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- Homework Statement
- Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve number. If we now multiply both of the vectors by a matrix ## W ## which has real entries, will the inner product of the 'transformed' vectors also be positive?

- Relevant Equations
- Inner product

Hi,

I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea.

Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve number. If we now multiply both of the vectors by a matrix ## W ## which has real entries, will the inner product of the 'transformed' vectors also be positive?

Intuitively I think along the lines of: if we imagine the operation as transforming a vector in some way, then the two vectors ## \vec{a}## and ## \vec{b}##, which were similar, should be transformed to similar vectors?

Mathematically, I can write the following:

[tex] <W \vec{a} , W \vec{b} > = (W \vec{a}) \cdot (W \vec{b}) = (W \vec{a})^{T} (W \vec{b}) = \vec{a}^{T} W^{T} W \vec{b} [/tex]

- ## W^{T} W ## is positive semi-definite.

- I suppose if ## \vec{a} ## and ## \vec{b} ##, then if/how positive the outcome will depend on some type of sensitivity of ## W ##? That is, we could view ## \vec{b} = \vec{a} + \vec{\epsilon} ## and think about that?

Any help is greatly appreciated.

I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea.

**Question:**Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve number. If we now multiply both of the vectors by a matrix ## W ## which has real entries, will the inner product of the 'transformed' vectors also be positive?

**Attempt:**Intuitively I think along the lines of: if we imagine the operation as transforming a vector in some way, then the two vectors ## \vec{a}## and ## \vec{b}##, which were similar, should be transformed to similar vectors?

Mathematically, I can write the following:

[tex] <W \vec{a} , W \vec{b} > = (W \vec{a}) \cdot (W \vec{b}) = (W \vec{a})^{T} (W \vec{b}) = \vec{a}^{T} W^{T} W \vec{b} [/tex]

- ## W^{T} W ## is positive semi-definite.

- I suppose if ## \vec{a} ## and ## \vec{b} ##, then if/how positive the outcome will depend on some type of sensitivity of ## W ##? That is, we could view ## \vec{b} = \vec{a} + \vec{\epsilon} ## and think about that?

Any help is greatly appreciated.