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 1980¹.

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. \operatorname{csgn}(z)= \begin{cases} 1 & \text{if } \mathrm{Re}(z) > 0, \\ -1 & \text{if } \mathrm{Re}(z) to the matrix valued analogue \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\in\C^{n\times n} , then \operatorname{csgn}(A)^2 = I . Theorem: Let A\in\C^{n\times n} , then \operatorname{csgn}(A) is diagonalizable and has eigenvalues that are \pm 1 . Theorem: Let A\in\C^{n\times n} , then (I+\operatorname{csgn}(A))/2 is a projector onto the invariant subspace associated with the eigenvalues in the right-half plane, and analogously for (I-\operatorname{csgn}(A))/2 and the left-half plane. Theorem: Let A\in\C^{n\times n} , and A = P\begin{bmatrix}J_+ & 0 \\ 0 & J_-\end{bmatrix}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 \operatorname{csgn}(A) = P\begin{bmatrix}I_+ & 0 \\ 0 & -I_-\end{bmatrix}P^{-1} , where I_+ and I_- are identity matrices of sizes corresponding to J_+ and J_- , respectively. The Newton iteration can be derived by observing that \operatorname{csgn}(x) = \sqrt{x^2}/x , which in terms of matrices can be written as \operatorname{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} \left(X_k + A^{2} X_k^{-1}\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 . Z_{k+1} = \frac{1}{2}Z_k\left(3I - Z_k^2\right) . Convergence is (still) quadratic, but only local (guaranteed for \|I-A^2\| ). 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 \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\in\Complex^{2n\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\in\Complex^{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 \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\in\Complex^{n\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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