HP49-50G : Eigenvalues & eigenvectors

[Gil]

Suppose that I have the special matrix
[[ 0 1 ]
[ 0 0 ]], which has no square root.

Its has only one distinct eigenvalue Lambda = 0.
The corresponding eigenvector is [t, 0], or [1,0].

Execute on the calculator
[[ 0 1 ]
[ 0 0 ]]
« EGV»

Nothing can be calculated.

Change now
the initial matrix into a real matrix
[[ 0. 1. ]
[ 0. 0. ]]

and execute again
« EGV
»

The answer is now:
[[ 1. 1. ]
[ 0. 0. ]] (2 repeated eigenvectors in column)

& [ 0. 0. ] (2 repeated eigenvalues, each = 0).

Why this separate handling of the EGV instruction?

By the way, Wolfram Alpha gives for
[[ 0 1 ]
[ 0 0 ]] 1/2 ^
the "answer"
[[ 0 0 ]
[ 0 0 ]] (the nil matrix).
Any idea why or how Wolfram finds this result
(even CHAT GPT can prove that such a Matrix has no square root) ?

Regards,

Gil