How to use QZ decomposition for single matrix in Matlab?

Question

Can I use QZ decomposition on a single square matrix in Matlab? Like,

[Aa,Q,Z]=qz(A);

Answer 1

As far as I know, QZ decomposition is given for two matrices, so that for $A,B\in \mathbb F^{n \times n}$: $$ A=QSZ^*, B=QTZ^* \tag{1} \label{QZ} $$ where $Q,Z^*$ are unitary, and $S,T$ are upper-triangular, and $\mathbb F$ is the field (real $\mathbb R$ or complex $\mathbb C$). QZ decomposition is usually called generalized Schur decomposition.

For a single matrix $A\in\mathbb F^{n\times n}$, one would simply computes Shur decomposition: $$ A=Q_\text{Schur}UQ^*_\text{Schur} \tag{2} \label{Schur} $$ where $Q_\text{Schur}$ is unitary, and $U$ is upper triangular.

So the Schur decomposition $\eqref{Schur}$, allows "more restricted" decomposition of $A$, since there is no matrix $B$, which presence would result in a generalized QZ decomposition $\eqref{QZ}$ and a presence of $Z$.

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In Matlab, Schur decomposition $\eqref{Schur}$ can be computed, as follows:

[Q,U] = schur(A,...)

Take a look at Matlab function help for some additional computation options. Notice a confusion between Wikipedia and this post notation choice, compared to the Matlab help page.