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An arbitrary function ''f(A)'' of a 2×2 matrix A has its [[Sylvester's formula]] simplify to
:<math>
f(A) = \frac{f(\lambda_+) + f(\lambda_-)}{2} I + \frac{A - \left (\frac{tr(A)}{2}\right )I}{\sqrt{\left (\frac{tr(A)}{2}\right )^2 - |A|}} \frac{f(\lambda_+) - f(\lambda_-)}{2} ~,
</math>
where <math>\lambda_\pm</math> are the eigenvalues of its characteristic equation, |A-λI|=0, and are given by
:<math>
\lambda_\pm = \frac{tr(A)}{2} \pm \sqrt{\left (\frac{tr(A)}{2}\right )^2 - |A|} .
</math>
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