  The matrix joint block diagonalization problem (JBDP) of a given matrix set
$\mathcal{A}=\{A_i\}_{i=1}^m$ is about finding a nonsingular matrix $W$ such
that all $W^{T} A_i W$ are block diagonal. It includes the matrix joint
diagonalization problem (JBD) as a special case for which all $W^{T} A_i W$ are
required diagonal. Generically, such a matrix $W$ may not exist, but there are
practically applications such as multidimensional independent component
analysis (MICA) for which it does exist under the ideal situation, i.e., no
noise is presented. However, in practice noises do get in and, as a
consequence, the matrix set is only approximately block diagonalizable, i.e.,
one can only make all $\widetilde{W}^{T} A_i\widetilde{W}$ nearly block
diagonal at best, where $\widetilde{W}$ is an approximation to $W$, obtained
usually by computation. This motivates us to develop a perturbation theory for
JBDP to address, among others, the question: how accurate this $\widetilde{W}$
is. Previously such a theory for JDP has been discussed, but no effort has been
attempted for JBDP yet. In this paper, with the help of a necessary and
sufficient condition for solution uniqueness of JBDP recently developed in [Cai
and Liu, {\em SIAM J. Matrix Anal. Appl.}, 38(1):50--71, 2017], we are able to
establish an error bound, perform backward error analysis, and propose a
condition number for JBDP. Numerical tests validate the theoretical results.
