Unlocking the Secrets of Nilpotent & Idempotent Matrices: The Weirdest Types in Linear Algebra 🔍

Nilpotent & Idempotent Matrices

Nilpotent Matrices Definition: A square matrix A is said to be nilpotent if A^n = 0 for some positive integer n.

Properties:
- Nilpotent matrices have all eigenvalues equal to 0.
- The determinant of a nilpotent matrix is 0.
- Nilpotent matrices are not invertible.

Idempotent Matrices Definition: A square matrix A is said to be idempotent if A^2 = A$.

Properties:
- Idempotent matrices have eigenvalues equal to 0 or 1.
- The determinant of an idempotent matrix is either 0 or 1.
- Idempotent matrices can be used in projection matrices.

*Key Differences*
- Nilpotent matrices have all eigenvalues equal to 0, while idempotent matrices have eigenvalues equal to 0 or 1.
- Nilpotent matrices are not invertible, while idempotent matrices can be invertible if they are full rank.

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