Elementary bidiagonal factorizations

Bidiagonal decompositions, minors and applications

Butterflies Solve Bidiagonal Toeplitz Systems

Here and hereafter T = (ti,j) n−1 i,j=0, ti,j = 1 for i − j = 0, ti,j = c 6= 0 for i − j = 1, and ti,j = 0 otherwise, x = (xi) n−1 i=0 , and b = (bi) n−1 i=0 . Note that the system is scaled so that the main diagonal is composed exclusively of ones, with no loss of generality. These systems are at the heart of problems as diverse as cubic spline and Bspline curve fitting [3], [11], precondition...

Penrose Inverse of Bidiagonal Matrices

Introduction. For a real m×n matrix A, the Moore–Penrose inverse A+ is the unique n×m matrix that satisfies the following four properties: AAA = A , AAA = A , (A+A)T = AA , (AA+)T = AA (see [1], for example). If A is a square nonsingular matrix, then A+ = A−1. Thus, the Moore–Penrose inversion generalizes ordinary matrix inversion. The idea of matrix generalized inverse was first introduced in ...

Bidiagonal Decompositions of Oscillating Systems of Vectors

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 decompo...

Accurate Singular Values of Bidiagonal Matrices

2 has nonzero entries only on its diagonal and first superdiagonal ) Compute orthogonal matrices P and Q such that Σ = P BQ is diagonal and nonnegat i 2 2 2 T 2 ive. The diagonal entries σ of Σ are the singular values of A . We will take them to be sorted in decreasing order: σ ≥ σ . The columns of Q= Q Q are the right singular vec i i + 1 1 2 t 1 2 ors, and the columns of P= P P are the left s...