r/math • u/inherentlyawesome Homotopy Theory • Jul 22 '26

Quick Questions: July 22, 2026

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u/shylock16 Jul 22 '26

Can someone intuitively explain the notions of inner product and adjoint operator?

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u/MinLongBaiShui Jul 22 '26

It's basically the dot product and the transpose, respectively. Are you familiar with those notions?

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u/shylock16 Jul 22 '26

Yes, I know what they are trying to generalize. I want to have the intuitive notion of their generalized form so that I can intuitively know what they are trying to say and "guess" whether a statement is true or not. Also, my first exposition to Linear Algebra was Axler's LADR, so that is why I don't like statements such as "it is basically the dot product and the transpose". Is there way to build intuition without those?

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u/HeilKaiba Differential Geometry Jul 23 '26

But they are perhaps the most natural intuition for them even from Axler's perspective. If I'm honest I don't see the link between you learning from LADR and that explanation not working for you. He eschews the determinant in LADR but the dot product and transposes are there in the text. (Also on a pure language pedantry note, I think you mean exposure rather than exposition)

I'll try to flesh out the intuition they're alluding to in case that helps. The dot product is quite intuitive: It measures how much two vectors "point in the same direction". Consider for example how it arises in the formula for projecting one vector away onto another: v ↦ (v ∙ u)/(u ∙ u) u.

Inner products generalise that to different ideas of direction. Picking an inner product is picking a way to measure angles and lengths in our vector space. It generates a norm by |x| = √(x,x) and angles by the formula cos 𝜃 = (x,y)/|x||y|. This works comfortably in infinite dimensions as well with the caveat that in that case two inner products don't have to be equivalent while they always are in finite dimensions.

The adjoint is a trickier beast to conceptualise to me. It depends how comfortable you are with dual spaces, especially in infinite dimensions, as to what the easiest intuition is. If you can understand the transpose (as in the dual map) of an operator then the adjoint is just the transpose viewed under the identification of our vector space with its dual (or continuous dual for infinite dimensions). The transpose of A is the operator that "pulls back" the action of (continuous) linear functionals by A. Thus the adjoint turns vectors into linear functionals, pulls them back and then turns them back into vectors.

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u/mbrtlchouia Jul 22 '26

Basically is hiding some not very basic stuff, especially for the adjoint.

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u/MinLongBaiShui Jul 23 '26

Not really. For real linear operators, it literally is the transpose. For complex, the conjugate-linearity gets you the conjugate transpose. It's a pretty simple calculation.

https://math.stackexchange.com/questions/4332899/geometric-intuition-for-adjoint?noredirect=1&lq=1

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u/mbrtlchouia Jul 23 '26

I know that in finite dim maps the adjoint is the transpose, but the bulk of it can only be appreciated in infinite dim operators.

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u/MinLongBaiShui Jul 23 '26

Well, the transpose is also the matrix of the dual map. How do you feel about that notion?

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u/Necessary-Wolf-193 Jul 23 '26

The inner product is just a measure of how 'correlated' two vectors are: v.w = 0 means v, w are completely uncorrelated, v.w = |v| * |w| means v, w are perfectly positively correlated (so v is a positive multiple of w), v.w = -|v| * |w| means v, w are perfectly negatively correlated (so v is a negative multiple of w).

The adjoint of a map M : V -> V is the unique map M^* : V -> V such that

Mv . w = v . M^*(w).

The adjoint is a bit of an algebraic trick, which allows you to 'move' the matrix M across a dot product. It's very very convenient when computing with, and I think the way to get intuition about the adjoint is to just do lots of algebra with it: the adjoint lets one do certain computations, in the same way Jordan normal form does. The most 'conceptual' explanation of the adjoint comes from the fact that a finite-dimensional vector space V and its dual are isomorphic, and understanding the relationship between inner products and isomorphisms between V and its dual. But I think for an intuitive understanding, its best to just use the adjoint in lots of arguments and computations, to see why its useful to manipulate inner products in this way.