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 ${\displaystyle x}$ are the vectors most turned by a matrix or operator ${\displaystyle A}$, that is to say those for which the angle between the original vector and its transformed image is greatest. The corresponding antieigenvalue ${\displaystyle \mu }$ is the cosine of the maximal turning angle. The maximal turning angle is ${\displaystyle \phi (A)}$ 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.

References

• Gustafson, Karl (1968), "The angle of an operator and positive operator products", Bulletin of the American Mathematical Society, 74 (3): 488–492, doi:10.1090/S0002-9904-1968-11974-3, ISSN 0002-9904, MR 0222668, Zbl 0172.40702
• Gustafson, Karl (2012), Antieigenvalue Analysis, World Scientific, ISBN 978-981-4366-28-1, archived from the original on 2012-05-19, retrieved 2012-01-31.