In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine transform (or [discrete] fourier transform) we had again orthogonal bases.

To describe any vector $x \in \mathbb{C}^N$, we need to have $N$ independent set of basis vectors but independent doesn't necessarily mean orthogonal. My intentions behind selecting orthogonal vectors are as follows:

• The solution is not unique for $x$ if the basis are not orthogonal.
• It is easy to find the solution numerically by projecting $x$ onto each vector and this solution doesn't depend on the order of the bases. Otherwise, it depends on the order.
• If we are talking about some set of vectors, they might be correlated in the original space, but uncorrelated in the transformd space which might be important when analyzing the data, in dimensionality reduction or compression.

I'm trying to understand the big picture. Do you think that I am right with these? Do you have any suggestions, what is the main reason for selecting orthogonal bases?