Jacobi matrix

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A square matrix with real entries such that for . If one writes (), , and (), then a Jacobi matrix has the form

Any minor of a Jacobi matrix is the product of certain principal minors of and certain elements of . A Jacobi matrix is completely non-negative (that is, all its minors are non-negative) if and only if all its principal minors and all elements and () are non-negative. If for , then the roots of the characteristic polynomial of are real and distinct.


[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)
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Jacobi matrix. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article