### A bound for the Moore-Penrose pseudoinverse of a matrix

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The construction of a well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition is proposed. A suitable parametrix is obtained by using a new unknown and an approximation of the transparency condition. We prove the well-posedness of the equation for any wavenumber. Finally, some numerical comparisons with well-tried method prove the efficiency of the new formulation.

The Demmel condition number is an indicator of the matrix condition, and its properties have recently found applications in many practical problems, such as in MIMO communication systems, in the analytical prediction of level-crossing and fade duration statistics of Rayleigh channels, and in spectrum sensing for cognitive radio systems, among others. As the characterizations of a probability distribution play an important role in probability and statistics, in this paper we study the characterizations...

a recurrence relation for computing the ${L}_{p}$-norms of an Hermitian matrix is derived and an expression giving approximately the number of eigenvalues which in absolute value are equal to the spectral radius is determined. Using the ${L}_{p}$-norms for the approximation of the spectral radius of an Hermitian matrix an a priori and a posteriori bounds for the error are obtained. Some properties of the a posteriori bound are discussed.

For a real square matrix $A$ and an integer $d\ge 0$, let ${A}_{\left(d\right)}$ denote the matrix formed from $A$ by rounding off all its coefficients to $d$ decimal places. The main problem handled in this paper is the following: assuming that ${A}_{\left(d\right)}$ has some property, under what additional condition(s) can we be sure that the original matrix $A$ possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number...