Matrix sign function

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

\operatorname{csgn}(z)= \begin{cases}

1 & \text{if } \mathrm{Re}(z) > 0, \\

-1 & \text{if } \mathrm{Re}(z) < 0,

\end{cases}

to the matrix valued analogue $\backslash operatorname\{csgn\}(A)$. 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).

Theorem: Let $A\backslash in\backslash C^\{n\backslash times\; n\}$, then $\backslash operatorname\{csgn\}(A)^2\; =\; I$.

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

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

Theorem: Let $A\backslash in\backslash C^\{n\backslash times\; n\}$, and be a Jordan decomposition such that $J\_+$ corresponds to eigenvalues with positive real part and $J\_-$ to eigenvalue with negative real part. Then , where $I\_+$ and $I\_-$ are identity matrices of sizes corresponding to $J\_+$ and $J\_-$, respectively.

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 $\backslash operatorname\{csgn\}(x)\; =\; \backslash sqrt\{x^2\}/x$, which in terms of matrices can be written as $\backslash operatorname\{csgn\}(A)\; =\; A^\{-1\}\backslash sqrt\{A^2\}$, 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 $X\_\{k+1\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(X\_k\; +\; A^\{2\}\; X\_k^\{-1\}\backslash right)$, and define the new iterate $Z\_k\; =\; A^\{-1\}X\_k$, we arrive at the iteration

Z_{k+1} = \frac{1}{2}\left(Z_k + Z_k^{-1}\right)

,

where typically $Z\_0=A$. Convergence is global, and locally it is quadratic.

The Newton iteration uses the explicit inverse of the iterates $Z\_k$.

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, $Z\_k^\{-1\}\backslash approx\; Z\_k\backslash left(2I-Z\_k^2\backslash right)$, derived by Schulz(de) in 1933.{{Cite journal|last=Schulz|first=Günther|date=1933|title=Iterative Berechung der reziproken Matrix|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.19330130111|journal=ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik|language=en|volume=13|issue=1|pages=57–59|doi=10.1002/zamm.19330130111|bibcode=1933ZaMM...13...57S |issn=1521-4001|url-access=subscription}} Substituting this approximation into the previous method, the new method becomes

Z_{k+1} = \frac{1}{2}Z_k\left(3I - Z_k^2\right)

.

Convergence is (still) quadratic, but only local (guaranteed for ).

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

\begin{bmatrix}

-I &2X\\ 0 & I

\end{bmatrix}

=

\operatorname{csgn}

\left(

\begin{bmatrix}

A &-C\\ 0 & -B

\end{bmatrix}

\right).

Proof sketch: The result follows from the similarity transform

\begin{bmatrix}

A &-C\\ 0 & -B

\end{bmatrix}

=

\begin{bmatrix}

I & X \\ 0 & I

\end{bmatrix}

\begin{bmatrix}

A & 0\\ 0 & -B

\end{bmatrix}

\begin{bmatrix}

I & X \\ 0 & I

\end{bmatrix}^{-1},

since

\operatorname{csgn}

\left(

\begin{bmatrix}

A &-C\\ 0 & -B

\end{bmatrix}

\right)

=

\begin{bmatrix}

I & X \\ 0 & I

\end{bmatrix}

\begin{bmatrix}

I & 0\\ 0 & -I

\end{bmatrix}

\begin{bmatrix}

I & -X \\ 0 & I

\end{bmatrix},

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 $A^T$.

There is a similar result applicable to the algebraic Riccati equation, $A^H\; P\; +\; P\; A\; -\; P\; F\; P\; +\; Q\; =\; 0$. Define $V,W\backslash in\backslash Complex^\{2n\backslash times\; n\}$ as

\begin{bmatrix}

V & W

\end{bmatrix}

=

\operatorname{csgn}

\left(

\begin{bmatrix}

A^H &Q\\ F & -A

\end{bmatrix}

\right)

-

\begin{bmatrix}

I &0\\ 0 & I

\end{bmatrix}.

Under the assumption that $F,Q\backslash in\backslash Complex^\{n\backslash times\; n\}$ 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

\begin{bmatrix}

A^H &Q\\ F & -A

\end{bmatrix}

=

\begin{bmatrix}

P & -I \\ I & 0

\end{bmatrix}

\begin{bmatrix}

-(A-FP) & -F\\ 0 & (A-FP)

\end{bmatrix}

\begin{bmatrix}

P & -I \\ I & 0

\end{bmatrix}^{-1},

and the stability of $A-FP$ implies that

\left(

\operatorname{csgn}

\left(

\begin{bmatrix}

A^H &Q\\ F & -A

\end{bmatrix}

\right)

-

\begin{bmatrix}

I &0\\ 0 & I

\end{bmatrix}

\right)

\begin{bmatrix}

X & -I \\ I & 0

\end{bmatrix}

=

\begin{bmatrix}

X & -I\\ I & 0

\end{bmatrix}

\begin{bmatrix}

0 & Y \\ 0 & -2I

\end{bmatrix},

for some matrix $Y\backslash in\backslash Complex^\{n\backslash times\; n\}$.

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 $A$.

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Category:Matrix theory

Category:Linear algebra