Conditions for Unique Eigenvalues and Solving Systems with Diagonal Matrices"

[iqjump123]

Homework Statement

Consider the matrix
A=[a d f; 0 b e; 0 0 c], where all elements are real numbers
(a) what condition(s) on the elements of A are sufficient to guarantee that A has 3 distinct eigenvalues?
(b) prove that any two eigenvectors x1 and x2 associated with two distinct eigenvalues e1=e2 must be linearly independent
(c) what condition(s) on the elements of A are sufficient to guarantee that the inverse A^-1 exists?
(d) consider diff eq
d/dt(u)=Au, u(0)=u0
where A is the matrix discussed above with three distinct eigenvalues, and u is a vector. write the general solution u(t) in terms of the eigenvalues and eigenvectors of A. do not solve for the actual eigen vectors.
(e) prove that a soln u(t) that is initially parallel to an eigenvector must remain so for all time.

Homework Equations

will involve diagonal matrices
If for a given matrix there exists a matrix B such that AB=I, then B=A^-1, if I is the identity matrix.

The Attempt at a Solution

(a) It seems that in order for A to have three distinct eigenvalues, a and b and c cannot be equal to each other- I think if that diagonal relationship is satisfied, the values will be distinct. Not sure though.
(b) Not sure how to approach this.
(c) I think I am supposed to use the relevant equation 2 that I wrote to prove this- would this be satisfied at all times if the matrix is a diagonal matrix? In that case, d, f, and e should be zero?
(d) Not sure
(e) Not sure

 

[vela]

The Attempt at a Solution

(a) It seems that in order for A to have three distinct eigenvalues, a and b and c cannot be equal to each other- I think if that diagonal relationship is satisfied, the values will be distinct. Not sure though.

This is correct. Can you prove it? Try calculating what the eigenvalues of A are.

(b) Not sure how to approach this.

One way you can do it is assume x₁ and x₂ are linearly dependent and show it leads to a contradiction.

(c) I think I am supposed to use the relevant equation 2 that I wrote to prove this- would this be satisfied at all times if the matrix is a diagonal matrix? In that case, d, f, and e should be zero?

This isn't correct. Hint: What can you say about the determinant of an invertible matrix?

(d) Not sure
(e) Not sure

You should review how to solve systems of linear differential equations.

 

[epenguin]

The Attempt at a Solution

(a) It seems that in order for A to have three distinct eigenvalues, a and b and c cannot be equal to each other- I think if that diagonal relationship is satisfied, the values will be distinct. Not sure though.

The equation for the eigenvalues det(A - λI) = 0 very easily gives you the condition you conjecture.