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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 =tr(A T B) be defined in the space of square real n×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, αI, for α>0, and the angle that A -1 forms with αI. 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 κ F (A) is given by the expression κ F (A)max(n,n cos 2 (A,I)) and its accuracy is evaluated in numerical experiments.
MSC:
15A12Conditioning of matrices
15A63Quadratic and bilinear forms, inner products
15A45Miscellaneous inequalities involving matrices
15A60Applications of functional analysis to matrix theory
15A48Positive matrices and their generalizations (MSC2000)
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