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\begin{document}
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\title{Special forms of generalized inverses of row block matrices\thanks{
Received by the editors 3 August 2005.
Accepted for publication 30 October 2005.  Handling Editor: Ravindra B. Bapat.}}

\author{Yongge Tian\thanks{School of  Economics, Shanghai University of Finance
and Economics, Shanghai 200433, China (yongge@mail.shufe.edu.cn).}}


\pagestyle{myheadings}

\markboth{Y.\ Tian}{Generalized Inverses of Block Matrices}

\maketitle

\begin{abstract}

Given a row block matrix $[\, A, \, B\,]$, this paper investigates the relations
between the generalized inverse $[\, A, \, B\,]^-$ and the column block matrix
$\left[\!\!\begin{array}{c} A^-  \\ B^- \end{array} \!\! \right]$ consisting of
two generalized inverses $A^-$ and $B^-$. The first step of the investigation is
to establish a formula for the minimal rank of the difference $[\, A, \, B\,]^- -
\left[\!\!\begin{array}{c} A^-  \\ B^- \end{array} \!\! \right]$, the second step
is to find a necessary and sufficient condition for $[\, A, \, B\,]^- =
 \left[\!\!\begin{array}{c} A^-  \\ B^- \end{array} \!\! \right]$ to hold by letting the
 minimal rank be zero.  Seven types of generalized inverses of matrices are taken into
 account.
\end{abstract}

\begin{keywords} Block matrix, Generalized inverse, Minimal rank formula,
Moore-Penrose inverse, Schur complement.
\end{keywords}
\begin{AMS}
15A03, 15A09.
\end{AMS}

%\section{{\bf Introduction}}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
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\section{Introduction}
Let $\mathbb C^{m \times n}$ denote the
set of all $m \times n$ matrices over
the field of complex numbers. The symbols $A^*$, $r(A)$ and
${\mathscr R}(A)$  stand for the conjugate transpose, the rank, the range
(column space) and the kernel of a matrix $A \in \mathbb
C^{m \times n}$, respectively; $[\, A, \, B\,]$ denotes a row block
matrix consisting of $A$ and $B$.


For a matrix $A \in \mathbb C^{m \times n}$, the Moore-Penrose inverse of $A$,
denoted by $A^{\dag}$, is the unique matrix $X \in \mathbb C^{n \times m}$
satisfying the four Penrose equations
\renewcommand{\theenumi}{\roman{enumi}}
\begin{enumerate}
\item[(i)]  \ $AXA = A$,

\item[(ii)] \ $XA X = X$,

\item[(iii)] \ $(AX)^* = AX$,

\item[(iv)] \   $(XA)^* =XA$.
\end{enumerate}
For simplicity, let $E_A = I_m - AA^{\dag}$ and $F_A = I_n - A^{\dag}A$.
Moreover, a matrix $X$ is called an $\{i,\ldots, j\}$-inverse of $A$, 
denoted by $A^{(i,\ldots, j)}$, if it satisfies the
 $i,\ldots, j$th equations. The collection of all $\{i,\ldots, j\}$-inverses
 of $A$ is denoted by $\{A^{(i,\ldots, j)} \}$. In particular, $\{1\}$-inverse
 of $A$ is often denoted by $A^-$. The seven frequently used
generalized inverses of $A$ are $A^{(1)}$, $A^{(1,2)}$, $
A^{(1,3)}$,  $A^{(1,4)}$, $A^{(1,2,3)}$, $A^{(1,2,4)}$ and  $A^{(1,3,4)}$, which have been
studied by many authors; see, e.g., \cite{BG, CM, RM, Ya}. The general expressions
of the seven generalized inverses of $A$ can be written as
\e
A^- \es A^{\dag} + F_AV + WE_A,
\label{11}
\\
A^{(1,2)} \es (\, A^{\dag} + F_AV\,)A(\, A^{\dag}  + WE_A \,),
\label{12}
\\
A^{(1,3)} \es A^{\dag} + F_AV,
\label{13}
\\
A^{(1,4)} \es A^{\dag} + WE_A,
\label{14}
\\
A^{(1,2,3)} \es A^{\dag} + F_AVAA^{\dag},
\label{15}
\\
A^{(1,2,4)} \es A^{\dag} + A^{\dag}AWE_A,
\label{16}
\\
A^{(1,3,4)} \es A^{\dag} + F_AVE_A,
\label{17}
\ee
where the two matrices $V$ and $W$ are arbitrary; see \cite{BG}. Obviously,
$$
\{A^{(1,3,4)}\} \subseteq \{A^{(1,3)}\}  \subseteq \{A^-\},  \ \
\{A^{(1,3,4)}\} \subseteq \{A^{(1,4)}\}  \subseteq \{A^-\},
$$
$$
\{A^{(1,2,3)}\} \subseteq \{A^{(1,2)}\}  \subseteq \{A^-\},  \ \
\{A^{(1,2,4)}\} \subseteq \{A^{(1,2)}\}  \subseteq \{A^-\},  \ \
$$
$$
\{A^{(1,2,3)}\} \subseteq \{A^{(1,3)}\}  \subseteq \{A^-\},  \ \
\{A^{(1,2,4)}\} \subseteq \{A^{(1,4)}\}  \subseteq \{A^-\}.
$$

One of the main studies in the theory of generalized inverses
is to find generalized inverses of block matrices and their
properties. For the simplest block matrix $M = [\, A, \, B \,]$,
 where $A \in \mathbb C^{m \times n}$ and $B \in \mathbb C^{m\times k}$,
 there have been many results on its generalized inverses.
 For example, the Moore-Penrose inverse of $[\, A, \, B \,]^{\dag}$ can be
 represented as
\e
[\, A, \, B \,]^{\dag} = \left[\!\! \begin{array}{c}
 A^*(\, AA^* + BB^* \,)^{\dag}  \\ B^*(\, AA^* + BB^* \,)^{\dag}
 \end{array} \!\!\right]\!,
\label{18}
\ee
which is derived from the equality $M^{\dag} = M^*(MM^*)^{\dag}$.
Another representation of $[\, A, \, B \,]^{\dag}$ in  Mih\'alyffy \cite{Mi} is
\en
[\, A, \, B \,]^{\dag} = \left[\!\! \begin{array}{c}
  (I_n + TT^*)^{-1}(\,  A^{\dag} - A^{\dag}BS^{\dag} \,)
 \\
 T^*(I_n + TT^*)^{-1}(\,  A^{\dag} - A^{\dag}BS^{\dag} \,) + S^{\dag}
 \end{array} \!\!\right]\!,
\een
where $S =E_AB$ and $T = A^{\dag}BF_S$. Moreover, it can easily be
verified by definition that
$$
\left\{ \left[ \begin{array}{c} (A - BB^-A)^-(I_m - BB^-) \\ B^- -
B^-A(A - BB^-A)^-(I_m- BB^-) \end{array} \!\right]  \right\} \subseteq
\{ [\, A, \, B \,]^- \},
$$ %\\[-8mm]
$$
\left\{ \left[ \begin{array}{c} A^- - A^-B(B - AA^-B)^-(I_m - AA^-)
\\
(B - AA^-B)^-(I_m - AA^-) \end{array} \!\right]  \right\} \subseteq
\{ [\, A, \, B \,]^- \},
$$   %\\[-8mm]
$$
\left[\!\! \begin{array}{c} (E_BA)^{\dag} \\ B^{\dag} -
B^{\dag}A(E_BA)^{\dag} \end{array} \!\!\right] \in
\{ [\, A, \, B \,]^{(1,2,4)}\}, \ \ \left[ \!\!\begin{array}{c}
A^{\dag} - A^{\dag}B(E_AB)^{\dag}
\\
(E_AB)^{\dag} \end{array} \!\!\right]  \in
\{ [\, A, \, B \,]^{(1,2,4)} \}
$$
hold true.


