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The Sylvester equation $$AX - XA = C$$ with symmetric $A$ and skew-symmetric $C$, admits non-unique symmetric solutions for $X$. See here for example. Of course, particular solutions can be obtained by solving the linear system $((I\otimes A) - (A\otimes I)){\mathrm{vec}}(X) = {\mathrm{vec}}(C)$.

I am interested in a special case where $A$ and $C$ have additional structure: $$A = A_1 - A_2, \quad C = A_{2}A_{1} - A_{1}A_{2}, \quad\text{where}\; A_{1}, A_{2}\;\text{are symmetric}.$$ Can anything further be said about the symmetric solution $X$ in such cases?

Abhishek Halder
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2 Answers2

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You may note that $X = -A_1$ and $X = -A_2$ are always solutions in such cases.

Robert Israel
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Since $A$ is symmetric, it is diagonalizable. We have two cases.

Case 1: $A$ has distinct eigenvalues

The minimal and characteristic polynomials coincide. Thus, by a theorem of Taussky and Zassenhaus, the solutions of $$ AX-XA=0$$ is always symmetric. Moreover, the dimension of the above space is $n$, the size of the matrix. Also, it is the space of polynomials of $A$. That is, $$ K=\{c_0 I + c_1 A + \cdots c_{n-1} A^{n-1} \ | \ c_i\in\mathbb{R}\}. $$ As we have a solution $X=-A_1$ by Robert Israel, we have the general solutions of $AX-XA=C$ given by the set $$ W=\{-A_1+c_0 I + c_1 A + \cdots c_{n-1} A^{n-1}\ | \ c_i\in\mathbb{R}\}. $$ Note that $-A_2$ is also in $W$.

Case 2: $A$ has repeated eigenvalues

By another theorem by Taussky and Zassenhaus, the solutions of $$AX-XA=0$$ may not be symmetric. Also, the dimension of the solution space is greater than $n$. Thus, the set $W$ above is contained in the solution set of $AX-XA=C$, but there are solutions outside the above $W$. Restricting $X$ to be symmetric, we still have solutions outside $W$ by comparing dimension. To see this, the space of $X$ satisfying $$AX-XA=0, \ X=X^T$$ has a dimension at least $n$ by rank-nullity theorem. However, the space of polynomials of $A$ has dimension less than $n$.

Sungjin Kim
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