(a) Show that if A is a matrix such that A^4=1, then the...

(a) Show that if $A$ is a matrix such that $A^4=1$, then the only possible eigenvalues of $A$ are $1,-1, i$, and $-\mathrm{i}$.
(b) Give an example of a real matrix that has all four numbers as eigenvalues.

Key Concepts

Matrix Polynomials and the Minimal Polynomial

When a matrix satisfies a polynomial equation, like A^4 = I, each eigenvalue ? must satisfy the corresponding scalar equation ?^4 = 1. The minimal polynomial of A, the smallest degree monic polynomial that A satisfies, must divide any such polynomial and thereby restricts the possible eigenvalues of A to the roots of that polynomial equation.

Eigenvalues

Eigenvalues are scalars ? for which there exists a nonzero vector v such that Av = ?v for a matrix A. They are fundamental in understanding linear transformations, and their properties are closely tied to how matrices behave under various operations, including taking powers or applying functions to the matrix.

Roots of Unity

The roots of unity are the solutions to the equation x^n = 1, and in the case n = 4, the solutions are 1, -1, i, and -i. These values directly arise as the only possible eigenvalues when the matrix A satisfies A^4 = I, since any eigenvalue ? must satisfy ?^4 = 1.

Real Matrices and Complex Eigenvalues

Even though a matrix might have entries in the real numbers, it can have complex eigenvalues, which must occur in conjugate pairs. This concept is important when constructing real matrices that exhibit a spectrum of both real and complex eigenvalues, ensuring that the matrix remains real while still permitting complex eigenvalues.

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