In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function.[1]
It was introduced by J.D. 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]
Definition
editThe 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).
Properties
editTheorem: Let , then .[1]
Theorem: Let , then is diagonalizable and has eigenvalues that are .[1]
Theorem: Let , then is a projector onto the invariant subspace associated with the eigenvalues in the right-half plane, and analogously for and the left-half plane.[1]
Theorem: Let , and be a Jordan decomposition such that corresponds to eigenvalues with positive real part and to eigenvalue with negative real part. Then , where and are identity matrices of sizes corresponding to and , respectively.[1]
Computational methods
editThe function can be computed with generic methods for matrix functions, but there are also specialized methods.
Newton iteration
editThe 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 , that is, the iteration , and define the new iterate , we arrive at the iteration
,
where typically . Convergence is global, and locally it is quadratic.[1][2]
The Newton iteration uses the explicit inverse of the iterates .
Newton–Schulz iteration
editTo 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 ).[1]
Applications
editSolutions of Sylvester equations
editTheorem:[2][3] Let and assume that and are stable, then the unique solution to the Sylvester equation, , is given by such that
Proof sketch: The result follows from the similarity transform
since
due to the stability of and .
The theorem is, naturally, also applicable to the Lyapunov equation. However, due to the structure the Newton iteration simplifies to only involving inverses of and .
Solutions of algebraic Riccati equations
editThere is a similar result applicable to the algebraic Riccati equation, .[1][2] Define as
Under the assumption that 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
and the stability of implies that
for some matrix .
Computations of matrix square-root
editThe 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 is the square root of .
References
edit- ^ a b c d e f g h Higham, Nicholas J. (2008). Functions of matrices : theory and computation. Society for Industrial and Applied Mathematics. Philadelphia, Pa.: Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104). ISBN 978-0-89871-777-8. OCLC 693957820.
- ^ a b c d Roberts, J. D. (October 1980). "Linear model reduction and solution of the algebraic Riccati equation by use of the sign function". International Journal of Control. 32 (4): 677–687. doi:10.1080/00207178008922881. ISSN 0020-7179.
- ^ a b c Denman, Eugene D.; Beavers, Alex N. (1976). "The matrix sign function and computations in systems". Applied Mathematics and Computation. 2 (1): 63–94. doi:10.1016/0096-3003(76)90020-5. ISSN 0096-3003.
- ^ Schulz, Günther (1933). "Iterative Berechung der reziproken Matrix". ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 13 (1): 57–59. Bibcode:1933ZaMM...13...57S. doi:10.1002/zamm.19330130111. ISSN 1521-4001.