Odd-Dimensional Real Matrices Have A Real Eigenvalue

Why Odd Dimension Forces a Real Eigenvalue

Some real matrices have no real eigenvalues. The matrix \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\), for example, rotates vectors in the plane and has no real eigenvalue. But this behaviour cannot occur for a real matrix of odd positive dimension: at least one eigenvalue must be real. The reason is not a special pattern in the matrix entries. It is the sign behaviour of its characteristic polynomial, whose degree is odd.

For a real \(n\times n\) matrix \(\mathbf{A}\), use the convention \(p_{\mathbf{A}}(\lambda)=\det(\mathbf{A}-\lambda\mathbf{I}_n)\). A real number \(\lambda\) is a real eigenvalue if there is a nonzero real vector \(\mathbf{x}\) such that \(\mathbf{A}\mathbf{x}=\lambda\mathbf{x}\). Equivalently, \((\mathbf{A}-\lambda\mathbf{I}_n)\mathbf{x}=\mathbf{0}\) has a nonzero solution, which holds exactly when \(\det(\mathbf{A}-\lambda\mathbf{I}_n)=0\). This is the standard determinant criterion for a square matrix to be singular.

The determinant is a sum of products of \(n\) entries. In \(p_{\mathbf{A}}(\lambda)\), the only term of degree \(n\) comes from taking \(-\lambda\) from every diagonal entry of \(\mathbf{A}-\lambda\mathbf{I}_n\). Consequently, \(p_{\mathbf{A}}\) has degree \(n\) and leading coefficient \((-1)^n\). When \(n\) is odd, that leading coefficient is \(-1\). The result about odd-degree polynomials gives the needed sign change.

The Odd-Degree Sign Argument

Proof. Write \(q(t)=c_mt^m+c_{m-1}t^{m-1}+\cdots+c_0\), where \(m\) is odd and \(c_m\ne0\). Dividing by \(t^m\) shows that \(q(t)/t^m\) tends to \(c_m\) as \(t\) tends to positive infinity, because each term \(c_kt^k/t^m\) with \(k<m\) tends to zero. Also \(q(-t)/t^m\) tends to \(-c_m\) as \(t\) tends to positive infinity, since \((-t)^m=-t^m\) for odd \(m\). Thus, for a sufficiently large positive \(R\), \(q(R)\) has the sign of \(c_m\) and \(q(-R)\) has the opposite sign. A polynomial is continuous, so the Intermediate Value Theorem applied on \([-R,R]\) gives a number \(r\in[-R,R]\) for which \(q(r)=0\). \(\square

The lemma requires a real polynomial of odd degree with a nonzero leading coefficient. It does not require the polynomial to have integer coefficients, a known factorization, or an easily calculated root. The Intermediate Value Theorem gives existence; it does not, by itself, give an exact formula for the zero.

Proof. Let \(\mathbf{A}\in\mathbb{R}^{n\times n}\), where \(n\) is a positive odd integer. Its characteristic polynomial \(p_{\mathbf{A}}(\lambda)=\det(\mathbf{A}-\lambda\mathbf{I}_n)\) is a real polynomial of degree \(n\), with leading coefficient \((-1)^n=-1\). The lemma therefore gives a real number \(\lambda\) such that \(p_{\mathbf{A}}(\lambda)=0\). By the determinant criterion, \(\mathbf{A}-\lambda\mathbf{I}_n\) has a nonzero vector \(\mathbf{x}\) in its null space. Hence \((\mathbf{A}-\lambda\mathbf{I}_n)\mathbf{x}=\mathbf{0}\), or \(\mathbf{A}\mathbf{x}=\lambda\mathbf{x}\), with \(\mathbf{x}\ne\mathbf{0}\). Thus \(\lambda\) is a real eigenvalue of \(\mathbf{A}\). \(\square

The proof has two distinct ingredients. Odd dimension makes the characteristic polynomial have odd degree; the polynomial lemma then supplies a real zero. The determinant criterion turns that zero into an eigenvalue. Neither step says that the real eigenvalue is unique, or that the other eigenvalues must be real.

Worked Examples

Common Pitfall: Losing the Leading Sign

A frequent sign error is to assume that \(\det(\mathbf{A}-\lambda\mathbf{I}_n)\) always has leading term \(+\lambda^n\). The sign depends on the convention and on the dimension: with this convention, the leading term is \((-1)^n\lambda^n\). For example, for \(\mathbf{A}=\mathbf{I}_3\), the incorrect calculation would claim a leading term \(+\lambda^3\) and hence that \(p_{\mathbf{A}}(\lambda)\) tends to positive infinity as \(\lambda\) tends to positive infinity. In fact,

so its leading term is \(-\lambda^3\), and the polynomial tends to negative infinity as \(\lambda\) tends to positive infinity. The correction is to track the diagonal terms \(-\lambda\) in \(\mathbf{A}-\lambda\mathbf{I}_n\): choosing all \(n\) of them gives \((-\lambda)^n\). For odd \(n\), the characteristic polynomial is positive for sufficiently large negative arguments and negative for sufficiently large positive arguments.

A related mistake is to apply the odd-degree argument to an even-dimensional matrix. The rotation matrix \(\begin{bmatrix}0&-1\\1&0\end{bmatrix}\) has characteristic polynomial \(\lambda^2+1\), which is positive for every real \(\lambda\) and has no real zero. Its dimension is even, so the odd-degree guarantee does not apply. The dimension hypothesis is essential.

What the Guarantee Does—and Does Not—Tell Us

The theorem is an existence result. It is useful when the characteristic polynomial is difficult to factor: if the dimension is odd, there must be a real zero even if no exact root is apparent. A sign check at two real inputs can sometimes locate an interval containing such a root, as in the three-dimensional example. For larger matrices, the same reasoning applies: the entries may be complicated, but the degree and leading coefficient still determine the eventual signs.

At the same time, the theorem does not identify the eigenvalue or provide its eigenvector. Once a zero \(\lambda\) is known, finding an eigenvector requires solving \((\mathbf{A}-\lambda\mathbf{I}_n)\mathbf{x}=\mathbf{0}\). Nor does the guarantee say how many real eigenvalues there are, counting multiplicities. The third example shows why those distinctions matter: the real eigenvalue is guaranteed, but the other characteristic roots need not be real.

For an odd-dimensional real matrix, the reliable chain of reasoning is: form \(p_{\mathbf{A}}(\lambda)=\det(\mathbf{A}-\lambda\mathbf{I}_n)\); identify its odd degree and leading coefficient; use the opposite signs at sufficiently large positive and negative inputs; apply the Intermediate Value Theorem; then use the determinant criterion to conclude that the real zero is an eigenvalue.