Statistics and characterization of matrices by determinant and trace
This paper presents results on different aspects concerning square matrices with integer and complex entries. The first part of the paper is about proving that the probability of a random \(n\times n\) matrix, whose entries are (positive) integer prime numbers less than or equal to \(x\), being singular goes to zero as \(x\) goes to infinity (Theorem 1). The second part of the work concerns unitary, Hermitian and normal matrices over the complex numbers. Theorems 2 and 3 state that a (real) complex \(n\times n\) square matrix \(A\) is unitary with respect to a positive definite (bilinear form or scalar product over \(\mathbb{R}\)) Hermitian form if and only if its determinant is \(\pm 1\) and the trace of \(AA^*\) equals \(n\), where \(A^*\) denotes the conjugate transpose of \(A\). Moreover, the authors give a characterization of non-singular commuting complex square matrices having the same characteristic polynomial in terms of determinant and trace. They also show that a complex square matrix being Hermitian is equivalent to it having the same eigenvectors as its conjugate adjoint. Finally, it is proved that a complex matrix \(A\) is normal if and only if \(AA^*\) and \(A^*A\) have the same eigenvectors corresponding to each of their (common) eigenvalues.
scientific article; zbMATH DE number 7040082