Linear Operators Related to Matrix Equation Solvers

L = lyapop(A; disc = false, her = false)

Define, for an n x n matrix A, the continuous Lyapunov operator L:X -> AX+XA' if disc = false or the discrete Lyapunov operator L:X -> AXA'-X if disc = true. If her = false the Lyapunov operator L:X -> Y maps general square matrices X into general square matrices Y, and the associated matrix M = Matrix(L) is $n^2 \times n^2$. If her = true the Lyapunov operator L:X -> Y maps symmetric/Hermitian matrices X into symmetric/Hermitian matrices Y, and the associated matrix M = Matrix(L) is $n(n+1)/2 \times n(n+1)/2$. For the definitions of the Lyapunov operators see:

M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov operators. Linear Algebra and its Applications 312:35–71, 2000.

L = lyapop(A, E; disc = false, her = false)

Define, for a pair (A,E) of n x n matrices, the continuous Lyapunov operator L:X -> AXE'+EXA' if disc = false or the discrete Lyapunov operator L:X -> AXA'-EXE' if disc = true. If her = false the Lyapunov operator L:X -> Y maps general square matrices X into general square matrices Y, and the associated matrix M = Matrix(L) is $n^2 \times n^2$. If her = true the Lyapunov operator L:X -> Y maps symmetric/Hermitian matrices X into symmetric/Hermitian matrices Y, and the associated M = Matrix(L) is a $n(n+1)/2 \times n(n+1)/2$. For the definitions of the Lyapunov operators see:

M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov operators. Linear Algebra and its Applications 312:35–71, 2000.

LINV = invlyapop(A; disc = false, her = false)

Define LINV, the inverse of the continuous Lyapunov operator L:X -> AX+XA' for disc = false or the inverse of the discrete Lyapunov operator L:X -> AXA'-X for disc = true, where A is an n x n matrix. If her = false the inverse Lyapunov operator LINV:Y -> X maps general square matrices Y into general square matrices X, and the associated matrix M = Matrix(LINV) is $n^2 \times n^2$. If her = true the inverse Lyapunov operator LINV:Y -> X maps symmetric/Hermitian matrices Y into symmetric/Hermitian matrices X, and the associated matrix M = Matrix(LINV) is $n(n+1)/2 \times n(n+1)/2$. For the definitions of the Lyapunov operators see:

M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov operators. Linear Algebra and its Applications 312:35–71, 2000.

LINV = invlyapop(A, E; disc = false, her = false)

Define LINV, the inverse of the continuous Lyapunov operator L:X -> AXE'+EXA' for disc = false or the inverse of the discrete Lyapunov operator L:X -> AXA'-EXE' for disc = true, where (A,E) is a pair of n x n matrices. If her = false the inverse Lyapunov operator LINV:Y -> X maps general square matrices Y into general square matrices X, and the associated matrix M = Matrix(LINV) is $n^2 \times n^2$. If her = true the inverse Lyapunov operator LINV:Y -> X maps symmetric/Hermitian matrices Y into symmetric/Hermitian matrices X, and the associated matrix M = Matrix(LINV) is $n(n+1)/2 \times n(n+1)/2$. For the definitions of the Lyapunov operators see:

M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov operators. Linear Algebra and its Applications 312:35–71, 2000.

LINV = invlyapsop(A; disc = false, her = false)

Define LINV, the inverse of the continuous Lyapunov operator L:X -> AX+XA' for disc = false or the inverse of the discrete Lyapunov operator L:X -> AXA'-X for disc = true, where A is an n x n matrix in Schur form. If her = false the inverse Lyapunov operator LINV:Y -> X maps general square matrices Y into general square matrices X, and the associated matrix M = Matrix(LINV) is $n^2 \times n^2$. If her = true the inverse Lyapunov operator LINV:Y -> X maps symmetric/Hermitian matrices Y into symmetric/Hermitian matrices X, and the associated matrix M = Matrix(LINV) is $n(n+1)/2 \times n(n+1)/2$. For the definitions of the Lyapunov operators see:

M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov operators. Linear Algebra and its Applications 312:35–71, 2000.

LINV = invlyapsop(A, E; disc = false, her = false)

Define LINV, the inverse of the continuous Lyapunov operator L:X -> AXE'+EXA' for disc = false or the inverse of the discrete Lyapunov operator L:X -> AXA'-EXE' for disc = true, where (A,E) is a pair of n x n matrices in generalized Schur form. If her = false the inverse Lyapunov operator LINV:Y -> X maps general square matrices Y into general square matrices X, and the associated matrix M = Matrix(LINV) is $n^2 \times n^2$. If her = true the inverse Lyapunov operator LINV:Y -> X maps symmetric/Hermitian matrices Y into symmetric/Hermitian matrices X, and the associated matrix M = Matrix(LINV) is $n(n+1)/2 \times n(n+1)/2$. For the definitions of the Lyapunov operators see:

M. Konstantinov, V. Mehrmann, P. Petkov. On properties of Sylvester and Lyapunov operators. Linear Algebra and its Applications 312:35–71, 2000.

M = sylvop(A, B; disc = false)

Define the continuous Sylvester operator M: X -> AX+XB if disc = false or the discrete Sylvester operator M: X -> AXB+X if disc = true, where A and B are square matrices.

M = sylvop(A, B, C, D)

Define the generalized Sylvester operator M: X -> AXB+CXD, where (A,C) and (B,D) a pairs of square matrices.

MINV = invsylvop(A, B; disc = false)

Define MINV, the inverse of the continuous Sylvester operator M: X -> AX+XB if disc = false or of the discrete Sylvester operator M: X -> AXB+X if disc = true, where A and B are square matrices.

MINV = invsylvop(A, B, C, D)

Define MINV, the inverse of the generalized Sylvester operator M: X -> AXB+CXD, where (A,C) and (B,D) a pairs of square matrices.

MINV = invsylvsop(A, B; disc = false)

Define MINV, the inverse of the continuous Sylvester operator M: X -> AX+XB if disc = false or of the discrete Sylvester operator M: X -> AXB+X if disc = true, where A and B are square matrices in Schur forms.

MINV = invsylvsop(A, B, C, D; DBSchur = false)

Define MINV, the inverse of the generalized Sylvester operator M: X -> AXB+CXD, with the pairs (A,C) and (B,D) in generalized Schur forms. If DBSchur = true, the pair (D,B) is in generalized Schur form.

M = sylvsysop(A, B, C, D)

Define the operator M: (X,Y) -> (AX+YB, CX+YD ), where (A,C) and (B,D) a pairs of square matrices.

MINV = invsylvsysop(A, B, C, D)

Define MINV, the inverse of the linear operator M: (X,Y) -> (AX+YB, CX+YD ), where (A,C) and (B,D) a pairs of square matrices.

MINV = invsylvsyssop(A, B, C, D)

Define MINV, the inverse of the linear operator M: (X,Y) -> (AX+YB, CX+YD), with the pairs (A,C) and (B,D) in generalized Schur forms.

M = trmatop(n, m)

Define the transposition operator M: X -> X' for all n x m matrices.

M = trmatop(A)

Define the transposition operator M: X -> X' of all matrices of the size of A.