Finding A to the power of n without using eigenvalues

  • Context: Undergrad 
  • Thread starter Thread starter dlevanchuk
  • Start date Start date
  • Tags Tags
    Eigenvalues
Click For Summary
SUMMARY

The discussion revolves around calculating A^5 * u for a given 3x3 matrix A = [4 -2 2; 2 4 -4; 1 1 0] and vector u = [1 3 2], without using eigenvalues. The user successfully verified that Au = 2u, which indicates that u is an eigenvector corresponding to the eigenvalue 2. The key insight provided is that A^n * u can be computed using the relationship A^n * u = 2^n * u, allowing for the calculation of A^5 * u as 2^5 * u = 32 * u.

PREREQUISITES
  • Understanding of matrix multiplication
  • Familiarity with eigenvectors and eigenvalues
  • Basic knowledge of linear algebra concepts
  • Ability to manipulate vectors and matrices
NEXT STEPS
  • Study the properties of eigenvectors and eigenvalues in linear algebra
  • Learn about matrix exponentiation techniques
  • Explore the Cayley-Hamilton theorem for polynomial matrix equations
  • Investigate applications of eigenvalues in systems of differential equations
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as software developers working on algorithms involving matrix computations.

dlevanchuk
Messages
29
Reaction score
0
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?? :(
 
Physics news on Phys.org
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.[/size]
 
Oh man! Thats simple! Didn't even think about relating problem a with problem b! :rolleyes: Thanks a ton! :biggrin:
 

Similar threads

Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K
Undergrad Getting eigenvalues of an arbitrary matrix with programming
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Find the Eigenvalues and eigenvectors of 3x3 matrix
  • · Replies 12 ·
Replies
12
Views
3K
Graduate First eigenvalue not matching, but all others are
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Eigenvalues of block matrix/Related non-linear eigenvalue problem
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad How can I test for positive semi-definiteness in matrices?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Simple Generalized Eigenvalue problem
  • · Replies 17 ·
Replies
17
Views
4K
Undergrad Meaning of Third Eigenvalue in a Tilted Ellipse in a 3x3 Matrix
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Eigenvalue Problem and the Calculus of Variations
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Numerically Calculating Eigenvalues
  • · Replies 12 ·
Replies
12
Views
2K