Orthogonal Vectors VS Conjugate Vectors?

[skyhr]

Umm... what's really the difference? Isn't the inner product for both 0? So I thought conjugate is very similar to orthogonal, but what's the difference? (This came up from conjugate gradient algorithm)

 

[mmm4444bot]

I think it's a subtle difference, that can be ignored in practice.

I checked two sources, and I saw the phrases "conjugate vectors", "orthogonal vectors", and "conjugate, orthogonal vectors" used (it seemed to me) interchangably.

Both sources stated the distinction as: conjugate means orthogonal with respect to an inner product with a weight matrix.

If you can understand what they mean by referencing the inner product that way, good for you.

I don't.

 

[skyhr]

Ah I see.

What they mean by "with respect to a weight matrix":

A set of vectors {pn} are conjugate with respect to matrix A if pj^T A pk = 0 j!=k

A set of vectors {tn} are orthogonal if tj^T tk = 0 j!=k

I think that is it.

 

[mmm4444bot]

What they mean by "with respect to a weight matrix"

I did not write "with respect to a weight matrix".

I wrote "with respect to an inner product".

The inner product involves the weight matrix A.

I don't fully understand the distinction; yet, I don't think such understanding is important, in practice.

I see this situation as similar to the distinction between roots and zeros. Polynomials have roots; functions have zeros. But these nouns are practically the same; they both represent the solutions to a polynomial equation set equal to zero. The distinction in terminology is generally not important, in practice.

If anybody here understands a clear and important distinction between the phrases "conjugate vectors", "orthogonal vectors", and "conjugate, orthogonal vectors", then I hope they post it.

Cheers ~ Mark