Prove that if A^2=O, then 0 is the only eigenvalue of A...

Prove that if $A^{2}=O,$ then 0 is the only eigenvalue of $A$ Getting Started: You need to show that if there exists a nonzero vector $\mathbf{x}$ and a real number $\lambda$ such that $A \mathbf{x}=\lambda \mathbf{x},$ then if $A^{2}=O, \lambda$ must be zero. (i) $A^{2}=A \cdot A,$ so you can write $A^{2} \mathbf{x}$ as $A(A \mathbf{x})$ (ii) Use the fact that $A \mathbf{x}=\lambda \mathbf{x}$ and the properties of matrix multiplication to show that $A^{2} \mathbf{x}=\lambda^{2} \mathbf{x}$ (iii) $A^{2}$ is a zero matrix, so you can conclude that $\lambda$ must be zero.

Prove that if $A^{2}=O,$ then 0 is the only eigenvalue of $A$ Getting Started: You need to show that if there exists a nonzero vector $\mathbf{x}$ and a real number $\lambda$ such that $A \mathbf{x}=\lambda \mathbf{x},$ then if $A^{2}=O, \lambda$ must be zero.
(i) $A^{2}=A \cdot A,$ so you can write $A^{2} \mathbf{x}$ as $A(A \mathbf{x})$
(ii) Use the fact that $A \mathbf{x}=\lambda \mathbf{x}$ and the properties of matrix multiplication to show that $A^{2} \mathbf{x}=\lambda^{2} \mathbf{x}$
(iii) $A^{2}$ is a zero matrix, so you can conclude that $\lambda$ must be zero.

Key Concepts

Eigenvalues and Eigenvectors

When a nonzero vector x satisfies the equation A x = ? x, ? is called an eigenvalue of matrix A and x is the corresponding eigenvector. This concept is central to understanding how matrices act on vectors, particularly in determining invariant directions and scalings. In this problem, it is used to relate the scalar ? to the effect of applying the transformation A repeatedly.

Nilpotent Matrices

A matrix A is termed nilpotent if there exists a positive integer k such that A^k is the zero matrix. In the case where k = 2, that is, A^2 = O, it implies that every eigenvalue ? of A must satisfy ?^2 = 0, thereby forcing ? to be zero. This characteristic of nilpotent matrices is a critical property in many areas of linear algebra, particularly in simplifying matrix structures and understanding their spectral properties.

Zero Matrix

The zero matrix, denoted by O, is a matrix in which every element is zero. Its significance in this context lies in its role as the outcome of the nilpotent condition A^2 = O. When A^2 is the zero matrix, applying A twice to any vector produces the zero vector, which imposes strict constraints on the possible eigenvalues of A.

Properties of Matrix Multiplication

Matrix multiplication is associative and respects scalar multiplication. This property allows us to write A^2 x as A (A x) and, using the eigenvalue equation A x = ? x, to conclude that A^2 x = ? (A x) = ? (? x) = ?^2 x. This step is crucial in linking the nilpotency condition to the eigenvalue, leading directly to the conclusion that ?^2 = 0 and thus ? = 0.

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