Hi marky82!
An orthonormal basis requires 2 things: orthogonal and normal.
Orthogonal means that he vectors are orthogonal.
Normal means that they have length 1.
Are your current vectors orthogonal?
And what is their length?
I've been at this for a while and I'm now stumped.
The matrix, A is
5 -4 0
-4 3 -4
0 - 4 1
Here's what I've got so far.
eigenvalues are λ = 9, λ = -3, λ = 3
corresponding eigen vectors are (2k, -2k, k), (k/2, k, k), (-k, k/2, k)
{(2, -2, 1)} is a basis for s(9)
{(1,2,2)} is a basis for s(-3)
{(-2,-1,2)} is a basis for s(3)
So that gives the set
E = {(2,-2,1),(1,2,2),(-2,-1,2)} which is an eigenvector basis of A (not sure if this is even needed as I think I need an orthonormal basis)
I think I need to find an orthonormal eigenvector basis next, but I can't understand how to do that (if that is actually next). As that should give me the transition matrix.
So looking for someone to hold my hand to the end of this process please
Hi marky82!
An orthonormal basis requires 2 things: orthogonal and normal.
Orthogonal means that he vectors are orthogonal.
Normal means that they have length 1.
Are your current vectors orthogonal?
And what is their length?
Two vectors are orthogonal if they are perpendicular.
This is equivalent to their dot product being zero.
What are the dot products of each pair of vectors in the basis?
The (regular) length of a vector is
What are the lengths of the vectors in your basis?
To normalize a vector, you divide each of its coordinates by its length...