# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On a solution of the quaternion matrix equation $X-A \widetilde{X} B=C$ and its application. (English) Zbl 1083.15019
The authors study the solvability and explicit representation by means of a characteristic polynomial of the solution to the matrix equation $X-AXB=C$. Then they study solvability of the matrix equation $(*)$ $X-A\widetilde{X}B=C$ where if $x=a+bi+cj+dk$ is a quaternion ($i^2=j^2=k^2=-1$, $ij=-ji=k$), then $\widetilde{x}=a-bi+cj-dk$. To solve equation $(*)$ explicitly a real representation of quaternion matrices is used. The results are applied to the resolution of the matrix equation $X-A\overline{X}B=C$ where $\overline{X}$ is the complex conjugate of $X$.

##### MSC:
 15A24 Matrix equations and identities 15B33 Matrices over special rings (quaternions, finite fields, etc.)
##### Keywords:
quaternion matrix equation; solution; real representation
Full Text:
##### References:
 [1] Barnett, S., Storey, C.: Matrix Methods in Stability Theory, Nelson, London, 1970 · Zbl 0243.93017 [2] Barnett, S.: Matrices in Control Theory with Applications to Linear Programming, Van Nostrand Reinhold, New York, 1971 · Zbl 0245.93002 [3] Jameson, A.: Solution of the equation AX XB = C by inversion of an M {$\times$}M or N {$\times$} N matrix. SIAM J. Appl. Math., 16, 1020--1023 (1968) · Zbl 0169.35202 · doi:10.1137/0116083 [4] Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications, 2nd ed., Academic Press, New York, 1985 · Zbl 0558.15001