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Let the eigenvalues be -1 1
make up two eigenvectors
and compute A = S^-1 L S

example
A=
-3 -4
2 3

eigs(A)
ans =
1
-1
eigs(A^2)
ans =
1
1

Hey!! At the end, it is T or F?

It only takes one counter-example to make it false.
The statement is false

Got you. Thanks a lot.