Oscillating matrix

oscillatory matrix

A totally non-negative matrix $A$ for which there exists a positive integer $\chi$ such that $A^\chi$ is a totally positive matrix; the matrix $A$ is called totally non-negative (totally positive) if all its minors, of whatever order, are non-negative (positive). The lowest exponent $\chi$ is called the exponent of the oscillating matrix. If $A$ is an oscillating matrix with exponent $\chi$, then for any integer $k\geq\chi$ the matrix $A^k$ is totally positive; an integer positive power of an oscillating matrix and the matrix $(A^+)^{-1}$ are also oscillating matrices. In order that a totally non-negative matrix $A=\|a_{ik}\|_1^n$ is an oscillating matrix, it is necessary and sufficient that: 1) $A$ is a non-singular matrix; and 2) for $i=1,\dots,n$, the following are fulfilled: $a_{i,i+1}>0$, $a_{i+1,i}>0$.

The basic theorem on oscillating matrices is: An oscillating matrix $A=\|a_{ik}\|_1^n$ always has $n$ different positive eigen values; for the eigen vector $u^1$ that corresponds to the largest eigen value $\lambda_1$, all coordinates differ from zero and are of the same sign; for an eigen vector $u^s$ that corresponds to the $s$-th eigen value $\lambda_s$ (arranged according to decreasing value) there are exactly $s-1$ changes of sign; for any real numbers $c_g,\dots,c_h$, $1\leq g\leq h\leq n$, $\sum_{k=g}^hc_k^2>0$, the number of changes of sign in the sequence of coordinates of the vector $u=\sum_{k=g}^hc_ku^k$ is between $g-1$ and $h-1$.

References

[1]	F.R. Gantmakher, M.G. Krein, "Oscillation matrices and kernels and small vibrations of mechanical systems" , Dept. Commerce USA. Joint Publ. Service (1961) (Translated from Russian)

References

[a1]	S. Karlin, "Total positivity" , Stanford Univ. Press (1960)
[a2]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 2 , Chelsea, reprint (1959) pp. Chapt. XIII, §9 (Translated from Russian)