It has been shown that the rank of matrix provides a powerful tool
for characterizing various equalities for generalized inverses of matrices.
  For example, Tian \cite{Ti-hjm} established the following three
  equalities:
\e
\label{19}
 \ \ \ r\!\left([\, A, \, B \,] -
 [\, A, \,  B \,]\!\left[\!\!\begin{array}{c} A^{\dag}  \\
B^{\dag} \end{array} \!\! \right]\![\, A, \, B \,] \right)
 \es r[ \, A, \, B \, ] + 2r(A^*B) - r(A) - r(B),
\\
\label{110}
r\!\left( [\,  A, \,  B \,]^{\dag}  -
\!\left[\!\!\begin{array}{c} A^{\dag}  \\ B^{\dag}
\end{array} \!\! \right] \right) \es r[\,  AA^*B, \,  BB^*A \,],
\\
\label{111}
r\!\left( [\,  A, \,  B \,]^{\dag}  -
\!\left[\!\!\begin{array}{c} (E_BA)^{\dag}  \\ (E_AB)^{\dag}
\end{array} \!\! \right] \right) \es r(A) + r(B) - r[\, A, \,  B \,].
\ee
Let the right-hand side of (\ref{19}) be zero, we see that
$\left[\!\!\begin{array}{c} A^{\dag}  \\ B^{\dag}
\end{array} \!\! \right] \in \{ [\, A, \,  B \,]^- \}$ if and only if
  $r[\, A, \, B \,] = r(A) + r(B) - 2r(A^*B)$. However, the inequality
  $r[ \, A, \, B \, ] \geq r(A) + r(B) - r(A^*B)$ always holds.
Hence $r[ \, A, \, B \, ] = r(A) + r(B) - 2r(A^*B)$ is equivalent to
$A^*B =0$. Let the right-hand side of (\ref{110}) be zero, we see that
$[\,  A, \,  B \,]^{\dag} = \!\left[\!\!\begin{array}{c} A^{\dag}
 \\ B^{\dag} \end{array} \!\! \right]$ if and only if $A^*B = 0$.
Let the right-hand side of (\ref{111}) be zero, we see that
$[\,  A, \,  B \,]^{\dag} = \left[\!\!\begin{array}{c} (E_BA)^{\dag}  \\
(E_AB)^{\dag} \end{array} \!\! \right]$ if and only if
$r[\, A, \,  B \,] =r(A) + r(B)$. Motivated by these results,
we shall establish a variety of new rank
formulas for the difference
$$
[\, A, \, B\,]^{(i,\ldots, j)} -
\left[\!\!\begin{array}{c} A^{(i,\ldots, j)}  \\ B^{(i,\ldots, j)}
\end{array} \!\! \right]\,,
$$
and then use the formulas to characterize the corresponding matrix
equalities for generalized inverses.

Some useful rank formulas for partitioned matrices  due to
Marsaglia  and Styan \cite{MS} are given in the following lemma.

\begin{lemma} \label{T11}
Let $A \in \mathbb C^{m \times n},$ $ B \in \mathbb C^{m \times k},$
$C \in \mathbb C^{l \times n}$ and $D \in \mathbb C^{l \times k}.$ Then
\e
\label{112}
  r[\, A, \, B \, ] &=& r(A)+ r[ \,(\, I_m - AA^- \,)B \,] = r(B) +
r[\,(\,I_m - BB^- \,)A \,], \ \ \
\\
\label{113}
r\!\left[ \begin{array}{cc} A \\ C \end{array} \!\right] &=& r(A) +
r[ \,C(\, I_n - A^-A \,) \,] =  r(C) + r[\, A(\, I_n - C^-C \,)
\,],
\\
\label{114}
 \ \ \ r\!\left[ \begin{array}{cc} A & B \\ C & 0 \end{array} \!\right] &=&
r(B) + r(C) + r[\, ( \,I_m - BB^- \,)A (\, I_n - C^-C \,) \,].
\ee
If ${\mathscr R}(B) \subseteq {\mathscr R}(A)$ and ${\mathscr R}(C^*)\subseteq
{\mathscr R}(A^*),$ then
\e
\label{115}
r\!\left[ \begin{array}{cc} A & B \\ C & D \end{array} \!\right] &=&
r(A) + r(\, D - CA^{\dag}B \,).
\ee
\end{lemma}


Applying (\ref{115}) and $A^*(A^*AA^*)^{\dag}A^* = A^{\dag}$ (see Zlobec \cite{Zl}) to
 a general Schur complement $D - CA^{\dag}B$ gives the following rank formula:
 \e
r(\,D - CA^{\dag}B\,) = r\left[\!\!\begin{array}{cc} A^*AA^* & A^*B  \\ CA^* &
 D \end{array} \!\!\right] - r(A).
\label{116}
\ee
This formula can be used to find the rank of any matrix expression involving
the Moore-Penrose inverse. In fact, the equalities in (\ref{19}), (\ref{110}) and
(\ref{111}) are derived from (\ref{116}).

Because generalized inverse of a matrix is not
necessarily unique, the rank of a matrix expression involving generalized inverses
 of matrices may vary with respect to the choice of the generalized inverses.
Suppose $f(A_1^-, \ldots, A^-_p)$ and $g(B_1^-, \ldots, B^-_q)$ are two expressions
consisting of generalized inverses of matrices. Then there are
$A_1^-, \ldots,\ A^-_p$, $B_1^-, \ldots, B^-_q$ such that
$f(A_1^-, \ldots, A^-_p) = g(B_1^-, \ldots, B^-_q)$ if and only if
$$
\min_{A_1^-, \ldots,\ A^-_p, B_1^-, \ldots, B^-_q}
r[\, f(A_1^-, \ldots, A^-_p) - g(B_1^-, \ldots, B^-_q)\,] =0.
$$
One of the simplest matrix expressions involving a generalized inverse is
the Schur complement $D - CA^-B$. In Tian \cite{Ti-amc}, a group of formulas
for the extremal ranks of the Schur complement $D - CA^{(i,\ldots, j)}B$
with respect to $A^{(i,\ldots, j)}$ are derived. The following lemma presents
 some special cases of these formulas.


\begin{lemma} \label{T12}
Let $A \in \mathbb C^{m \times n}$ and $G \in \mathbb C^{n\times m}.$  Then
\e
\ \ \ \ \min_{A^-} r(\, A^- - G \,) \es  r(\, A - AGA \,),
\label{117}
\\
 \min_{A^{(1,2)}} r(\, A^{(1,2)} - G \,) \es
\max \{ \, r(\,  A - AGA \,), \ r(G) +
r(A)  -   r(GA) - r(AG) \, \},
\label{118}
\\
 \ \ \ \ \min_{A^{(1,3)}} r(\, A^{(1,3)} - G \,) \es
r(\, A^*AG - A^* \,),
\label{119}
\\
  \ \ \ \ \ \ \ \min_{A^{(1,4)}} r(\,  A^{(1,4)} - G \,) \es
 r(\, GAA^* - A^* \,),
\label{120}
\ee  \\[-12mm]
\e
 \ \ \min_{A^{(1,2,3)}} r(\, A^{(1,2,3)} - G \,) \es
 r(A^*AG - A^*) + r\!\left[\!\!\begin{array}{c}  A^*
 \\ G  \end{array} \!\! \right]\! -
 r\!\left[\!\!\begin{array}{c}  A^*  \\ AG  \end{array} \!\! \right]\!,
\label{121}
\\
 \ \ \min_{A^{(1,2,4)}} r(\, A^{(1,2,4)} - G \,) \es
 r(\, GAA^* - A^* \,) + r[\, A^*, \, G \,] -
r[\, A^*,\, GA \,],
\label{122}
\\
 \ \ \ \min_{A^{(1,3,4)}} r(A^{(1,3,4)} - G ) \es
 r(A^*AG - A^*)  +  r(GAA^* - A^*) - r(A - AGA),
\label{123}
\\
\ \ \ \ r( A^{\dag} - G \,) \es r \!\left[\!\!\begin{array}{cc} A^*AA^* & A^*
 \\ A^* & G \end{array}\!\!\right]- r(A).
\label{124}
\ee
\end{lemma}

