Further Study and Generalization of Kahan’s Matrix Extension Theorem
Dongzhe Zheng · SIAM Journal on Matrix Analysis and Applications · 1996
In 1967, Kahan obtained a matrix extension theorem: Suppose $H \in \mathbb{C}^{l \times l} $ is Hermitian and $B \in \mathbb{C}^{s \times l} $. Denote the spectral norm of \[ R = \begin{bmatrix} H \\ B \end{bmatrix} \] by $ \| R \|_2 $. Then there exists a $W \in \mathbb{C}^{s \times s} $ such that \[ A = \begin{bmatrix} H & {B^* } \\ B & W \end{bmatrix} \] is Hermitian and $ \| A \|_2 = \| R \|_2 $. Kahan did not give an explicit expression for W. We show that one may take\[ ( 1 )\qquad W = - BH( \varrho^2 I - H^2 )^\dag B^ * , \] where $A^\dag $ denotes the Moore–Penrose generalized inverse of A. Furthermore, the inequality \[ ( 2 )\qquad B ( \varrho I + H )^\dag B^* - \varrho I \leq W \leq \varrho I - B ( \varrho I - H )^\dag B^* \] gives the “general solution formula” for W in Kahan’s theorem, where $A \geq B$ means A and B are Hermitian and $A - B$ is positive semidefinite. A by-product of (2) is the inequality \[ ( 3 )\qquad 2\varrho I \geq B [ ( \varrho I + H )^\dag + ( \varrho I - H )^\dag ] B . \] In this paper we also consider the following problem: Suppose $H \in \mathbb{C}^{l \times l} $ is normal, $B \in \mathbb{C}^{s \times l} $, and \[ R = \begin{bmatrix} H \\ B \end{bmatrix} . \] How can we find a Hermitian W and a matrix $B_1 $ such that $\| B_1 \|_2 = \| B \|_2$ and $\| A \|_2 = \| R \|_2 $, where \[ A = \begin{bmatrix} H & {B_1^* } \\ B & W \end{bmatrix}? \]