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Geometrical properties of the Frobenius condition number for positive definite matrices. (English) Zbl 1153.15009
Let the Frobenius inner product $(A, B)_F = \text{tr}(A^{T}B)$ be defined in the space of square real $n \times n$ matrices. The geometrical properties of the Frobenius condition number of positive definite matrices in such an inner product space are studied with the aim to get a bound for the ratio between the angle that a matrix $A$ forms with the identity ray, $\alpha I$, for $\alpha > 0$, and the angle that $A^{-1}$ forms with $\alpha I$. As a result new lower bounds for the condition number of $A$ which only require the trace of $A$ and the Frobenius norm of $A$ are found. A new practical lower bound for the Frobenius condition number $\kappa_F(A)$ is given by the expression $\kappa_F(A) \geq \max(n, \frac{\sqrt{n}}{\cos^{2}(A, I)})$ and its accuracy is evaluated in numerical experiments.

15A12Conditioning of matrices
15A63Quadratic and bilinear forms, inner products
15A45Miscellaneous inequalities involving matrices
15A60Applications of functional analysis to matrix theory
15B48Positive matrices and their generalizations; cones of matrices
Full Text: DOI
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