Using local variables in ExternalEvaluate Python

If you are willing to setup and use WolframClientForPython you could do:

With Mathematica

linearMap = 1. {{1, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0}, 
                {0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1/4, 0, 0, 0}, 
                {0, 0, 0, 0, 1/2, 0, 0}, {0, 0, 0, 0, 0, 1/2, 0}, 
                {0, 0, 0, 0, 0, 0, 1/4}};
Export[FileNameJoin[{"C:", "temp", "linearMap.wxf"}], "WXF"]

then in Python

import numpy as np
import os
from wolframclient.evaluation import WolframLanguageSession
from wolframclient.serializers import export

math_kernel = r'C:\Program Files\Wolfram Research\Mathematica\11.3\MathKernel.exe'
output_path = r'C:\temp'

session = WolframLanguageSession(math_kernel)
session.start()

linear_map = session.evaluate('Import[FileNameJoin[{"C:", "temp", "linearMap.wxf"}]]')

linear_map = np.array(linear_map)

out = np.linalg.eigvalsh(linear_map)

export(out, os.path.join(output_path, 'out.wxf'), target_format='wxf')

session.terminate()

finally back in Mathematica

Import[FileNameJoin[{"C:", "temp", "out.wxf"}]] // Normal
(* {0.25, 0.25, 0.5, 0.5, 1., 1., 1.} *)

You may use the Association syntax for ExternalEvaluate.

If numpy is installed in your Python instance then you should have a "Python-NumPy" external evaluator. Check by evaluating FindExternalEvaluators[].

Initialise the connection with

ExternalEvaluate["Python-NumPy", "1+1"]

2

Then

ExternalEvaluate["Python-NumPy",
 <|
  "Command" -> "numpy.linalg.eigvalsh",
  "Arguments" -> {linearMap}
 |>
]

{0.25, 0.25, 0.5, 0.5, 1., 1., 1.}

If you need to use this often then create a function

numpyEigvalsh[m_?MatrixQ] :=
 ExternalEvaluate["Python-NumPy",
  <|
   "Command" -> "numpy.linalg.eigvalsh",
   "Arguments" -> {m}
  |>
 ]

Then

numpyEigvalsh@linearMap

{0.25, 0.25, 0.5, 0.5, 1., 1., 1.}

Why it may be slower

Note that when using Rationals that Mathematica will take longer as it works to preserve the infinite precision of rationals.

Eigenvalues@linearMap

{1, 1, 1, 1/2, 1/2, 1/4, 1/4}

You can speed things up by using Reals. All you need to do is multiply by 1.

Eigenvalues[1. linearMap]

{1., 1., 1., 0.5, 0.5, 0.25, 0.25}

Note that the output is now with reals instead of rationals

Hope this helps.