In mathematics, a jacket matrix is a square symmetric matrix ${\displaystyle A=(a_{ij})}$ of order n if its entries are non-zero and real, complex, or from a finite field, and

${\displaystyle \ AB=BA=I_{n}}$

where I_n is the identity matrix, and

${\displaystyle \ B={1 \over n}(a_{ij}^{-1})^{T}.}$

where T denotes the transpose of the matrix.

In other words, the inverse of a jacket matrix is determined its element-wise or block-wise inverse. The definition above may also be expressed as:

${\displaystyle \forall u,v\in \{1,2,\dots ,n\}:~a_{iu},a_{iv}\neq 0,~~~~\sum _{i=1}^{n}a_{iu}^{-1}\,a_{iv}={\begin{cases}n,&u=v\\0,&u\neq v\end{cases}}}$

The jacket matrix is a generalization of the Hadamard matrix; it is a diagonal block-wise inverse matrix.