Invertibility of Matrices of Field Elements

Invertibility of Matrices of Field Elements

In this paper the theory of invertibility of matrices of field elements (see e.g. [5], [6]) is developed. The main purpose of this article is to prove that the left invertibility and the right invertibility are equivalent for a matrix of field elements. To prove this, we introduced a special transformation of matrix to some canonical forms. Other concepts as zero vector and base vectors of fiel...

Invertibility of symmetric random matrices

We study n × n symmetric random matrices H, possibly discrete, with iid abovediagonal entries. We show that H is singular with probability at most exp(−nc), and ‖H−1‖ = O(√n). Furthermore, the spectrum of H is delocalized on the optimal scale o(n−1/2). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of T...

Calculation of Matrices of Field Elements . Part

The articles [8], [3], [10], [11], [4], [1], [5], [2], [13], [6], [7], [12], and [9] provide the notation and terminology for this paper. In this paper i denotes a natural number. Let K be a field and let M1, M2 be matrices over K. The functor M1 −M2 yielding a matrix over K is defined by: (Def. 1) M1 − M2 = M1 + −M2. One can prove the following propositions: (1) For every field K and for every...

Determinant of Some Matrices of Field Elements

Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that t...

Invertibility of Random Matrices Lecture Notes