Abstract
We give a complete solution of the matrix equation AX + BX ⋆ = 0, where A, B ∈ ℂ m×n are two given matrices, X ∈ ℂ n×n 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 + λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A + λB to the canonical form by strict equivalence.
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