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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.

##### MSC:
 15A12 Conditioning of matrices 15A63 Quadratic and bilinear forms, inner products 15A45 Miscellaneous inequalities involving matrices 15A60 Applications of functional analysis to matrix theory 15B48 Positive matrices and their generalizations; cones of matrices
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##### References:
 [1] Bai, Z.; Fahey, M.; Golub, G.: Some large-scale matrix computation problems, J. comput. Appl. math. 74, 71-89 (1996) · Zbl 0870.65035 · doi:10.1016/0377-0427(96)00018-0 [2] Bekas, C.; Kokiopoulou, E.; Saad, Y.: An estimator for the diagonal of a matrix, Appl. numer. Math. 57, 1214-1229 (2007) · Zbl 1123.65026 · doi:10.1016/j.apnum.2007.01.003 [3] Benzi, M.: Preconditioning techniques for large linear systems: a survey, J. comput. Phys. 182, 418-477 (2002) · Zbl 1015.65018 · doi:10.1006/jcph.2002.7176 [4] Crouzeix, J. P.; Seeger, A.: New bounds for the extreme values of a finite sample of real numbers, J. math. Anal. appl. 197, 411-426 (1996) · Zbl 0869.49017 · doi:10.1006/jmaa.1996.0029 [5] Datta, B. N.: Numerical linear algebra and applications, (1995) · Zbl 1182.65001 [6] Hill, R. D.; Waters, S. R.: On the cone of positive semidefinite matrices, Linear algebra appl. 90, 81-88 (1987) · Zbl 0615.15008 · doi:10.1016/0024-3795(87)90307-7 [7] Iusem, A. N.; Seeger, A.: Measuring the degree of pointedness of a closed convex cone: a metric approach, Math. nachr. 279, 599-618 (2006) · Zbl 1110.15015 · doi:10.1002/mana.200310380 [8] Iusem, A. N.; Seeger, A.: On pairs of vectors achieving the maximal angle of a convex cone, Math. program., ser. B 104, 501-523 (2005) · Zbl 1087.52005 · doi:10.1007/s10107-005-0626-z [9] Merikoski, J. K.; Urpala, U.; Virtanen, A.; Tam, T. -Y.; Uhlig, F.: A best upper bound for the 2-norm condition number of a matrix, Linear algebra appl. 254, 355-365 (1997) · Zbl 0877.15006 · doi:10.1016/S0024-3795(96)00474-0 [10] Saad, Y.: Iterative methods for sparse linear systems, (2003) · Zbl 1031.65046 [11] Tarazaga, P.: Eigenvalue estimates for symmetric matrices, Linear algebra appl. 135, 171-179 (1990) · Zbl 0701.15012 · doi:10.1016/0024-3795(90)90120-2 [12] Tarazaga, P.: More estimates for eigenvalues and singular values, Linear algebra appl. 149, 97-110 (1991) · Zbl 0723.15014 · doi:10.1016/0024-3795(91)90328-T [13] Wolkowicz, H.; Styan, G. P. H.: Bounds for eigenvalues using traces, Linear algebra appl. 29, 471-506 (1980) · Zbl 0435.15015 · doi:10.1016/0024-3795(80)90258-X