3 Linear DAEs with Constant Coefficients

Definition 3.1 Let \(E\in \mathbb C^{n,n}\) and \(\nu = \operatorname{ind}(E)\). A matrix \(X\in \mathbb C^{n,n}\) that fulfills

\[\begin{align} EX & = XE, \tag{3.1} \\ XEX & = X, \tag{3.2} \\ XE^{\nu+1} & = E^{\nu}, \tag{3.3} \end{align}\]

is called a Drazin inverse of \(E\).

With the following theorem we confirm that a Drazin inverse to a matrix \(E\) is unique so that we can write \(E^D\) for it.

Theorem 3.1 Every matrix \(E\in\mathbb{C}^{n,n}\) has one, and only one, Drazin inverse.

Proof. Uniqueness: Let \(X_1\) and \(X_2\) be two Drazin inverses of \(E\). Then by repeated application of the identities in (3.1)(3.3) one derives that

\[\begin{align*} X_1 EX_1 E X_2 = &X_1EX_2 = X_1EX_2EX_2 \\ X_1^2 E^2 X_2 = \dotsm= &X_1EX_2 = \dotsm= X_1E^2X_2^2 \\ X_1^{\nu+1}E^{\nu+1} X_2 =\dotsm=\dotsm= &X_1EX_2 =\dotsm=\dotsm= X_1E^{\nu+1}X_2^{\nu+1} \\ X_1^{\nu+1}E^{\nu+1} X_1 =\dotsm=\dotsm=\dotsm= &X_1EX_2 =\dotsm=\dotsm=\dotsm= X_2E^{\nu+1}X_2^{\nu+1} \\ X_1 =\dotsm=\dotsm=\dotsm=\dotsm= &X_1EX_2 =\dotsm=\dotsm=\dotsm=\dotsm= X_2, \\ \end{align*}\]

where in the second last step we used the identities

\[ E^{\nu+1}X_1=X_1E^{\nu+1}=E^{\nu}=X_2E^{\nu+1}=E^{\nu+1}X_2. \]