Bidiagonal Decompositions of Oscillating Systems of Vectors Articles uri icon

publication date

  • June 2008

start page

  • 2536

end page

  • 2548


  • 11-12


  • 428

International Standard Serial Number (ISSN)

  • 0024-3795

Electronic International Standard Serial Number (EISSN)

  • 1873-1856


  • We establish necessary and sufficient conditions, in the language of bidiagonal decompositions, for a matrix V to be an eigenvector matrix of a totally positive matrix. Namely, this is the case if and only if V and V-T are lowerly totally positive. These conditions translate into easy positivity requirements on the parameters in the bidiagonal decompositions of V and V-T. Using these decompositions we give elementary proofs of the oscillating properties of V. In particular, the fact that the jth column of V has j-1 changes of sign. Our new results include the fact that the Q matrix in a QR decomposition of a totally positive matrix belongs to the above class (and thus has the same oscillating properties).