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nested matrix; tridiagonal matrix; inverse $M$-matrix; principal minor; determinant; $QR$-factorization
Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the $LU$- and $QR$-factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse $M$-matrices with symmetric, irreducible, tridiagonal inverses.
[1] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9 SIAM, Philadelphia (1994). MR 1298430 | Zbl 0815.15016
[2] The On-Line Encyclopedia of Integer Sequences. (2010).
[3] Weisstein, E. W.: Fibonorial. From MathWorld---A Wolfram Web Resource,\
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