The solution of the equation AX+BX*=0 Articles uri icon

publication date

  • December 2013

start page

  • 1605

end page

  • 1628

issue

  • 12

volume

  • 61

International Standard Serial Number (ISSN)

  • 0308-1087

Electronic International Standard Serial Number (EISSN)

  • 1563-5139

abstract

  • We give a complete solution of the matrix equation AX+BX=0, where A, BCmxn are two given matrices, XCnxn is an unknown matrix, and denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A+B, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A+B to the canonical form by strict equivalence.

keywords

  • matrix equations ; matrix pencils ; kronecker canonical form ; transpose; conjugate transpose sylvester equation