Solving Matrix Orthonormalization Problem with MATLAB

[morry]

Ok, this should be a simple problem, but its not working out for me.

I need to use MATLAB to show that the columns of a 3x3 matrix are orthonormal. I called each of the columns separate vectors, because I thought it would be easier. So now I have 3 3x1 vectors. I want to multiply them together to show they are orthogonal.
But MATLAB keeps returning an error. Saying that internal dimensions must match or something. Anyone know why this is happening?

Also, is there an easy way to show that the vectors are normal? I don't know of any commands to show this.

Thanks guys.

 

[TD]

Matrix calculates everything with matrices. When you have 2 vectors with dimensions 1x4 (e.g. X and Y) and you want to multiply them, you cannot do X*Y since that would be multiplying a (1x4) with a (1x4) and that's impossible. You can use the dot-operator so that Matlab multiplies them element-by-element rather then seeing it as a matrix multiplication.

Instead of doing x*y, try x.*y

 

[SGT]

To prove that the vectors are orthogonal, their scalar product must be zero. Calling your vectors $$V_1, V_2, V_3$$, you must find the products:
$$V_1 * V_2^'$$, $$V_2 * V_3^'$$ and $$V_3 * V_1^'$$.

 

[Werg22]

Talking about matlab, how do you intergrate area or distance from a point to another with a function?

 

[SGT]

Talking about matlab, how do you intergrate area or distance from a point to another with a function?

If you mean the area below a function, you can use the functions:
quad Use adaptive Simpson quadrature
quadl Use adaptive Lobatto quadrature

 

[uart]

Ok, this should be a simple problem, but its not working out for me.

I need to use MATLAB to show that the columns of a 3x3 matrix are orthonormal. I called each of the columns separate vectors, because I thought it would be easier. So now I have 3 3x1 vectors. I want to multiply them together to show they are orthogonal.
But MATLAB keeps returning an error. Saying that internal dimensions must match or something. Anyone know why this is happening?

Also, is there an easy way to show that the vectors are normal? I don't know of any commands to show this.

Thanks guys.

Think about this Morry, if you pre-multiple the matrix by it's transpose (M'*M), what does the occurrence of zeros in the off diagonal positions tell you?

 

[morry]

Think about this Morry, if you pre-multiple the matrix by it's transpose (M'*M), what does the occurrence of zeros in the off diagonal positions tell you?

This would show that its orthog wouldn't it? I have to show that the vectors are orthonormal as well though.

Thanks a lot for your help guys. I knew there was a little trick I had to do. Cheers.

 

[SGT]

In order for the vectors to form an orthonormal basis, they must be orthogonal and unit.
A unit vector has modulus 1.
|V| = V'*V

 

[morry]

Thanks SGT. Doing V'*V gives me 1s on the diagonal.

How would I actually go about orthog. diagonalising this matrix? If I was doing this by hand, I would just divide by its modulus, but I can't find the moduli of these vectors using matlab.

I have multiplied the 3x1 vectors to try and get them to equal 0, but they are not equalling 0.

Thanks guys.

 

[SGT]

Thanks SGT. Doing V'*V gives me 1s on the diagonal.

How would I actually go about orthog. diagonalising this matrix? If I was doing this by hand, I would just divide by its modulus, but I can't find the moduli of these vectors using matlab.

I have multiplied the 3x1 vectors to try and get them to equal 0, but they are not equalling 0.

Thanks guys.

If $$P$$ is an orthogonal matrix and $$B = P^{-1}AP = P'AP$$, then $$B$$ is said orthogonally similar to $$A$$.
If $$A$$ is real and symetric, it is orthogonally similar to a diagonal matrix whose diagonal elements are the eigenvalues of $$A$$.
In Matlab the command $$[V,D] = eig(A)$$ returns two matrices. $$D$$ is a diagonal matrix containing the eigenvalues of $$A$$ and is orthogonally similar to $$A$$. $$V$$ is a matrix containing in its columns the eigenvectors of $$A$$.
We have $$D = V^{-1}AV = V'AV$$

 

[morry]

Thanks again SGT.

I am still unsure about how to show that my eigenvectors are orthonormal? I tried multiplying them like you mentioned, but they come out as numbers, not zero. Also, is there a command that finds the modulus of the vectors?

 

[SGT]

The eigenvectors are not necessarilly orthonormal. All it is required is that they are linearly independent in order to form a basis.
To my knowledge there is no single command to calculate the modulus of a vector, but the command V´*V is so simple that I think any other command would be longer to type.

 

[neurocomp2003]

v'v or vv' depending on if you used columns or vectors. or i believe there is a norm function. use the condition v'v or vv' < 1+e where e is a sufficiently small threshold

orthonormal system:
vi'vi < 1+e
abs(vi'vj)< 0+e

 

[morry]

Cheers everyone, I finally got the q out. I think I was being a bit of a dumbarse. :)