Finding A to the power of n without using eigenvalues

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In summary, The conversation discusses a problem involving a 3x3 matrix and a vector. The first part, a), asks to verify that Au = 2u, which the person was able to do without difficulty. The second part, b), asks to calculate the vector A^5*u without forming A^5. The person was stuck on this part and asked for a hint. The hint provided was to use the fact that Au = 2u and relate it to A(Au), which would simplify the calculation. The person also mentioned trying to use eigenvalues but not fully understanding them. The responder then explains that they are, in fact, using eigenvalues in this problem.
  • #1
dlevanchuk
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Oh it gives me headache... been thinking on this problem for a while, and don't even know where to begin! Could anyone give me a hint at least?? :(

Problem:
Let A be (3x3) matrix : [ 4 -2 2; 2 4 -4; 1 1 0] and u (vector) = [1 3 2].
a) Verify that Au = 2u
I got this one without a problem.

b) Without forming A^5, calculate the vector A^5*u.
This is where I get stuck.. I've tried to search, but keep coming up with some equations that involve eigenvalues (which I haven't studied yet..). So, I am assuming that i don't have to use any of eigenvalues.. Is there any other way?
I tried to replace A matrix with [a b c; d...] values, and take first 3 powers of that, but its way too hard to keep track of everything..

So, any hint?? :(
 
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  • #2
You know that Au = 2u.

Now what is A2u = A (A u)?

Note that numbers commute with matrices, so for example A(2u) = 2(Au).

Actually you are using eigenvalues here: when Av = c v for some (non-trivial) vector v and some number c, then we say that v is an eigenvector for the matrix A, with corresponding eigenvalue c.
 
  • #3
Oh man! Thats simple! Didn't even think about relating problem a with problem b! :rolleyes: Thanks a ton! :biggrin:
 

1. What is a matrix without eigenvalues?

A matrix without eigenvalues is a square matrix that does not have any real or complex numbers that satisfy the equation Ax = λx, where A is the matrix and λ is the eigenvalue. In simpler terms, it is a matrix that does not have a special set of numbers that can be multiplied by a vector to give a scaled version of that vector.

2. Why is it important to study matrices without eigenvalues?

Studying matrices without eigenvalues is important because it helps us understand the properties and behavior of matrices. It also has practical applications in fields such as physics, engineering, and computer science.

3. How do you calculate the power of a matrix without eigenvalues?

To calculate the power of a matrix without eigenvalues, you can use the Cayley-Hamilton theorem. This theorem states that a matrix satisfies its own characteristic equation, which allows you to calculate its powers without knowing its eigenvalues.

4. Can a matrix without eigenvalues have eigenvectors?

Yes, a matrix without eigenvalues can still have eigenvectors. However, these eigenvectors will be complex numbers rather than real numbers. This is because the eigenvalues of a matrix without eigenvalues are complex numbers.

5. How can matrices without eigenvalues be diagonalized?

A matrix without eigenvalues cannot be diagonalized. This is because diagonalizing a matrix involves finding its eigenvalues and eigenvectors, but since a matrix without eigenvalues does not have any eigenvalues, it cannot be diagonalized.

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