A letter matrix is an n-by-n matrix whose entries are n symbols, each appearing n times. The row (column) distribution of a letter matrix is an n-by-n nonnegative integer matrix that tells how many of each letter are in each row (column). A row distribution R and a column distribution C are compatible if there exits a letter matrix A whose row distribution is R and whose column distribution is ...

The ith row of A represents a ±1-valued “square wave” function of j with halfperiod i. Thus xA, where x is a (row) N -vector, gives a picture of the waveform that is a linear combination of square waves with weights in x. Similarly yA finds weights so that a given waveform y is a weighted linear combination of these square waves. This matrix has a surprisingly simple inverse. More generally, we...

An efficient method for the computation to high relative accuracy of the LDU decomposition of an n × n row diagonally dominant M–matrix is presented, assuming that the off–diagonal entries and row sums are given. This method costs an additional O(n) elementary operations over the cost of Gaussian elimination, and leads to a lower triangular, column diagonally dominant matrix and an upper triang...

A new method for feature extraction and recognition, namely the fuzzy bidirectional weighted sum criterion (FBWSC) is proposed in this paper. FBWSC defines the row directional fuzzy image optimal image projection matrix. Subsequently, each sample in the original training sample set is transformed using the row directional optimal image projection matrix, and the row directional feature training...

8 Elementary operations and the rank of a matrix 25 8.1 Gauss transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 8.2 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.3 The augmented matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.4 Row reducing a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 8.5 Elem...

in this paper, we consider a class of column-weight two quasi-cyclic low-density paritycheck codes in which the girth can be large enough, as an arbitrary multiple of 8. then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. finally, we show that the free distance of the convolution...

The row-by-row frontal method may be used to solve general large sparse linear systems of equations. By partitioning the matrix into (nearly) independent blocks and applying the frontal method to each block, a coarse-grained parallel frontal algorithm is obtained. The success of this approach depends on preordering the matrix. This can be done in two stages, (1) order the matrix to bordered blo...