Question

If T restricted to a generalized eigenspace has all eigenvalues equal to each other, is it nilpotent:

a) True
b) False

Answer 1

False. If T restricted to a generalized eigenspace has all eigenvalues equal to each other, it does not necessarily mean that it is nilpotent.

Step-by-step explanation:

False. If T restricted to a generalized eigenspace has all eigenvalues equal to each other, it does not necessarily mean that it is nilpotent.

Nilpotent matrices are matrices that can be raised to a certain power to become the zero matrix. In other words, there exists a positive integer k such that T^k = 0.

Having all eigenvalues equal to each other is one property of nilpotent matrices, but it is not the only property. Therefore, it is possible for a matrix to have all eigenvalues equal to each other without being nilpotent.