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## Matrix equations and the Kronecker product

### 4.0    Motivation

#### 4.0.1    Lyapunov’s equation

In Section (2.2), the equation

arose in connection with the study of matrix stability. The matrix A ϵ Mn is given and the right-hand side H ϵ Mn is Hermitian. The unknown matrix X ϵ Mn is to be determined if possible. We are interested in the solvability of (4.0.1.1) in terms of properties of A and/or H.

It is important to note that the left-hand side of the matrix equation (4.0.1.1) is linear in the unknown X. This linearity may be thought of in two ways: If we define the mapping L: MnMn by L(X) ≡ XA + A*X, then L(αX + βY) = αL(X) + βL

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