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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

There are a few things at work here..
1. EGV in approx mode returns the characteristic or generalised eigenvectors for repeated eigenvalues - except that here, it doesn't ;-) that's a bug IMHO
2. EGV in exact mode doesn't do that because people kept complaining about getting 'wrong' eigenvectors. Bernard Parisse changed the behaviour of EGV in exact mode somewhere along the line, see last comment here
3.In case of doubt, use JORDAN that will give you the correct result, and tags the vectors with :Char: for characteristic or :Eigen: for eigenvector (on my 49G).

Cheers, Werner