It is shown that a sufficient and necessary condition for a large eigen-gap is that is a "hub" matrix in the sense that it has dominant columns.
In this study, for the flexural dominant columns with damage concentrated near the base, only the portion of the columns with severe damage, and the region immediately adjacent to it, were repaired.
The reason is that the dominant columns are the same, and moving closer to, or farther away from each column changes the corresponding weights only slightly, and keeps the extent and elongation of the receptive fields more or less the same.
The dominant right (column) eigenvector of M 2 can be written u = [1, m 12/m 21]′, with the first element of u scaled to one.
Thus the matrix B is strictly diagonally dominant by columns, and (vert Avert = vert Bvert neq 0).
(langle Arangle) stands for the comparison (Ostrowski) matrix of the matrix A, and (|A|) is the absolute matrix of the matrix A. In what follows, when A is a strictly diagonally dominant matrix in column, A is called a strictly diagonally dominant matrix.
Thus, we can write (2.8) equivalently as ∑ k ≠ i | b k i | x k b i i x i = ρ ( J ) ∀ i ∈ N. Set X = diag ( x 1, x 2, …, x n ) and B ˜ = X B. It is easy to check that B ˜ is a strictly diagonally dominant matrix by column.
(b) If A = ( a i j ) is an n × n strictly diagonally dominant matrix by column, that is, | a i i | > ∑ j ≠ i | a j i | for any i ∈ N, then A − 1 = ( β i j ) exists, and | β i j | ≤ ∑ k ≠ j | a k j | | a j j | | β i i |, for all j ≠ i. Proof We give a simple proof of (a) which is different from that in [8].
The reason is that the highest density of afferent arborization in the center makes the isotropic, non-oriented receptive field of the column dominant, preventing any receptive field elongation.
