Solve the discrete Lyapunov equation :

Solve a discrete Lyapunov equation:

Verify the solution:

Solve an equation with symbolic matrices:

Solve for coefficient matrices having different dimensions:

Solve :

Solve :

Solve the discrete Lyapunov equation with symbolic coefficients:

Obtain the symbolic solution of :

Test the stability of by checking if the solution of is positive definite for a negative definite :

As expected, the eigenvalues are inside the unit circle:

An unstable system:

Compute the controllability Gramian of a stable discrete-time system:

Compute the observability Gramian of a stable discrete-time system:

The equation , with a negative definite , yields a unique positive definite solution if and only if the eigenvalues of are within the unit circle:

An unstable system:

The indefinite sum is the solution to if is asymptotically stable:

Compute the infinite-horizon quadratic cost for the asymptotically stable system :

Compute the same using direct summation:

Solve the matrix equation :

LinearSolve gives the same solution:

Solve the equation using LinearSolve:

DiscreteLyapunovSolve gives the same solution: