In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function.
Roberts in 1971 as a tool for model reduction and for solving Lyapunov and Algebraic Riccati equation in a technical report of Cambridge University, which was later published in a journal in 1980.
[2][3] The matrix sign function is a generalization of the complex signum function
to the matrix valued analogue
Although the sign function is not analytic, the matrix function is well defined for all matrices that have no eigenvalue on the imaginary axis, see for example the Jordan-form-based definition (where the derivatives are all zero).
is a projector onto the invariant subspace associated with the eigenvalues in the right-half plane, and analogously for
corresponds to eigenvalues with positive real part and
to eigenvalue with negative real part.
are identity matrices of sizes corresponding to
[1] The function can be computed with generic methods for matrix functions, but there are also specialized methods.
The Newton iteration can be derived by observing that
, which in terms of matrices can be written as
, where we use the matrix square root.
If we apply the Babylonian method to compute the square root of the matrix
Convergence is global, and locally it is quadratic.
[1][2] The Newton iteration uses the explicit inverse of the iterates
To avoid the need of an explicit inverse used in the Newton iteration, the inverse can be approximated with one step of the Newton iteration for the inverse,
, derived by Schulz(de) in 1933.
[4] Substituting this approximation into the previous method, the new method becomes
Convergence is (still) quadratic, but only local (guaranteed for
are stable, then the unique solution to the Sylvester equation,
Proof sketch: The result follows from the similarity transform
The theorem is, naturally, also applicable to the Lyapunov equation.
However, due to the structure the Newton iteration simplifies to only involving inverses of
There is a similar result applicable to the algebraic Riccati equation,
are Hermitian and there exists a unique stabilizing solution, in the sense that
is stable, that solution is given by the over-determined, but consistent, linear system
Proof sketch: The similarity transform
The Denman–Beavers iteration for the square root of a matrix can be derived from the Newton iteration for the matrix sign function by noticing that
is a degenerate algebraic Riccati equation[3] and by definition a solution