Let the right-hand sides of (\ref{117})--(\ref{124}) be zero, we can obtain
 necessary and sufficient conditions for $G \in \{ A^- \}$,
$\{A^{(1,2)} \}$, $\{A^{(1,3)}\}$, $\{ A^{(1,4)}\}$,  $\{A^{(1,2,3)}\}$,
$\{A^{(1,2,4)}\}$, $\{A^{(1,3,4)}\}$ and $G=A^{\dag}$ to hold.  If $A$ and $G$ in
(\ref{117})--(\ref{124}) are taken as matrix products, matrix sums and
partitioned matrices,  then many valuable results can be derived from the
corresponding rank formulas.

Some other rank formulas used in the context are given below:
\e
\min_{X}r(\, A - BXC \,) =  r[\, A, \, B \,] +
r \!\left[\!\! \begin{array}{c} A \\ C \end{array} \!\! \right] -
 r \!\left[\!\! \begin{array}{cc} A & B \\ C & 0 \end{array} \!\!\right]\!,
\label{125}
\ee
where $A \in \mathbb C^{m \times n}$, $B \in \mathbb C^{m \times k}$ and
  $C \in \mathbb C^{l \times n}$; see Tian \cite{Ti-se}. In particular,
\e
\min_{X}r(\, A - BX \,) = r[\, A, \, B \,] - r(A)
 = r(\, B - AA^{\dag}B \,).
\label{126}
\ee
The formula for minimal rank of $A - B_1X_1C_1 - B_2X_2C_2$ is
\e
\label{127}
 \ \ \ \ \ \min_{X_1, \, X_2} r( \, A - B_1X_1C_1 - B_2X_2C_2\,) =
r \! \left[\!\! \begin{array}{c} A \\ C_1 \\ C_2
 \end{array} \!\!
\right] + r[ \, A, \, B_1, \,  B_2 \, ]  +  \max\{ \, r_1, \  r_2 \, \},
\ee
where
\en
r_1 \es r\!\left[\!\!\begin{array}{cc} A & B_1\\ C_2 & 0 \end{array}
 \!\! \right] - r \!\left[ \!\! \begin{array}{ccc} A   & B_1 & B_2
\\ C_2   & 0 & 0  \end{array} \!\! \right] -
r\!\left[\!\!\begin{array}{cc} A   & B_1 \\ C_1 & 0 \\ C_2 & 0
 \end{array} \!\! \right]\!,
\\
 r_2 \es r \!\left[\!\!\begin{array}{cc}A & B_2 \\ C_1 & 0 \end{array}\!\!
\right]  - r\!\left[\!\! \begin{array}{ccc} A   & B_1 & B_2  \\ C_1
 & 0 & 0  \end{array} \!\!\right] - r\!\left[\! \!\begin{array}{cc} A
 & B_2 \\ C_1   & 0 \\ C_2 & 0  \end{array} \!\!\right]\!;
\een
see Tian \cite{Ti-laa02}. The following is also given \cite{Ti-laa02}:
\e
\min_{A^-}r(\, CA^-B \,) =  r \!\left[\!\! \begin{array}{cc} A & B \\ C & 0
\end{array} \!\!\right]
- r\!\left[\!\! \begin{array}{c} A \\ C \end{array} \!\! \right]
 - r[\, A, \, B \,] + r(A).
\label{128}
\ee


\section{Main results}

We first establish a set of rank formulas for the difference $[\, A, \, B\,]^{(i,\ldots,j)} -
\left[\!\!\begin{array}{c} A^{\dag}  \\ B^{\dag}\end{array} \!\! \right]\!.$


\begin{theorem} \label{T21}
Let $A \in \mathbb C^{m \times n},$ $B \in \mathbb C^{m\times k}$ and let
$M = [\, A, \, B\,],$ $G= \left[\!\!\begin{array}{c} A^{\dag}
 \\ B^{\dag} \end{array} \!\! \right].$ Then
\e
\min_{M^-} r(\, M^- - G \,) \es r(M) + 2r(A^*B) - r(A) - r(B),
\label{21}
\\
\min_{M^{(1,2)}} r( \, M^{(1,2)} - G \,) \es  r(M) + 2r(A^*B)
 - r(A) - r(B),
\label{22}
\\
\min_{M^{(1,3)}} r(\, M^{(1,3)} - G \,)  \es r(M) + 2r(A^*B)
 - r(A) - r(B),
\label{23}
\\
\min_{M^{(1,4)}} r(\, M^{(1,4)} - G\,) \es   r[\,  AA^*B, \,  BB^*A \,],
 \label{24}
 \\
\min_{M^{(1,2,3)}} r(\, M^{(1,2,3)} - G\,)  \es r(M) + 2r(A^*B)
 - r(A) - r(B),
\label{25}
\\
\min_{M^{(1,2,4)}} r(\, M^{(1,2,4)} - G\,) \es r[\,  AA^*B, \,  BB^*A \,],
\label{26}
\\
\min_{M^{(1,3,4)}} r(\, M^{(1,3,4)} - G\,) \es  r[\,  AA^*B, \,  BB^*A \,],
\label{27}
\\
r(\, M^{\dag} - G\,)  \es  r[\,  AA^*B, \,  BB^*A \,].
\label{28}
\ee
Hence
\e
\label{29}
G \in \{ M^-\} \Leftrightarrow  M^{\dag} = G \Leftrightarrow  A^*B = 0.
\ee
\end{theorem}

