Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding

Jiří Rohn

Displaying similar documents to “Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding”

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

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Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) $\sum_{i = 1}^{n} X ₂_{i} - d \sum_{i = 1}^{n} Y ²_{i} = - I$ with d ∈ N for nonsingular $X_{i}, Y_{i} \in M ₂ (Z)$ , i=1,...,n.

Nested matrices and inverse $M$ -matrices

Jeffrey L. Stuart (2015)

Czechoslovak Mathematical Journal

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Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the $L U$ - and $Q R$ -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.

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$

Daniel Uzcátegui Contreras, Dardo Goyeneche, Ondřej Turek, Zuzana Václavíková (2021)

Communications in Mathematics

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It is known that a real symmetric circulant matrix with diagonal entries $d \geq 0$ , off-diagonal entries $\pm 1$ and orthogonal rows exists only of order $2 d + 2$ (and trivially of order $1$ ) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d \geq 0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider...

G-matrices, $J$ -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_{1}$ and $D_{2}$ such that $A^{- T} = D_{1} A D_{2}$ , where $A^{- T}$ denotes the transpose of the inverse of $A$ . Denote by $J = diag (\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+ 1$ or $- 1$ . A nonsingular real matrix $Q$ is called $J$ -orthogonal if $Q^{T} J Q = J$ . Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

A direct solver for finite element matrices requiring $O (N log N)$ memory places

Vejchodský, Tomáš

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We present a method that in certain sense stores the inverse of the stiffness matrix in $O (N log N)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O (N^{3 / 2})$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O (N log N)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular...

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an $2 n \times 2 n$ Brualdi-Li tournament matrix $B_{2 n}$ . We show that $B_{2 n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2 n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.

Analytic aspects of the circulant Hadamard conjecture

Teodor Banica, Ion Nechita, Jean-Marc Schlenker (2014)

Annales mathématiques Blaise Pascal

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We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $| q_{0} | = ... = | q_{N - 1} | = 1$ the quantity $Φ = \sum_{i + k = j + l} \frac{q_{i} q_{k}}{q_{j} q_{l}}$ satisfies $Φ \geq N^{2}$ , with equality if and only if $q = (q_{i})$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $Φ$ , (2) the study of the critical points of $Φ$ , and (3) the computation of the moments of $Φ$ . We explore here...

Displaying similar documents to “Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding”

On the matrix negative Pell equation

Nested matrices and inverse M -matrices

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

G-matrices, J -orthogonal matrices, and their sign patterns

A direct solver for finite element matrices requiring O ( N log N ) memory places

Some properties complementary to Brualdi-Li matrices

Analytic aspects of the circulant Hadamard conjecture

Nested matrices and inverse $M$ -matrices

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$

G-matrices, $J$ -orthogonal matrices, and their sign patterns

A direct solver for finite element matrices requiring $O (N log N)$ memory places