np.linalg.pinv of a DataArray results in mismatched coordinates

What happened?

In general, np.linalg.pinv takes an array of shape (..., n, m) and returns an array of shape (..., m, n). Something isn't recognizing that the last two axes get switched around, so it incorrectly retains the coordinates in the same order.

In the attached example, m1 is an DataArray with shape (3,4) and defined coordinates, and m2 is the result of np.linalg.pinv(m1). m2 has shape (4,3), but maintains the coordinates from m1, which are 3 and 4 in length.

>>> m1
<xarray.DataArray (foo: 3, bar: 4)> Size: 96B
...
Coordinates:
  * foo      (foo) <U1 12B 'x' 'y' 'z'
  * bar      (bar) <U1 16B 'a' 'b' 'c' 'd'
>>> m2
<xarray.DataArray (foo: 4, bar: 3)> Size: 96B
...
Coordinates:
  * foo      (foo) <U1 12B 'x' 'y' 'z'
  * bar      (bar) <U1 16B 'a' 'b' 'c' 'd'

What did you expect to happen?

That the coordinates of m2 would have the same length as their respective axis's shape. Or at least an error message or warning or something.

Minimal Complete Verifiable Example

# /// script
# requires-python = ">=3.11"
# dependencies = [
#   "xarray[complete]@git+https://github.com/pydata/xarray.git@main",
#   "numpy"
# ]
# ///
#
# This script automatically imports the development branch of xarray to check for issues.
# Please delete this header if you have _not_ tested this script with `uv run`!

import xarray as xr
xr.show_versions()
# your reproducer code ...
import numpy as np
m1 = xr.DataArray(np.arange(12).reshape(3,4), 
                  coords={'foo': ['x', 'y', 'z'],
                          'bar': ['a', 'b', 'c', 'd']})
m2 = np.linalg.pinv(m1)

Steps to reproduce

No response

MVCE confirmation

• Minimal example — the example is as focused as reasonably possible to demonstrate the underlying issue in xarray.
• Complete example — the example is self-contained, including all data and the text of any traceback.
• Verifiable example — the example copy & pastes into an IPython prompt or Binder notebook, returning the result.
• New issue — a search of GitHub Issues suggests this is not a duplicate.
• Recent environment — the issue occurs with the latest version of xarray and its dependencies.

Relevant log output

Anything else we need to know?

This probably affects np.linalg.inv too, although less visibly.

Environment

Start by running the provided MVCE with xarray and NumPy, focusing on the np.linalg.pinv DataArray dispatch path and how the final two dimensions are handled. Add a regression test for a 3×4 DataArray with named coordinates; done means the result has shape 4×3 and coordinates whose lengths match their respective dimensions, or a documented error is raised.