\begin{proof}
Applying (\ref{117}) to the difference  $M^- -  G$ gives
\e
\min_{M^-} r(\, M^- - G \,) = r(\, M - MGM \,),
\label{210}
\ee
where
$$
M - MGM = M - (\,  AA^{\dag} +  BB^{\dag} \,)M
 = - [\, BB^{\dag}A, \ AA^{\dag}B \,].
$$
It is shown in \cite{Ti-hjm} that
\e
r[\, BB^{\dag}A , \, AA^{\dag}B \,] =  r[ \, A, \, B \, ]
 + 2r(A^*B) - r (A) - r(B).
\label{211}
\ee
Substituting (\ref{211}) into (\ref{210}) yields (\ref{21}).
Applying (\ref{118}) to $M^{(1,2)} - G$ gives
$$
\min_{M^{(1,2)}} r(\, M^{(1,2)} - G \,) =
\max \{ \, r(\,  M - MGM \,), \ r(G) +
r(M)  -   r(GM) - r(MG) \, \}.
$$
It is easy to verify that
$$
r(G)  =  r\!\left[\!\!\begin{array}{c} A^*  \\ B^* \end{array}
\!\! \right] = r(M), \ \ r(GM) =
r\!\left(\left[\!\!\begin{array}{c} A^* \\ B^* \end{array}
\!\! \right]\![\,  A, \,  B \,] \right) = r(M),
$$  %\\[-10mm]
$$
r(MG) =  r(\, AA^{\dag} + BB^{\dag} \,) =
 r[ \, AA^{\dag}, \,  BB^{\dag} \,]  = r(M).
$$
Hence,
$$
\min_{M^{(1,2)}} r(\, M^{(1,2)} - G \,) =
r(\, M - MGM \,) = \min_{M^-} r(\, M^- - G \,).
$$
Applying (\ref{119}) to $M^{(1,3)} - G$ gives
$$
\min_{M^{(1,3)}} r(\, M^{(1,3)} - G \,) = r(\,  M^*MG - M^* \,),
$$
where
$$
M^*MG - M^* =
\left[\!\!\begin{array}{c} A^* + A^*BB^{\dag} \\
 B^* + B^*AA^{\dag} \end{array} \!\! \right] -
\left[\!\!\begin{array}{c} A^* \\ B^* \end{array} \!\! \right] =
\left[\!\!\begin{array}{c} A^*BB^{\dag} \\  B^*AA^{\dag} \end{array}
 \!\!\right] = [\, BB^{\dag}A , \, AA^{\dag}B \,]^*.
$$
Hence, (\ref{23}) follows from (\ref{211}). From (\ref{120})
$$
\min_{M^{(1,4)}} r(\, M^{(1,4)} - G \,) = r(\, GMM^* - M^* \,),
$$
where $GMM^* - M^* = \left[\!\!\begin{array}{c} A^{\dag}BB^*
\\ B^{\dag}AA^* \end{array} \!\! \right]\!.$ Hence
$$
r(\, GMM^* - M^* \,) =
r\! \left[\!\!\begin{array}{c} A^{\dag}BB^*
\\ B^{\dag}AA^* \end{array} \!\! \right] =
r\! \left[\!\!\begin{array}{c} A^*BB^*
\\ B^*AA^* \end{array} \!\! \right] = r[\,  AA^*B, \,  BB^*A \,],
$$
establishing (\ref{24}).   Equalities (\ref{25})--(\ref{27})
can be shown by (\ref{121})--(\ref{123}), and
 details are omitted. Eq.(\ref{28}) is (\ref{110}).
\end{proof}

A set of rank formulas for the difference $[\, A, \, B\,]^{\dag} -
  \left[\!\!\begin{array}{c} A^{(i,\ldots,j)}  \\ B^{(i,\ldots,j)}
\end{array} \!\! \right]$ are given below.

\begin{theorem} \label{T22}
Let
$A \in \mathbb C^{m \times n}$ and $B \in \mathbb C^{m\times k},$ and let
$M= [\, A, \, B \,].$ Then
\e
\min_{A^-,\,  B^- } r\!\left( M^{\dag}  - \!\left[\!\!\begin{array}{c}
  A^-  \\ B^-  \end{array} \!\! \right] \right) \es 2r(A) + 2r(B) -
 2r(M),
\label{212}
\\
\min_{A^{(1,2)},\,  B^{(1,2)} } r\!\left( M^{\dag}
- \!\left[\!\!\begin{array}{c}
  A^{(1,2)}  \\ B^{(1,2)}  \end{array} \!\! \right] \right)
\es 2r(A) + 2r(B) - 2r(M),
\label{213}
\\
\min_{A^{(1,3)},\,  B^{(1,3)} } r\!
\left( M^{\dag}  - \!\left[\!\!\begin{array}{c}
  A^{(1,3)}  \\ B^{(1,3)}  \end{array} \!\! \right] \right)
 \es r[\, BB^*A, \, AA^*B \,],
\label{214}
\\
\min_{A^{(1,4)},\,  B^{(1,4)} } r\!
\left( M^{\dag}  - \!\left[\!\!\begin{array}{c}
  A^{(1,4)}  \\ B^{(1,4)}  \end{array} \!\! \right] \right)
\es 2r(A) + 2r(B) - 2r(M),
\label{215}
\\
\min_{A^{(1,2,3)},\,  B^{(1,2,3)} } r\!
\left( M^{\dag}  - \!\left[\!\!\begin{array}{c}
  A^{(1,2,3)}  \\ B^{(1,2,3)}  \end{array} \!\! \right] \right)
 \es r[\, BB^*A , \, AA^*B \,],
\label{216}
\\
\min_{A^{(1,2,4)},\,  B^{(1,2,4)} } r\!
\left( M^{\dag}  - \!\left[\!\!\begin{array}{c}
  A^{(1,2,4)}  \\ B^{(1,2,4)}  \end{array} \!\! \right] \right)
\es 2r(A) + 2r(B) - 2r(M),
\label{217}
\\
\min_{A^{(1,3,4)},\,  B^{(1,3,4)} } r\!
\left( M^{\dag}  - \!\left[\!\!\begin{array}{c}
  A^{(1,3,4)}  \\ B^{(1,3,4)}  \end{array} \!\! \right] \right)
\es
r[\, BB^*A , \, AA^*B \,].
\label{218}
\ee
Hence$,$
\begin{enumerate}
\item[{\rm (a)}]  There are $A^-$ $(A^{(1,2)}, \, A^{(1,2,4)})$  and $B^-$
$(B^{(1,2)}, \, B^{(1,2,4)})$  such that
$$
M^{\dag} = \left[\!\!\begin{array}{c}
 A^-  \\ B^-  \end{array} \!\! \right] \left(  M^{\dag} = \left[\!\!\begin{array}{c}
 A^{(1,2)} \\ B^{(1,2)}  \end{array} \!\! \right]\!, \,
 M^{\dag} = \left[\!\!\begin{array}{c}
 A^{(1,2,4)} \\ B^{(1,2,4)}  \end{array} \!\! \right] \right)
$$
if and only if $r(M) =r(A) + r(B),$ i.e.$,$ ${\mathscr R}(A) \cap
{\mathscr R}(B) = \{ 0 \}.$

\item[{\rm (b)}] There are $A^{(1,3)}$ $(A^{(1,2,3)}, \ A^{(1,3,4)})$ and
$B^{(1,3)}$ $(B^{(1,2,3)},$ $B^{(1,3,4)})$ such that
$$
 M^{\dag} = \left[\!\!\begin{array}{c}
A^{(1,3)}  \\ B^{(1,3)} \end{array} \!\! \right]
\left( M^{\dag} = \left[\!\!\begin{array}{c}
A^{(1,2,3)}  \\ B^{(1,2,3)} \end{array} \!\! \right]\!,
 \ M^{\dag} = \left[\!\!\begin{array}{c}  A^{(1,3,4)}  \\ B^{(1,3,4)}
\end{array} \!\! \right] \right)
$$
if and only if $A^*B =0.$
\end{enumerate}
\end{theorem}

