Question

Help with Linear Algebra Problem

Answer 1

Your diagonal matrix D for A = PDP^(-1) will have entries greater than 1 so some of the entries of D^n will approach infinity so it can’t exist (since P, P^(-1), and v have real entries).

Answer 2

Let's denote the eigenvectors as **x** , **y** and **z**, with associated eigenvalues l, m and n. (Weird notation, I know, but I can't be bothered to write greek)

The spectral theorem tells us that the eigenvectors of a diagonalizable matrix from a basis, so for any vector v, we can say v=a**x**+b**y**+b**z** for some numbers a b and c. So that means for any vector v, Av=A(a**x**+b**y**+c**z**)

Since matrix multiplication is linear, we have A(a**x**+b**y**+c**z**)=aA**x**+bA**y**+cA**z**=al**x**+bm**y**+cn**z**.

What happens when you multiply with A again?

Another approach: since A is diagonalizable, it can be written as UDU^-1 . So A^2 =UDU^-1 UDU^-1 = UDIDU^-1 =UD^2 U^-1.

So A^n **v**=UD^n U^-1 **v** . That should be enough to get you started

Answer 3

*Suggestion:* Try working with a concrete example of a 3×3 matrix with these eigenvalues. See what kind of behavior is has in the context of your question, then see whether you can generalize your result to an arbitrary 3×3 matrix with these eigenvalues.

So, in particular, the 3×3 matrix *A* given by

    A :=

    [ 7.4      0     0 ]

    [   0    0.7     0 ]

    [   0      0   2.3 ]

has the indicated eigenvalues, with corresponding eigenvectors **v**\_1 := [1 0 0]^(T), **v**\_2 := [0 1 0]^(T), and **v**\_3 := [0 0 1]^(T). (I.e., the eigenvectors are the column vectors that are the transposes of the row vectors [1 0 0], [0 1 0], and [0 0 1], respectively.)

A natural starting place would be to consider, for each *j*, the limits

- lim\_[*t*→∞] *A*^(*t*) **v**\_*j*

over the eigenbasis vectors {**v**\_1, **v**\_2, **v**\_3}.

After working with this example, of course, you'd still need to consider the general case. Can you show how this specific case is enough to describe the behavior of the general case?

I hope something here points you in a useful direction. Good luck!