Antieigenvalue theory

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In applied mathematics, antieigenvalue theory was developed by Karl Gustafson from 1966 to 1968. The theory is applicable to numerical analysis, wavelets, statistics, quantum mechanics, finance and optimization. The antieigenvectors [math]\displaystyle{ x }[/math] are the vectors most turned by a matrix or operator [math]\displaystyle{ A }[/math], that is to say those for which the angle between the original vector and its transformed image is greatest. The corresponding antieigenvalue [math]\displaystyle{ \mu }[/math] is the cosine of the maximal turning angle. The maximal turning angle is [math]\displaystyle{ \phi(A) }[/math] and is called the angle of the operator. Just like the eigenvalues which may be ordered as a spectrum from smallest to largest, the theory of antieigenvalues orders the antieigenvalues of an operator A from the smallest to the largest turning angles.