\begin{proof}
From (\ref{11})
\e
\label{219}
M^{\dag}  - \left[\!\!\begin{array}{c} A^-
 \\ B^-  \end{array} \!\! \right] \es  M^{\dag} -
 \left[\!\!\begin{array}{c} A^{\dag} + F_AV_1 + W_1E_A
 \\ B^{\dag} + F_BV_2 + W_2E_B \end{array} \!\! \right]
\\
\es  M^{\dag} - \left[\!\!\begin{array}{c}  A^{\dag} \\ B^{\dag}
\end{array} \!\! \right] - \left[\!\!\begin{array}{cc} F_A & 0
\\  0 & F_B \end{array} \!\! \right]\!\left[\!\!\begin{array}{c}
V_1 \\ V_2 \end{array} \!\! \right] - \left[\!\!\begin{array}{c}
W_1E_A \\ W_2E_B \end{array} \!\! \right]\!, \nb
\ee
where $V_1$, $W_1$, $V_2$ and $W_2$ are arbitrary. From (\ref{126})
\e
\label{220}
& & \min_{V_1,\, V_2} r\!\left( M^{\dag} - \left[\!\!\begin{array}{c}  A^{\dag}
 \\ B^{\dag} \end{array} \!\! \right] - \left[\!\!\begin{array}{cc} F_A & 0
\\  0 & F_B \end{array} \!\! \right]\!\left[\!\!\begin{array}{c}
V_1 \\ V_2 \end{array} \!\! \right] - \left[\!\!\begin{array}{c}
W_1E_A \\ W_2E_B \end{array} \!\! \right]\right)
\\
& & =  r\!\left\{ \left[\!\!\begin{array}{cc} A^{\dag}A & 0
\\  0 & B^{\dag}B  \end{array} \!\! \right]\!\left( M^{\dag} -
\left[\!\!\begin{array}{c}  A^{\dag}
 \\ B^{\dag} \end{array} \!\! \right] - \left[\!\!\begin{array}{c}
W_1E_A \\ W_2E_B \end{array} \!\! \right] \right) \right \} \nb
\\
& & =  r\!\left(M^{\dag} -
\left[\!\!\begin{array}{c}  A^{\dag}
 \\ B^{\dag} \end{array} \!\! \right] - \left[\!\!\begin{array}{c}
 A^{\dag}AW_1E_A \\  B^{\dag}BW_2E_B \end{array} \!\! \right] \right) \nb
\\
& & =  r\!\left(M^{\dag} -
\left[\!\!\begin{array}{c}  A^{\dag}
 \\ B^{\dag} \end{array} \!\! \right] -
\left[\!\!\begin{array}{c}
 A^{\dag}A \\ 0 \end{array} \!\! \right] W_1E_A -
\left[\!\!\begin{array}{c} 0  \\  B^{\dag}B \end{array}
\!\! \right]W_2E_B \right)\!. \nb
\ee
Let $ N =\left[\!\!\begin{array}{c}  A^{\dag}
 \\ B^{\dag} \end{array} \!\! \right]$,  $P_1 = \left[\!\!\begin{array}{c}
 A^{\dag}A \\ 0 \end{array} \!\! \right]$ and
$P_2 =\left[\!\!\begin{array}{c} 0  \\  B^{\dag}B
 \end{array}\!\!\right]$. Then applying (\ref{127}) gives
\e
\label{221}
& & \min_{W_1,\, W_2} r( \, M^{\dag} - N - P_1W_1E_A -
P_2W_2E_B \, )
\\
& & =  r[\, M^{\dag} - N, \, P_1, \, P_2 \,] +
r\!\left[\!\!\begin{array}{c}  M^{\dag} - N \\ E_A \\ E_B \end{array}
\!\! \right] + \max\{ r_1, \ \ r_2 \}, \nb
\ee
where
\en
r_1 \es r\!\left[\!\!\begin{array}{cc}
M^{\dag} - N  & P_1 \\ E_B  & 0 \end{array} \!\! \right] -
r\!\left[\!\!\begin{array}{ccc}  M^{\dag} - N  & P_1  & P_2 \\ E_B  &
  0 & 0 \end{array} \!\! \right] - r\!\left[\!\!\begin{array}{cc}
M^{\dag} - N  & P_1 \\ E_A  & 0  \\  E_B  & 0 \end{array} \!\! \right]\!,
\\
r_2 \es r\!\left[\!\!\begin{array}{cc}
M^{\dag} - N  & P_2 \\ E_A  & 0 \end{array} \!\! \right] -
r\!\left[\!\!\begin{array}{ccc}  M^{\dag} - N  & P_1  & P_2 \\ E_A  &
  0 & 0 \end{array} \!\! \right] - r\!\left[\!\!\begin{array}{cc}
M^{\dag} - N  & P_2 \\ E_A  & 0  \\  E_B  & 0 \end{array} \!\! \right]\!.
\een
Substituting (\ref{18}) into the eight partitioned matrices in (\ref{221}) and
simplifying by (\ref{112}), (\ref{113}) and (\ref{114}) give us the
following results:
\en
&& r[\, M^{\dag} - N, \, P_1, \, P_2 \,] = r(A) +  r(B), \ \ \
r\!\left[\!\!\begin{array}{c} M^{\dag} - N \\ E_A \\ E_B \end{array}
\!\! \right] = m +  r(A) +  r(B) - r(M),
\\
& & r\!\left[\!\!\begin{array}{cc}  M^{\dag} - N  & P_1 \\ E_B  & 0
\end{array} \!\! \right]  = m  + 2r(A) -  r(M),   \ \ \
r\!\left[\!\!\begin{array}{ccc}  M^{\dag} - N  & P_1  & P_2 \\ E_B  &
  0 & 0 \end{array} \!\! \right] = m + r(A),
\\
& & r\!\left[\!\!\begin{array}{cc}
M^{\dag} - N  & P_1 \\ E_A  & 0  \\  E_B  & 0 \end{array} \!\! \right]
= m + r(A),  \ \ \
r\!\left[\!\!\begin{array}{cc}  M^{\dag} - N  & P_2 \\ E_A  & 0
\end{array} \!\! \right]  = m  + 2r(B) -  r(M),
\\
& & r\!\left[\!\!\begin{array}{ccc}  M^{\dag} - N  & P_1  & P_2 \\ E_A  &
  0 & 0 \end{array} \!\! \right] = m + r(B),  \ \ \
r\!\left[\!\!\begin{array}{cc}
M^{\dag} - N  & P_2 \\ E_A  & 0  \\  E_A  & 0 \end{array} \!\! \right]
= m + r(B).
\een
The details are quite tedious, and therefore are omitted here. Substituting these
 eight results into (\ref{221}) and then (\ref{221}) into (\ref{220}) yields
 (\ref{212}). From (\ref{13})
\e
\label{222}
 \ \ \ \ M^{\dag}  - \left[\!\!\begin{array}{c} A^{(1,3)}
 \\ B^{(1,3)}  \end{array} \!\! \right] \es  M^{\dag} -
 \left[\!\!\begin{array}{c} A^{\dag} + F_AV_1
 \\ B^{\dag} + F_BV_2 \end{array} \!\! \right] = M^{\dag} -
 \left[\!\!\begin{array}{c}  A^{\dag} \\ B^{\dag}
\end{array} \!\! \right] - \left[\!\!\begin{array}{cc} F_A & 0
\\  0 & F_B \end{array} \!\! \right]\!\left[\!\!\begin{array}{c}
V_1 \\ V_2 \end{array} \!\! \right]\!,
\ee
where $V_1$ and $V_2$ are arbitrary.  Applying (\ref{126}) to this expression gives
\en
\min_{V_1, \, V_2}  r\!\left( M^{\dag} - N  - \left[\!\!\begin{array}{cc} F_A & 0
\\  0 & F_B \end{array} \!\! \right] \!\left[\!\!\begin{array}{c}
V_1 \\ V_2 \end{array} \!\! \right] \right) \es
r\!\left( \left[\!\!\begin{array}{cc} A^{\dag}A & 0 \\  0 & B^{\dag}B
\end{array} \!\! \right]\! ( \, M^{\dag} - \,N)  \right)
\\
\es r(\, M^{\dag} - \,N)
\\
\es r[\, BB^*A , \, AA^*B \,]  \ \ \ ({\rm by} \ (\ref{110})),
\een
as required for (\ref{214}). From (\ref{14})
\e
\label{223}
M^{\dag}  - \left[\!\!\begin{array}{c} A^{(1,4)}
 \\ B^{(1,4)}   \end{array} \!\! \right] \es  M^{\dag} -
 \left[\!\!\begin{array}{c} A^{\dag} +  W_1E_A
 \\ B^{\dag} + W_2E_B \end{array} \!\! \right]
\\
\es  M^{\dag} - N  - \left[\!\!\begin{array}{c}
  I_n \\ 0 \end{array} \!\! \right] \!W_1E_A -
\left[\!\!\begin{array}{c} 0  \\  I_k \end{array}
\!\! \right]\!W_2E_B, \nb
\ee
where $W_1$ and $W_2$ are arbitrary.  Applying (\ref{126}) to (\ref{223})
and simplifying by (\ref{112}), (\ref{113}) and (\ref{114})
yield (\ref{215}). The details are omitted here.
Observe that
$$
\min_{\begin{array}{c} A^{(1,3)}  \\[-1mm] B^{(1,3)}  \end{array} }
\!\!r\!\left( M^{\dag}  - \!\left[\!\!
\begin{array}{c} A^{(1,3)}  \\ B^{(1,3)}  \end{array} \!\! \right]
 \right) \leqslant  \min_{\begin{array}{c} A^{(1,2,3)}
 \\[-1mm] B^{(1,2,3)}  \end{array} }\!\! r\!\left( M^{\dag}  -
 \!\left[\!\!\begin{array}{c} A^{(1,2,3)}  \\ B^{(1,2,3)}  \end{array}
\!\! \right] \right) \leqslant  r(\, M^{\dag} - N \,)
$$
and
$$
\min_{\begin{array}{c} A^{(1,3)}  \\[-1mm] B^{(1,3)}  \end{array} }\!\!
r\!\left( M^{\dag}  - \!\left[\!\!\begin{array}{c} A^{(1,3)}  \\ B^{(1,3)}
\end{array} \!\! \right]
 \right) \leqslant  \min_{\begin{array}{c} A^{(1,3,4)}\\[-1mm] B^{(1,3,4)}
  \end{array} }
 \!\!
r\!\left( M^{\dag}  - \!\left[\!\!\begin{array}{c} A^{(1,3,4)}  \\ B^{(1,3,4)}
 \end{array}
\!\! \right] \right) \leqslant  r(\, M^{\dag} - N \,).
$$
Applying (\ref{110}) and (\ref{214}) to these two inequalities gives (\ref{216})
and (\ref{218}). From (\ref{16})
\e \label{224}
M^{\dag}  - \left[\!\!\begin{array}{c} A^{(1,2,4)}
 \\ B^{(1,2,4)} \end{array} \!\! \right] \es  M^{\dag} -
 \left[\!\!\begin{array}{c} A^{\dag} + A^{\dag}A W_1E_A
 \\ B^{\dag} +  B^{\dag}BW_2E_B \end{array} \!\! \right]
\\
\es  M^{\dag} - N  - \left[\!\!\begin{array}{c}
  A^{\dag}A \\ 0 \end{array} \!\! \right]\! W_1E_A -
\left[\!\!\begin{array}{c} 0  \\  B^{\dag}B \end{array}
\!\! \right]\!W_2E_B, \nb
\ee
where $W_1$ and $W_2$ are arbitrary. Applying (\ref{127}) to (\ref{224})
and simplifying  by (\ref{112}), (\ref{113}) and (\ref{114})
yield (\ref{217}). Also  note that
$$
\min_{A^-,\,  B^- }\! r\!\left( M^{\dag}  - \!\left[\!\!
\begin{array}{c} A^-  \\ B^-  \end{array} \!\! \right]
 \right) \leqslant \!\! \min_{\begin{array}{c} A^{(1,2)} \\[-1mm] B^{(1,2)}
 \end{array} }\!\!\!\! r\!\left( M^{\dag}
 - \!\left[\!\!\begin{array}{c} A^{(1,2)}  \\ B^{(1,2)}  \end{array}
\!\! \right] \right) \leqslant \!\! \min_{\begin{array}{c} A^{(1,2,4)}
 \\[-1mm] B^{(1,2,4)}  \end{array} } \!\!\!\!
r\!\left( M^{\dag} - \!\left[\!\!\begin{array}{c} A^{(1,2,4)}
\\
B^{(1,2,4)}  \end{array} \!\! \right] \right)\!.
$$
Applying (\ref{212}) and (\ref{217}) to this gives (\ref{213}).
\end{proof}

