(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant 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.

3. 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

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# Homework Help: Eigenvalue questions

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