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

$\mathrm{csgn}(z)=\{\begin{array}{cc}1& \text{if }\mathrm{R}\mathrm{e}(z)>0,\\ -1& \text{if }\mathrm{R}\mathrm{e}(z)<0,\end{array}$

to the matrix valued analogue $\mathrm{csgn}(A)$. 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 $A\in {ℂ}^{n\times n}$, then $\mathrm{csgn}(A{)}^2=I$.

Theorem: Let $A\in {ℂ}^{n\times n}$, then $\mathrm{csgn}(A)$ is diagonalizable and has eigenvalues that are $\pm 1$.

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

Theorem: Let $A\in {ℂ}^{n\times n}$, and $A=P\left[\begin{array}{cc}{J}_{+}& 0\\ 0& J_{-}\end{array}\right]P^{-1}$ be a Jordan decomposition such that $J_{+}$ corresponds to eigenvalues with positive real part and $J_{-}$ to eigenvalue with negative real part. Then $\mathrm{csgn}(A)=P\left[\begin{array}{cc}{I}_{+}& 0\\ 0& -I_{-}\end{array}\right]P^{-1}$, 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 $\mathrm{csgn}(x)=\sqrt{x^2}/x$, which in terms of matrices can be written as $\mathrm{csgn}(A)=A^{-1}\sqrt{A^2}$, 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 $X_{k+1}=\frac{1}{2}(X_k+A{X}_k^{-1})$, and define the new iterate $Z_k=A^{-1}{X}_k$, we arrive at the iteration

$Z_{k+1}=\frac{1}{2}(Z_k+Z_k^{-1})$,

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, $Z_k^{-1}\approx Z_k(2I-Z_k^2)$, derived by Schulz(de) in 1933. Substituting this approximation into the previous method, the new method becomes

$Z_{k+1}=\frac{1}{2}{Z}_k(3I-Z_k^2)$.

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

Applications

Solutions of Sylvester equations

Theorem: Let $A,B,C\in {ℝ}^{n\times n}$ 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

$\left[\begin{array}{cc}-I& 2X\\ 0& I\end{array}\right]=\mathrm{csgn}\left(\left[\begin{array}{cc}A& -C\\ 0& -B\end{array}\right]\right).$

Proof sketch: The result follows from the similarity transform

$\left[\begin{array}{cc}A& -C\\ 0& -B\end{array}\right]=\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right]\left[\begin{array}{cc}A& 0\\ 0& -B\end{array}\right]{\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right]}^{-1},$

since

$\mathrm{csgn}\left(\left[\begin{array}{cc}A& -C\\ 0& -B\end{array}\right]\right)=\left[\begin{array}{cc}I& X\\ 0& I\end{array}\right]\left[\begin{array}{cc}I& 0\\ 0& -I\end{array}\right]\left[\begin{array}{cc}I& -X\\ 0& I\end{array}\right],$

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, $A^{H}P+PA-PFP+Q=0$. Define $V,W\in {ℂ}^{2n\times n}$ as

$\left[\begin{array}{cc}V& W\end{array}\right]=\mathrm{csgn}\left(\left[\begin{array}{cc}{A}^H& Q\\ F& -A\end{array}\right]\right)-\left[\begin{array}{cc}I& 0\\ 0& I\end{array}\right].$

Under the assumption that $F,Q\in {ℂ}^{n\times n}$ 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

$\left[\begin{array}{cc}{A}^H& Q\\ F& -A\end{array}\right]=\left[\begin{array}{cc}P& -I\\ I& 0\end{array}\right]\left[\begin{array}{cc}-(A-FP)& -F\\ 0& (A-FP)\end{array}\right]{\left[\begin{array}{cc}P& -I\\ I& 0\end{array}\right]}^{-1},$

and the stability of $A-FP$ implies that

$(\mathrm{csgn}\left(\left[\begin{array}{cc}{A}^H& Q\\ F& -A\end{array}\right]\right)-\left[\begin{array}{cc}I& 0\\ 0& I\end{array}\right])\left[\begin{array}{cc}X& -I\\ I& 0\end{array}\right]=\left[\begin{array}{cc}X& -I\\ I& 0\end{array}\right]\left[\begin{array}{cc}0& Y\\ 0& -2I\end{array}\right],$

for some matrix $Y\in {ℂ}^{n\times n}$.

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 $A$.