Jacket Matrix and Its Applications to Signal Processing

The Hadamard transform is an orthogonal transform with highly practical values for signal sequence transforms and data processing. Jacket matrices, which are motivated by the center weight Hadamard matrices, are a class of matrices with their inverse being determined by the element-wise of the matrix. Mathematically, let A = (a_kt) be a matrix, if A^-1 = (a_kt^-1)^T, then the matrix A is a Jacket matrix, where (·) denotes a matrix and T denotes the transpose. Since inverse of the Jacket matrix can be calculated easily, it is very helpful to employ this kind of matrix in the signal processing, encoding, mobile communication, image compression, cryptography, etc. Especially, the interesting orthogonal matrices, such as Hadamard, Haar, DFT, slant matrices, belong to the Jacket matrix family. In addition, Jacket matrices are associated with many kinds of matrices, such as unitary matrices and Hermitian matrices which are very important in signal processing, communication (e.g., encoding), mathematics, and physics.