Some rank formulas for $[\, A, \, B\,]^{(i,\ldots,j)}-
 \left[\!\!\begin{array}{c} A^{(i,\ldots,j)}  \\ B^{(i,\ldots,j)}
\end{array} \!\! \right]$ are given below.

\begin{theorem} \label{T23}
Let
$A \in \mathbb C^{m \times n}$ and $B \in \mathbb C^{m\times k},$ and let
$M = [\, A, \, B \,].$ Then
\e
 \ \ \ \min_{M^-, \, A^-, \, B^-} r\!\left( M^- - \left[\!\!\begin{array}{c} A^-
 \\ B^- \end{array} \!\! \right] \right) \es  r(A) + r(B) - r(M),
\label{225}
\\
 \ \ \ \min_{M^{(1,3)}, \, A^{(1,3)}, \, B^{(1,3)} } r\!\left( M^{(1,3)} -
\left[\!\!\begin{array}{c} A^{(1,3)}  \\ B^{(1,3)} \end{array} \!\! \right]
\right) \es  r(M) + 2r(A^*B) - r (A) - r(B), \ \ \
\label{226}
\\
 \ \ \ \min_{M^{(1,4)}, \, A^{(1,4)},\, B^{(1,4)}} r\!\left( M^{(1,4)} -
\left[\!\!\begin{array}{c} A^{(1,4)}  \\ B^{(1,4)} \end{array} \!\! \right]
\right) \es r(A) + r(B) - r(M),
\label{227}
\\
 \ \ \ \min_{\begin{array}{c} M^{(1,2,3)} \\ A^{(1,2,3)}, \, B^{(1,2,3)}
 \end{array}} \!\! r\!\left( M^{(1,2,3)} -
 \left[\!\!\begin{array}{c} A^{(1,2,3)}  \\ B^{(1,2,3)} \end{array}
 \!\! \right] \right) \es r(M) + 2r(A^*B) - r (A) - r(B). \ \ \
\label{228}
\ee
Hence$,$
\begin{enumerate}
\item[{\rm(a)}]  The following statements are equivalent$:$
\begin{enumerate}
\item[{\rm(i)}] There are $A^-$ and $B^-$ such that
$\left[\!\!\begin{array}{c} A^- \\  B^- \end{array} \!\!\right] \in \{M^-\}.$

\item[{\rm(ii)}] There are $A^{(1,4)}$ and $B^{(1,4)}$ such that
$\left[\!\!\begin{array}{c} A^{(1,4)} \\  B^{(1,4)}  \end{array} \!\!\right] \in
\{M^{(1,4)}\}.$

\item[{\rm(iii)}] ${\mathscr R}(A) \cap {\mathscr R}(B) = \{ 0 \}.$
\end{enumerate}

