Finding an orthogonal basis for the column space of a matrix

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Bea 2023-9-7

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回答： Dheeraj 2023-9-20

How do you find an orthogonal basis for the column space of a matrix? Because if you use orth, you get the ortonormal basis...

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Torsten 2023-9-7

Orthonormal is also orthogonal, isn't it ?

John D'Errico 2023-9-7

If you just want an orthogonal basis, then start with orth. Then multiply the column basis that results by a set of random numbers. They will still be orthogonal, but almost certainly no longer orthonormal.

Or, you can just recognize that orthonormal IS orthogonal, just a special case of orthogonal.

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Answer 1

Dheeraj 2023-9-20

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Hi,

I understand that you want ways to find orthogonal basis for a column matrix. It's possible but not with “orth” function as it generates orthonormal basis which you do not intend to find.

To find an orthogonal basis for the column space of a matrix without normalizing the vectors to make them orthonormal, you can use orthogonalization techniques such as the Gram-Schmidt process. The Gram-Schmidt process takes a set of linearly independent vectors and transforms them into a set of orthogonal vectors.

Gram-Schmidt process can be implemented in MATLAB from scratch, or you could use the package that implements Gram-Schmidt process, through MATLAB File exchange with the link below.

https://mathworks.com/matlabcentral/fileexchange/55881-gram-schmidt-orthogonalization

Also, you can refer to the below link to have a better understanding about Gram-Schmidt process.

https://en.wikipedia.org/wiki/Gram–Schmidt_process

Hope this helps!