Orthogonality by Singular value decomposition "svd"

How to use the Singular value decomposition "svd" to perform the Orthogonalization between vectors?

[M]

Could you be more specific ?

The svd function outputs orthogonal matrices such that:

[U,S,V] = svd(X);

with , i.e. The columns of U are orthogonal (and same for V).

[M]

@M I read in in the following link https://en.wikipedia.org/wiki/Orthogonalization

that one of the Methods for performing orthogonalization include:

• Symmetric orthogonalization, which uses the Singular value decomposition

I want to perform it for example between a matrix and vector to achieving the Orthogonality. But I dont know how to perform that.

Do you have any idea plaese?

[Torsten]

Never heard of this method. According to the literature link, it seems to be strongly coupled with a physics application. So I suggest you read the article referenced if you think it might be of help for your application, too.

Löwdin, Per-Olov (1970). "On the nonorthogonality problem". Advances in quantum chemistry. Vol. 5. Elsevier. pp. 185–199.

Answer 1

[John D'Errico]

Seems easy enough.

A = randi(5,[5,3])

A = 5×3

3 1 2 1 2 3 2 4 5 4 5 2 2 2 5

rank(A)

ans = 3

So A is a simple matrix, with rank 3. There are 3 vectors that would form a basis for the column space of A.

Using the SVD, can we find such a set of vectors? Yes.

[U,S,V] = svd(A,0);
U

U = 5×3

-0.2851 0.1392 0.8395 -0.3135 -0.2300 -0.1791 -0.5693 -0.2524 -0.4040 -0.5261 0.7822 -0.1560 -0.4687 -0.5022 0.2749

The matrix U is exactly what you want. The vectors are orthonormal (so both orthogonal and having unit norm.)

U'*U

ans = 3×3

1.0000 -0.0000 -0.0000 -0.0000 1.0000 -0.0000 -0.0000 -0.0000 1.0000

We get an identity matrix. (To within floating point trash. the -0.0000 elements are all essentially on the order of +/- eps.)

And U has the property that they span the column space of A. So ANY of the columns of A can be represented as some linear combination of the columns of U. The SVD essentially provides the transformation to orthogonality you seem to be asking to get. If you want, it has orthognalized the columns of A.

[M]

@John D'Errico Thank you so much.

But what if there are a matrix orthogonalized with a vector , and I want to do the orthogonalization test between them using SVD, How to do that?

[John D'Errico]

What does it mean to have a matrix orthogonalized with a vector? I'm sorry, but that statement makes little mathematical sense. As well, what are you talking about in terms of an orthogonalization test?

If you want better help, then you need to ask meaningful questions. Sorry, but true. It seems as if you are throwing around a lot of jargon that you don't really understand. Perhaps this is what you are asking:

X = 1:5;
Xnull = null(X)

Xnull = 5×4

-0.2697 -0.4045 -0.5394 -0.6742 0.9359 -0.0961 -0.1282 -0.1602 -0.0961 0.8558 -0.1923 -0.2403 -0.1282 -0.1923 0.7437 -0.3204 -0.1602 -0.2403 -0.3204 0.5995

V is a vector. And the matrix Vnull is a set of vectors that span the nullspace of V. Two vectors ar e orthogonal if their dot product is zero. Is that the test you were asking about?

X*Xnull

ans = 1×4

1.0e-15 * 0.4441 0.4441 0.8882 0.8882

As you should see, the dot products of X with each of the vectors in Xnull are zero, to within floating point trash.

Xnull'*Xnull

ans = 4×4

1.0000 -0.0000 -0.0000 -0.0000 -0.0000 1.0000 -0.0000 -0.0000 -0.0000 -0.0000 1.0000 -0.0000 -0.0000 -0.0000 -0.0000 1.0000

As well, you should see this is the 4x4 identity matrix, so we see that Xnull is indeed a set of orthonormal vectors.

I used NULL to do the work. But if you look carefully at the code for NULL (it is not built-in), you would see it just calls SVD. I could also have done this:

[U,S,V] = svd(X');

Now the set of nullspace vectors are just given as

U(:,2:end)

ans = 5×4

-0.2697 -0.4045 -0.5394 -0.6742 0.9359 -0.0961 -0.1282 -0.1602 -0.0961 0.8558 -0.1923 -0.2403 -0.1282 -0.1923 0.7437 -0.3204 -0.1602 -0.2403 -0.3204 0.5995

As you can see, this is what null returns. And null just uses SVD anyway. So just use null, if that is what you need.

I'm only making random guesses here, since I have no idea what you are really asking.