One of the most effective methods for solving the matrix equation AX+XB=C is the Bartels-Stewart algorithm. Key to this technique is the orthogonal reduction of A and B to triangular form using the QR algorithm for eigenvalues. A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form. The resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B . The stability of the new method is demonstrated through a roundoff error analysis and supported by numerical tests. Finally, it is shown how the techniques described can be applied and generalized to other matrix equation problems.

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A Hessenberg-Schur method for the problem AX + XB= C

conditions and an accurate semidefinite programming solver," The Journal of the SDPA family for solving large-scale SDPs," in Handbook on Semidefinite. Semidefinite Programming, Combinatorial. Optimization and Primal semidefinite programming problem Handbook of Semidefinite Programming. Kluwer. Solving Euclidean distance matrix completion problems via semidefinite programming. AY Alfakih, A Khandani, H Wolkowicz. Computational optimization.

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Handbook of Semidefinite Programming

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

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Computational Methods for Linear Matrix Equations

The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣∂(vecX)/∂(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ∂(vechX)/∂(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics.

Vec and vech operators for matrices, with some uses in jacobians and multivariate statistics

Deterministic and stochastic linear periodic systems

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Remarks on inertia theorems for matrices

The equation of the title is attacked via Taylor's formula for matrix functions. This approach contrasts with that of Lancaster [1], who used methods of contour integration suggested by functional analysis. The paper is designed as a supplement to his, being organized around a number of generalizations of his theorems. Solutions of the target equation are expressed in terms of component matrices of A and B and in terms of contour integrals.

Solving the Matrix Equation $\sum _{\rho = 1}^r f_\rho (A)Xg_\rho (B) = C$

Es sei A eine komplexe n x n Matrix mit den Eigenwerten It, . . ., In. Die Funktionen fij(x), i , j = 1, . . ., m seien analytisch auf dem Spektrum von A . F (x) sei die m x m Matrix (fij (5)). Fur die nm-reihige Blockmatrix P ( A ) gilt der folgende Satz: Sstz 1 (WILLIAMSON [4]): Die Eigenwerte der Matrix . . P ( A ) = ( f , , (A)) % , 3 = 1, . . . , m sind die nm Eigenwerte der Matrizen F(&), k = 1, . . ., n. In dieser Note geben wir ein System von Hauptvektoren von P ( A ) fur den Fall an, daS A von einfacher Struktur ist. Dieses Ergebnis liefert einen anderen Beweis fur den Satz von WILLIAMSON und zudem als Spezialfall einen Satz von EGERVLLRY uber die Spektralzerlegung von P ( A ) bei HERMITEschem A und E’(s). Im weiteren wird die Inverse von F ( A ) bestimmt, und fur die Gleichung Zf,(A) X B , = C werden Losungen in expliziter Form angegeben.

Blockmatrizen und lineare Matrizengleichungen

This paper is an exposition of results achieved by Garrets Birkhoff and coworkers in their study of the linear differential equation ${\bf x}' = A(t){\bf x}$, in which the off-diagonal entries of the coefficient matrix are nonnegative. Of interest is the direction of a solution vector, not its length. Conditions are established guaranteeing that solutions starting in a certain half-space will eventually wind up in the positive orthant, and it is shown that all such solutions are “eventually parallel”. Analogies between length and direction problems are pointed out, including a brief discussion of a directionally stable solution of a perturbed equation ${\bf x}' = A{\bf x} + {\bf f}(t,{\bf x})$.

Allen D. Ziebur

Papers

Remarks on inertia theorems for matrices

Solving the Matrix Equation $\sum _{\rho = 1}^r f_\rho (A)Xg_\rho (B) = C$

Blockmatrizen und lineare Matrizengleichungen

New Directions in Linear Differential Equations

Age-dependent models of population growth.