Oscillating matrix
oscillatory matrix
A totally non-negative matrix for which there exists a positive integer
such that
is a totally positive matrix; the matrix
is called totally non-negative (totally positive) if all its minors, of whatever order, are non-negative (positive). The lowest exponent
is called the exponent of the oscillating matrix. If
is an oscillating matrix with exponent
, then for any integer
the matrix
is totally positive; an integer positive power of an oscillating matrix and the matrix
are also oscillating matrices. In order that a totally non-negative matrix
is an oscillating matrix, it is necessary and sufficient that: 1)
is a non-singular matrix; and 2) for
, the following are fulfilled:
,
.
The basic theorem on oscillating matrices is: An oscillating matrix always has
different positive eigen values; for the eigen vector
that corresponds to the largest eigen value
, all coordinates differ from zero and are of the same sign; for an eigen vector
that corresponds to the
-th eigen value
(arranged according to decreasing value) there are exactly
changes of sign; for any real numbers
,
,
, the number of changes of sign in the sequence of coordinates of the vector
is between
and
.
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) |
Comments
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) |
Oscillating matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillating_matrix&oldid=34294