\item[{\rm(b)}]  The following statements are equivalent$:$
\begin{enumerate}
\item[{\rm(i)}] There are $A^{(1,3)}$ and $B^{(1,3)}$ such that
$\left[\!\!\begin{array}{c}  A^{(1,3)} \\  B^{(1,3)} \end{array} \!\!\right]
\in \{M^{(1,3)}\}.$

\item[{\rm(ii)}] There are $A^{(1,2,3)}$ and $B^{(1,2,3)}$ such that
$\left[\!\!\begin{array}{c} A^{(1,2,3)} \\  B^{(1,2,3)}  \end{array} \!\!\right]
\in \{M^{(1,2,3)}\}.$

\item[{\rm(iii)}] $M^{\dag} =\left[\!\!\begin{array}{c}
A^{\dag} \\  B^{\dag} \end{array} \!\!\right]\!.$

\item[{\rm(iv)}] $A^*B =0.$
\end{enumerate}
\end{enumerate}
\end{theorem}

\begin{proof}
From (\ref{117})
\e
\min_{M^-} r\!\left( M^- - \left[\!\!\begin{array}{c} A^-
 \\ B^- \end{array} \!\! \right] \right) \es
r\!\left( M - M\!\left[\!\!\begin{array}{c} A^-
 \\ B^- \end{array} \!\! \right]\!M \right)\!.
\label{229}
\ee
Expanding the difference on the right-hand side of (\ref{229}) yields
$$
M - M\!\left[\!\!\begin{array}{c} A^- \\ B^- \end{array} \!\! \right]\!M
= [\,  BB^-A, \,  AA^-B \,] =
[\,  BB^{\dag}A, \,  AA^{\dag}B \,] + [\, BW_1E_BA , \, AW_2E_AB \,],
$$
where $W_1$ and $W_2$ are two arbitrary matrices. Applying (\ref{127}) to
the right-hand side  and simplifying by (\ref{112}), (\ref{113}) and (\ref{114})
give us
\e
\label{230}
& & \min_{A^-, \, B^-}r\!\left( M - M\!\left[\!\!\begin{array}{c} A^-
 \\ B^- \end{array} \!\! \right]\!M \right)
 \\
 && = \min_{V_1, \, V_2}r(\,[\,  BB^{\dag}A, \,  AA^{\dag}B \,]
 + [\, BV_1E_BA , \, AW_2E_AB \,] \,) \nb
\\
&& = r(A) + r(B) - r(M). \nb
\ee
Combining (\ref{229}) and (\ref{230}) results in (\ref{225}). Eqs.(\ref{226}),
(\ref{227}) and (\ref{228}) can be shown by (\ref{119}),
(\ref{120}), (\ref{121}) and (\ref{127}). The details are omitted here.
\end{proof}


To find
$$
\min_{\begin{array}{c} M^{(1,2)} \\ A^{(1,2)}, \,  B^{(1,2)} \end{array}}
\!\!\!\!r\!\left( M^{(1,2)} -
\left[\!\!\begin{array}{c} A^{(1,2)}  \\ B^{(1,2)} \end{array} \!\!
 \right] \right)\!, \ \ \ \min_{\begin{array}{c} M^{(1,2,4)} \\ A^{(1,2,4)},
 \, B^{(1,2,4)} \end{array}} \!\!\!\! r\!\left( M^{(1,2,4)} -
\left[\!\!\begin{array}{c} A^{(1,2,4)}  \\ B^{(1,2,4)}
\end{array} \!\! \right] \right)\!,
$$ %\\[-8mm]
$$
\min_{\begin{array}{c} M^{(1,3,4)} \\ A^{(1,3,4)},
 \,  B^{(1,3,4)} \end{array}} \!\!\!\! r\!\left( M^{(1,3,4)} -
\left[\!\!\begin{array}{c} A^{(1,3,4)}
 \\ B^{(1,3,4)} \end{array} \!\! \right] \right),
$$
a formula for the minimal rank of $A + B_1X_1C_1 +  B_2X_2C_2 +  B_3X_3C_3$
with respect to $X_1$, $X_2$ and $X_3$ is needed. However, such a formula
has not been established at present; see \cite{Ti-lama}.

In addition to $\left[\!\!\begin{array}{c} A^- \\  B^-\end{array} \!\!\right]$,
 $[\, A, \, B\,]$ may have generalized inverses like
$\left[\!\!\begin{array}{c} A^- \\  X
\end{array} \!\!\right]$ or $\left[\!\!\begin{array}{c} Y \\  B^-
\end{array} \!\!\right]$ for some matrices $X$ and $Y$.


\begin{theorem} \label{T24}
Let $A \in \mathbb C^{m \times n},$ $B \in \mathbb C^{m \times
k}$, $X \in \mathbb C^{k\times m},$  $Y \in \mathbb C^{n \times m},$
and let $M =[\, A, \, B\,].$ Then
\e
\label{231}
\min_X r\!\left( M - M \!\left[\!\!\begin{array}{c} A^{\dag} \\  X
\end{array} \!\!\right]\!M \right) & =&
\min_Y r\!\left( M - M \!\left[\!\!\begin{array}{c} Y \\  B^{\dag}
\end{array} \!\!\right]\!M \right)
\\
& = & r[\, A, \, B \,] + r(A^*B) - r(A) -
r(B). \nb
\ee
Hence the following statement are equivalent$:$
\begin{enumerate}
\item[{\rm(a)}] There is $X$ such that $\left[\!\!\begin{array}{c} A^{\dag} \\  X
\end{array} \!\!\right] \in \{M^-\}.$

\item[{\rm(b)}] There is $Y$ such that $\left[\!\!\begin{array}{c} Y \\ B^{\dag}
\end{array} \!\!\right] \in \{M^-\}.$

\item[{\rm(c)}]  $r[\, A, \, B \,]  =  r(A) + r(B) - r(A^*B).$
\end{enumerate}
\end{theorem}

\begin{proof}
Applying (\ref{125}) and simplifying by elementary block matrix operations give us
\en
& & \min_X r\!\left( M - M \left[\!\!\begin{array}{c} A^{\dag} \\  X
\end{array} \!\!\right]\!M
 \right) \\
& & = \min_X r\!\left( [\, 0, \, B - AA^{\dag}B \,] -
  BX[\, A, \, B \, ] \right)
\\
& & = r[\, B -  AA^{\dag}B, \, B \,] +  r\!\left[\! \begin{array}{cc}
 0 & B - AA^{\dag}B \\  A & B \end{array} \!\right]
- r\!\left[\! \begin{array}{ccc} 0 & B - AA^{\dag}B  & B \\
A & B & 0 \end{array}\!\right]\\
& & = r[\, AA^{\dag}B, \, B \,] +  r\!\left[\! \begin{array}{cc}
 0 & 0 \\  A & B \end{array} \!\right]
- r\!\left[\! \begin{array}{ccc} 0 & 0 & B \\
A & B & 0 \end{array}\!\right]
\\
& &  = r[\, AA^{\dag}B, \, B \,]  - r(B)
\\
& & =  r[\, AA^{\dag}B, \, (\, I_m  - AA^{\dag} \,) B \,]  - r(B)
\\
& & =  r(AA^{\dag}B) + r[\, (\, I_m  - AA^{\dag} \,) B \,] - r(B)
\\
& & =  r[\, A, \, B \, ] + r(A^*B) -  r(A) -  r(B).
\een
Equality
$$
\min_X r\!\left( M - M \!\left[\!\!\begin{array}{c} X \\  B^{\dag}
\end{array} \!\!\right]\!M \right) = r[\, A, \, B \,] + r(A^*B) - r(A) -
r(B)
$$
can be shown similarly. The equivalence of (a), (b) and (c) follows from
 (\ref{231}).
\end{proof}


