Matrix Sign Function

In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function.

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.

Definition

The matrix sign function is a generalization of the complex signum function

$\operatorname(z)= \begin 1 & \text \mathrm(z) > 0, \\ -1 & \text \mathrm(z) < 0, \end$

to the matrix valued analogue $\operatorname(A)$. 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

Theorem: Let $A\in\C^$, then $\operatorname(A)^2 = I$.

Theorem: Let $A\in\C^$, then $\operatorname(A)$ is diagonalizable and has eigenvalues that are $\pm 1$.

Theorem: Let $A\in\C^$, then $(I+\operatorname(A))/2$ is a projector onto the invariant subspace associated with the eigenvalues in the right-half plane, and analogously for $(I-\operatorname(A))/2$ and the left-half plane.

Theorem: Let $A\in\C^$, and $A = P\beginJ_+ & 0 \\ 0 & J_-\endP^$ be a Jordan decomposition such that $J_+$ corresponds to eigenvalues with positive real part and $J_-$ to eigenvalue with negative real part. Then $\operatorname(A) = P\beginI_+ & 0 \\ 0 & -I_-\endP^$, where $I_+$ and $I_-$ are identity matrices of sizes corresponding to $J_+$ and $J_-$, respectively.

Computational methods

The function can be computed with generic methods for matrix functions, but there are also specialized methods.

Newton iteration

The Newton iteration can be derived by observing that $\operatorname(x) = \sqrt/x$, which in terms of matrices can be written as $\operatorname(A) = A^\sqrt$, where we use the matrix square root. If we apply the Babylonian method to compute the square root of the matrix $A^2$, that is, the iteration $X_ = \frac \left(X_k + A X_k^\right)$, and define the new iterate $Z_k = A^X_k$, we arrive at the iteration

$Z_ = \frac\left(Z_k + Z_k^\right)$,

where typically $Z_0=A$. Convergence is global, and locally it is quadratic.

The Newton iteration uses the explicit inverse of the iterates $Z_k$.

Newton–Schulz iteration

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, $Z_k^\approx Z_k\left(2I-Z_k^2\right)$, derived by Schulz( de) in 1933. Substituting this approximation into the previous method, the new method becomes

$Z_ = \fracZ_k\left(3I - Z_k^2\right)$.

Convergence is (still) quadratic, but only local (guaranteed for $\, I-A^2\, <1$).

Applications

Solutions of Sylvester equations

Theorem: Let $A,B,C\in\R^$ and assume that $A$ and $B$ are stable, then the unique solution to the Sylvester equation, $AX +XB = C$, is given by $X$ such that

$\begin -I &2X\\ 0 & I \end = \operatorname \left( \begin A &-C\\ 0 & -B \end \right).$

''Proof sketch:'' The result follows from the similarity transform

$\begin A &-C\\ 0 & -B \end = \begin I & X \\ 0 & I \end \begin A & 0\\ 0 & -B \end \begin I & X \\ 0 & I \end^,$

since

$\operatorname \left( \begin A &-C\\ 0 & -B \end \right) = \begin I & X \\ 0 & I \end \begin I & 0\\ 0 & -I \end \begin I & -X \\ 0 & I \end,$

due to the stability of $A$ and $B$.

The theorem is, naturally, also applicable to the Lyapunov equation. However, due to the structure the Newton iteration simplifies to only involving inverses of $A$ and $A^T$.

Solutions of algebraic Riccati equations

There is a similar result applicable to the algebraic Riccati equation, $A^H P + P A - P F P + Q = 0$. Define $V,W\in\Complex^$ as

$\begin V & W \end = \operatorname \left( \begin A^H &Q\\ F & -A \end \right) - \begin I &0\\ 0 & I \end.$

Under the assumption that $F,Q\in\Complex^$ are Hermitian and there exists a unique stabilizing solution, in the sense that $A-FP$ is stable, that solution is given by the over-determined, but consistent, linear system

$VP=-W.$

''Proof sketch:'' The similarity transform

$\begin A^H &Q\\ F & -A \end = \begin P & -I \\ I & 0 \end \begin (-A-FP) & -F\\ 0 & (A-FP) \end \begin P & -I \\ I & 0 \end^,$

and the stability of $A-FP$ implies that

$\left( \operatorname \left( \begin A^H &Q\\ F & -A \end \right) - \begin I &0\\ 0 & I \end \right) \begin X & -I \\ I & 0 \end = \begin X & -I\\ I & 0 \end \begin 0 & Y \\ 0 & -2I \end,$

for some matrix $Y\in\Complex^$.

Computations of matrix square-root

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 $A - PIP=0$ is a degenerate algebraic Riccati equation and by definition a solution $P$ is the square root of $A$.

References

{{reflist

Matrix theory
Linear algebra