Category InnerProductSpaces

[mkoeppe]

(follow-up from #30092)

We define a category InnerProductSpaces with

• parent method dual (which returns self),
• element method inner_product.

A complication for complex vector spaces: FreeModule defines:

• dot_product - which ignores inner_product_matrix and does not conjugate
• hermitian_inner_product - which ignores inner_product_matrix but conjugates the left factor (self)
• inner_product - which uses inner_product_matrix but does not conjugate.

Perhaps we need to add an extra parameter inner_product_hermitian to FreeModule. For a complex base ring, the module would only be added to the category of InnerProductSpaces if this is True.

MatrixSpace elements currently do not have inner_product.

References:

• Hermitian inner product #10683 ... where hermitian_inner_product was introduced
• numpy.inner and numpy.dot, numpy.tensordot, numpy.einsum do not conjugate.
• numpy.vdot conjugates the left factor; and also handles the inner product of matrices in the same way.
• SymPy's Matrix.dot has parameters hermitian and conjugate_convention that control if and how factors are conjugated.

See also: https://en.wikipedia.org/wiki/Pseudo-Euclidean_space

CC: @tscrim @orlitzky @egourgoulhon @mjungmath @kcrisman

Component: linear algebra

Issue created by migration from https://trac.sagemath.org/ticket/30218

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,3 +1,5 @@
 (follow-up from #30092)
 
 
+We define a category `InnerProductSpaces` with element methods `inner_product`.
+

[mkoeppe]

comment:3

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[orlitzky]

comment:5

This is half of #2711, and probably obsoletes #29437.

[mkoeppe]

Description changed:

--- 
+++ 
@@ -3,3 +3,22 @@
 
 We define a category `InnerProductSpaces` with element methods `inner_product`.
 
+**A complication for complex vector spaces:** `FreeModule` defines:
+- `dot_product` - which ignores `inner_product_matrix` and does not conjugate
+- `hermitian_inner_product` - which ignores `inner_product_matrix` but conjugates the left factor (`self`)
+- `inner_product` - which uses `inner_product_matrix` but does not conjugate.
+
+Perhaps we need to add an extra parameter `inner_product_hermitian` to `FreeModule`. For a complex base ring, the module would only be added to the category of `InnerProductSpaces` if this is `True`.
+
+
+`MatrixSpace` elements currently do not have `inner_product`. 
+
+
+
+References:
+- #10683 ... where `hermitian_inner_product` was introduced
+- [numpy.inner](https://numpy.org/doc/stable/reference/generated/numpy.inner.html) and [numpy.dot](https://numpy.org/doc/stable/reference/generated/numpy.dot.html#numpy.dot), [numpy.tensordot](https://numpy.org/doc/stable/reference/generated/numpy.tensordot.html#numpy.tensordot), [numpy.einsum](https://numpy.org/doc/stable/reference/generated/numpy.einsum.html#numpy.einsum) do not conjugate.
+- [numpy.vdot](https://numpy.org/doc/stable/reference/generated/numpy.vdot.html#numpy.vdot) conjugates the left factor; and also handles the inner product of matrices in the same way.
+- SymPy's [Matrix.dot](https://docs.sympy.org/latest/modules/matrices/matrices.html#sympy.matrices.matrices.MatrixBase.dot) has parameters `hermitian` and `conjugate_convention` that control if and how factors are conjugated.
+
+

[slel]

comment:8

Replying to @orlitzky:

This is half of #2711, and probably obsoletes #29437.

On the other hand #29437 has a commit. I don't see one here.

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,7 +1,9 @@
 (follow-up from #30092)
 
 
-We define a category `InnerProductSpaces` with element methods `inner_product`.
+We define a category `InnerProductSpaces` with 
+- parent method `dual` (which returns `self`),
+- element method `inner_product`.
 
 **A complication for complex vector spaces:** `FreeModule` defines:
 - `dot_product` - which ignores `inner_product_matrix` and does not conjugate