On the extreme values of the roots of matrices
Tetsuro Yamamoto · Journal of the Mathematical Society of Japan · 1967
In this paper we shall investigate some properties concerning the behavior of the eigenvalues and singular values of complex matrices.Let $A$ be an nsquare matrix and $p$ be any positive integer.Let the eigenvalues of $A$ and the singular values of $A^{p}$ be denoted by $\lambda_{i}$ and $\alpha_{i}^{(p)}(1\leqq i\leqq n)$ respectively, which are so arranged that $|\lambda_{1}|\geqq|\lambda_{2}|\geqq\ldots\geqq|\lambda_{n}|$ and $\alpha_{1}^{(p)}\geqq\alpha_{2}^{(p)}\geqq\ldots\geqq\alpha_{n}^{(p)}$ .Then in \S 1 we shall prove that $\lim_{p\rightarrow\infty}\alpha_{i}^{(p)^{\frac{1}{p}}}=|\text{{\it \'{A}}}_{i}|,$ $i=1,2,$ $\cdots$ , $n$ .This generalizes a Gautschi's result ([3] p. 138).In \S 2 we shall treat non-negative matrices and state some properties which improve some results obtained by Gautschi [3] and Brauer [1].