Matrix inverses in MIMO wireless communication
Matrix inversion plays significant role in the MIMO (Multiple–Input, Multiple–Output) technology in wireless communications.
The MIMO system consists of N transmit and M receive antennas. Unique signals, occupying the same frequency band, are sent via N transmit
antennas and are received via M receive antennas.
The signal arriving at each receive antenna will be a linear combination of the N
transmitted signals forming a N × M transmission matrix H.
It is crucial for the matrix H to be invertible for the receiver to be
able to figure out the transmitted information.
Let A be a square matrix of order n × n and let In be an identity matrix of same order.
If there exists a square matrix B, such that AB = BA = I, then the matrix A is said to be invertible. The matrix B is called inverse of A.
It is represented as B = A−1
Stated differently AA−1 = In = A−1A
(The symbol A−1 is read as "A inverse").
The formula for calculating the inverse of a square matrix A is given by:
Consider a 2 × 2 matrix A =
where the elements are :
a11 = a; a12 = b; a21 = c; a22 = d
The minors of the elements are :
M11 = d; M12 = c; M21 = b; M22 = a
The corresponding co-factors are :
C11 = d; C12 = –c; C21 = – b; C22 = a
Adjoint matrix is obtained by first replacing the elements with their respective co-factors and then transposing the matrix.