Throughout this paper, let (A=(a_{n,k})) be a matrix with non-negative real entries i.e., (a_{n,k}ge0), for all n, k.
However, for a class of matrices, A−1can be a matrix with off-diagonal decay properties, i.e., |Aij−1| decays fast to 0 with respect to the increase of a properly defined distance between i and j.
For the moment we will exclude the outer DFT matrices and determine the expectation value of the inner matrix product. is a circular Toeplitz matrix based on the channel impulse response, and was shown in Section 3.2 to be a matrix with zero entries except for the submatrix found in its upper right corner.
This means that if the precoder matrix had dimensions equal to the number of beams (i.e., interferences from all beams are taken into account) it would be a matrix with very small, decreasing entries away from the main diagonal, resulting in an ill-conditioned precoding matrix that cannot be accurately handled.
and E ( x ¯ ) is a local solution of the problem P. Proof Let x ¯ be a local solution of the problem P E, then there is no vector d such that ∇ ( f ∘ E ) ( x ¯ ) d < 0 and ∇ ( g i ∘ E ) ( x ¯ ) d < 0. Let A be a matrix with rows ∇ ( f ∘ E ) ( x ¯ ) and ∇ ( g i ∘ E ) ( x ¯ ).
Let A = ( a i j ) n × n be a matrix with nonpositive off-diagonal elements, then the following statements are equivalent: (i) A is an M-matrix; (ii) there exists a vector η > 0 such that A η > 0 ; (iii) there exists a vector ξ > 0 such that ξ T A > 0 ; (iv) there exists a positive definite n × n diagonal matrix D such that A D + D A T > 0. .
