Question

find all \\(n \times n\\) matrices \\(a\\) such that \\(a^2 + a = 0\\).

Explanation:

Analyze the matrix equation

We are given the matrix equation:

$$ A^2 + A = 0 $$

where \(A\) is an \(n \times n\) matrix over a field (typically \(\mathbb{R}\) or \(\mathbb{C}\)).
This equation can be factored as:

$$ A(A + I) = 0 $$

where \(I\) is the \(n \times n\) identity matrix.

Determine the minimal polynomial

The equation \(A(A + I) = 0\) implies that the minimal polynomial \(p(x)\) of \(A\) must divide:

$$ x(x + 1) = 0 $$

Since the polynomial \(x(x + 1)\) has distinct linear factors, the matrix \(A\) must be diagonalizable.
The only possible eigenvalues of \(A\) are the roots of this polynomial, which are \(\lambda = 0\) and \(\lambda = -1\).

Characterize diagonalizable matrices

Since \(A\) is diagonalizable, there exists an invertible matrix \(P\) such that:

$$ A = P D P^{-1} $$

where \(D\) is a diagonal matrix whose diagonal entries are the eigenvalues of \(A\).
Since the only possible eigenvalues are \(0\) and \(-1\), the diagonal matrix \(D\) must be of the form:

$$ D =
LATEXBLOCK0
$$

for some integer \(k\) where \(0 \le k \le n\). Here, \(I_k\) is the \(k \times k\) identity matrix, and \(0_{n-k}\) is the \((n-k) \times (n-k)\) zero matrix.

Relate to projection matrices

Let \(P = -A\). Substituting this into the original equation:

$$ (-P)^2 + (-P) = 0 \implies P^2 - P = 0 \implies P^2 = P $$

A matrix \(P\) satisfying \(P^2 = P\) is a projection matrix (or idempotent matrix).
Thus, any solution \(A\) can be written in terms of a projection matrix \(P\) as:

$$ A = -P $$

where \(P^2 = P\).

State the final classification

The set of all such matrices \(A\) consists of all matrices that are similar to a diagonal matrix with diagonal entries in \(\{0, -1\}\).
Equivalently, they are of the form \(A = -P\), where \(P\) is any \(n \times n\) idempotent matrix (\(P^2 = P\)).

Answer:

The matrices \(A\) satisfying \(A^2 + A = 0\) are all matrices of the form:

$$ A = -P $$

where \(P\) is any \(n \times n\) projection (idempotent) matrix satisfying \(P^2 = P\).

Equivalently, these are all diagonalizable matrices whose eigenvalues belong to the set \(\{0, -1\}\). Any such matrix can be written as:

$$ A = -P
LATEXBLOCK0
P^{-1} $$

for some invertible matrix \(P\) and some integer \(0 \le k \le n\).