\begin{theorem} \label{T25}
Let $A \in \mathbb C^{m \times n},$ $B \in \mathbb C^{m \times
k}$, $X \in \mathbb C^{k\times m},$  $Y \in \mathbb C^{n \times m},$
and let $M =[\, A, \, B\,].$ Then
\e
\label{232}
\min_{A^-, X} r\!\left( M - M \!\left[\!\!\begin{array}{c} A^- \\  X
\end{array} \!\!\right]\!M \right) =
\min_{Y, \, B^-} r\!\left( M - M \!\left[\!\!\begin{array}{c} Y \\  B^-
\end{array} \!\!\right]\!M \right) =0.
\ee
Hence$,$
\begin{enumerate}
\item[{\rm(a)}] There are $A^-$ and $X$ such that $\left[\!\!\begin{array}{c} A^- \\  X
\end{array} \!\!\right] \in \{M^-\}.$

\item[{\rm(b)}] There are  $Y$  and $B^-$ and such that
$\left[\!\!\begin{array}{c} Y \\ B^- \end{array} \!\!\right] \in \{M^-\}.$
\end{enumerate}
\end{theorem}


\begin{proof}
Applying (\ref{125}) and simplifying by elementary block  matrix operations give us
\e
\label{233}
  \ \  & & \min_X r\!\left( M - M \left[\!\!\begin{array}{c} A^- \\  X
\end{array} \!\!\right]\!M
 \right)
 \\
& & = \min_X r\!\left( [\, 0, \, B - AA^-B \,] -
  BX[\, A, \, B \, ] \right) \nb
\\
& & = r[\, B -  AA^-B, \, B \,] +  r\!\left[\! \begin{array}{cc}
 0 & B - AA^-B \\  A & B \end{array} \!\right]
- r\!\left[\! \begin{array}{ccc} 0 & B - AA^-B  & B
\\
A & B & 0 \end{array}\!\right] \nb
\\
& &  = r[\, AA^-B, \, B \,]  - r(B) \nb
\\
& & =  r[\, AA^-B, \, (\, I_m  - AA^- \,) B \,]  - r(B) \nb
\\
& & =  r(AA^-B) + r[\, (\, I_m  - AA^- \,) B \,] - r(B) \nb
\\
& & =  r(AA^-B) + r[\, A, \, B \, ] -  r(A) -  r(B). \nb
\ee
Also from (\ref{128})
$$
\min_{A^-} r(AA^-B) = r(A) +  r(B) - r[\, A, \, B \,].
$$
Combining this with (\ref{233}) leads to
$$
\min_{A^-, X} r\!\left( M - M \!\left[\!\!\begin{array}{c} A^- \\  X
\end{array} \!\!\right]\!M \right) = 0.
$$
The second equality in (\ref{232}) can be shown similarly.
\end{proof}


Minimal rank formulas for column block matrices can be written out
 by symmetry. They are left for the reader. Some other topics on minimal
 ranks of generalized inverses of row block matrices can also be
  considered, for instance, to find
the minimal ranks of
$$
[\, A, \,  B \,]^{(i,\ldots,j)} - \left[\!\!\begin{array}{c}
(E_BA)^{(i,\ldots,j)}   \\
(E_AB)^{(i,\ldots,j)}  \end{array} \!\! \right] \ \ {\rm and } \ \
[\, A_1, \, \ldots, \,  A_p \,]^{(i,\ldots,j)} - \left[\!\!\begin{array}{c}
 A_1^{(i,\ldots,j)}   \\ \vdots \\
A_p^{(i,\ldots,j)}  \end{array} \!\! \right]\!,
$$
and then to use the minimal ranks to characterize the corresponding matrix
equalities.

For $2 \times 2$ partitioned matrices, it is of interest to establish
minimal rank formulas for the differences
$$
\left[\!\!\begin{array}{cc} A  & B  \\ C & 0 \end{array}
 \!\! \right]^{(i,\ldots,j)} -
\left[\!\!\begin{array}{cc} A^{(i,\ldots,j)}   & C^{(i,\ldots,j)}
 \\ B^{(i,\ldots,j)}  & 0 \end{array} \!\! \right]\!,
  \ \ \  \left[\!\!\begin{array}{cc} A  & B  \\ C & D \end{array}
 \!\! \right]^{(i,\ldots,j)} -
\left[\!\!\begin{array}{cc} A^{(i,\ldots,j)}   & C^{(i,\ldots,j)}
 \\ B^{(i,\ldots,j)}  & D^{(i,\ldots,j)}  \end{array} \!\! \right]\!.
$$
It is expected that the rank formulas obtained can be used to characterize
the corresponding equalities for generalized inverses of $2 \times 2$
partitioned matrices.




\begin{thebibliography}{99}
        %\def\itemsep{-0.5ex}
        %\small

\bibitem{BG}
A.~Ben-Israel and T.N.E.~Greville.
\newblock {\em Generalized Inverses: Theory and Applications}.
\newblock 2nd Edn., Springer-Verlag, New York, 2003.

\bibitem{CM} S.L.~Campbell and C.D.~Meyer.
{\em Generalized Inverses of Linear Transformations}.
Corrected reprint of the 1979 original, Dover Publications, Inc., New York, 1991.


\bibitem{MS} G.~Marsaglia and G.P.H.~Styan.
Equalities and inequalities for ranks of matrices.
{\em Linear and Multilinear Algebra}, 2:269--292, 1974.


\bibitem{Mi} L. Mih\'alyffy.
An alternative representation of the generalized inverse  of partitioned matrices.
  {\em Linear Algebra Appl.}, 4:95--100, 1971.

\bibitem{RM} C.R.~Rao and S.K.~Mitra.
{\em Generalized Inverse of Matrices and Its Applications}.
Wiley, New York, 1971.

\bibitem{Ti-laa02} Y.~Tian.
Upper and lower bounds for ranks of matrix expressions using generalized
inverses.
{\em Linear Algebra Appl.}, 355:187--214, 2002.

\bibitem{Ti-lama} Y.~Tian.
The minimal rank completion of a $3 \times 3$ partial block matrix.
{\em Linear and Multilinear Algebra}, 50:125--131, 2002.

\bibitem{Ti-se} Y.~Tian.
The maximal and minimal ranks of  some expressions of generalized
inverses of matrices.
{\em Southeast Asian Bull. Math.}, 25:745--755, 2002.


\bibitem{Ti-hjm} Y.~Tian.
Rank equalities for block matrices and their Moore-Penrose inverses.
{\em Houston J. Math.}, 30:483--510, 2004.


\bibitem{Ti-amc} Y.~Tian.
More on maximal and minimal ranks of Schur complements with applications.
{\em Appl. Math. Comput.}, 152:675--692, 2004.


\bibitem{Ya} H. Yanai.
Some generalized forms of least squares $g$-inverse, minimum norm $g$-inverse,
and Moore-Penrose inverse matrices.
{\em Comput. Statist. Data Anal.}, 10:251--260, 1990.

\bibitem{Zl} S. Zlobec.
An explicit form of the Moore-Penrose inverse of an arbitrary complex matrix.
{\em SIAM Review}, 12:132--134, 1970.


\end{thebibliography}
\end{document}


