.
- Suppose A is an n × n matrix and it satisfies A
^{m}= A for some m a positive integer larger than 1. Show that if λ is an eigenvalue of A thenequals either 0 or 1. - Show that if Ax = λx and Ay = λy, then whenever a,b are scalars,
Does this imply that ax + by is an eigenvector? Explain.

- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix .Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix .
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix .
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the complex eigenvalues and eigenvectors of the matrix .
- Find the eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the complex eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the complex eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the complex eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Find the complex eigenvalues and eigenvectors of the matrix . Determine whether the matrix is defective.
- Here is a matrix.
Find values of a,b,c for which the matrix is defective and values of a,b,c for which it is nondefective.

- Here is a matrix.
where a,b,c are numbers. Show this is sometimes defective depending on the choice of a,b,c. What is an easy case which will ensure it is not defective?

- Suppose A is an n × n matrix consisting entirely of real entries but a + ib is a complex eigenvalue having the eigenvector, x + iy. Here x and y are real vectors. Show that then a − ib is also an eigenvalue with the eigenvector, x − iy. Hint: You should remember that the conjugate of a product of complex numbers equals the product of the conjugates. Here a + ib is a complex number whose conjugate equals a − ib.
- Recall an n×n matrix is said to be symmetric if it has all real entries and if A = A
^{T}. Show the eigenvalues of a real symmetric matrix are real and for each eigenvalue, it has a real eigenvector. - Recall an n×n matrix is said to be skew symmetric if it has all real entries and if A = −A
^{T}. Show that any nonzero eigenvalues must be of the form ib where i^{2}= −1. In words, the eigenvalues are either 0 or pure imaginary. - Is it possible for a nonzero matrix to have only 0 as an eigenvalue?
- Show that the eigenvalues and eigenvectors of a real matrix occur in conjugate pairs.
- Suppose A is an n × n matrix having all real eigenvalues which are distinct. Show there
exists S such that S
^{−1}AS = D, a diagonal matrix. Ifdefine e

^{D}byand define

Next show that if A is as just described, so is tA where t is a real number and the eigenvalues of At are tλ

_{k}. If you differentiate a matrix of functions entry by entry so that for the ij^{th}entry of A^{′}you get a_{ij}^{′}where a_{ij}is the ij^{th}entry of A, showNext show det

≠0. This is called the matrix exponential. Note I have only defined it for the case where the eigenvalues of A are real, but the same procedure will work even for complex eigenvalues. All you have to do is to define what is meant by e^{a+ib}. - Find the principle directions determined by the matrix . The eigenvalues are,1, andlisted according to multiplicity.
- Find the principle directions determined by the matrix
The eigenvalues are 1 ,2, and 1. What is the physical interpretation of the repeated eigenvalue?
- Find oscillatory solutions to the system of differential equations, x
^{′′}= Ax where A =The eigenvalues are −1,−4, and −2. - Let A and B be n × n matrices and let the columns of B be
and the rows of A are

Show the columns of AB are

and the rows of AB are

- Let M be an n × n matrix. Then define the adjoint of M, denoted by M
^{∗}to be the transpose of the conjugate of M. For example,A matrix M, is self adjoint if M

^{∗}= M. Show the eigenvalues of a self adjoint matrix are all real. - Let M be an n×n matrix and suppose x
_{1},,x_{n}are n eigenvectors which form a linearly independent set. Form the matrix S by making the columns these vectors. Show that S^{−1}exists and that S^{−1}MS is a diagonal matrix (one having zeros everywhere except on the main diagonal) having the eigenvalues of M on the main diagonal. When this can be done the matrix is said to be diagonalizable. - Show that a n × n matrix M is diagonalizable if and only if F
^{n}has a basis of eigenvectors. Hint: The first part is done in Problem 38. It only remains to show that if the matrix can be diagonalized by some matrix S giving D = S^{−1}MS for D a diagonal matrix, then it has a basis of eigenvectors. Try using the columns of the matrix S. - Let
and let

Multiply AB verifying the block multiplication formula. Here A

_{11}=,A_{12}=,A_{21}=and A_{22}=. - Suppose A,B are n×n matrices and λ is a nonzero eigenvalue of AB. Show that then it is
also an eigenvalue of BA. Hint: Use the definition of what it means for λ to be an
eigenvalue. That is,
where x≠0. Maybe you should multiply both sides by B.

- Using the above problem show that if A,B are n × n matrices, it is not possible that AB −BA = aI for any a≠0. Hint: First show that if A is a matrix, then the eigenvalues of A − aI are λ − a where λ is an eigenvalue of A.
- Consider the following matrix.
Show det

= a_{0}+ λa_{1}+a_{n−1}λ^{n−1}+ λ^{n}. This matrix is called a companion matrix for the given polynomial. - A discreet dynamical system is of the form
where A is an n × n matrix and x

is a vector in ℝ^{n}. Show first thatfor all k ≥ 1. If A is nondefective so that it has a basis of eigenvectors,

whereyou can write the initial condition x

_{0}in a unique way as a linear combination of these eigenvectors. ThusNow explain why

which gives a formula for x

, the solution of the dynamical system. - Suppose A is an n×n matrix and let v be an eigenvector such that Av = λv. Also suppose
the characteristic polynomial of A is
Explain why

If A is nondefective, give a very easy proof of the Cayley Hamilton theorem based on this. Recall this theorem says A satisfies its characteristic equation,

- Suppose an n × n nondefective matrix A has only 1 and −1 as eigenvalues. Find
A
^{12}. - Suppose the characteristic polynomial of an n×n matrix A is 1 −λ
^{n}. Find A^{mn}where m is an integer. Hint: Note first that A is nondefective. Why? - Sometimes sequences come in terms of a recursion formula. An example is the Fibonacci
sequence.
Show this can be considered as a discreet dynamical system as follows.

Now use the technique of Problem 44 to find a formula for x

_{n}. - Let A be an n × n matrix having characteristic polynomial
Show that a

_{0}=^{n}det.

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