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The non-real roots of a real polynomial with real coefficients can be grouped into pairs of [[complex conjugate]]s, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the [[intermediate value theorem]] at least one of the roots is real. Therefore, any [[real matrix]] with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.
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An important quantity associated with the spectrum is the maximum absolute value of any eigenvalue. This is known as the [[Spectral_radius#Matrices|spectral radius]] of the matrix.
=== Algebraic multiplicity ===
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