Applications in Matrix Theory and Spectral Theory
The logarithmic norm is related to the extreme values of the Rayleigh quotient. It holds that
and both extreme values are taken for some vectors . This also means that every eigenvalue of satisfies
- .
More generally, the logarithmic norm is related to the numerical range of a matrix.
A matrix with is positive definite, and one with is negative definite. Such matrices have inverses. The inverse of a negative definite matrix is bounded by
Both the bounds on the inverse and on the eigenvalues hold irrespective of the choice of vector (matrix) norm. Some results only hold for inner product norms, however. For example, if is a rational function with the property
then, for inner product norms,
Thus the matrix norm and logarithmic norms may be viewed as generalizing the modulus and real part, respectively, from complex numbers to matrices.
Read more about this topic: Logarithmic Norm
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