editing

approved

The row sums are P(n, 1) = (-1)^(n-1)*(n-1)*2^(n-2) = A085750(n), for n >= 1, and for n=0 the row sum is 0. The alternating row sums are P(n, -1) = 2 for n=1, = -1 for n = 2 , and zero otherwise.

approved

editing

editing

approved

Therefore the spectral radius (absolute value of the maximal eigenvectoreigenvalue) is rho_n = sqrt(n), and the spectral norm of M_n (square root of the maximal eigenvalue of (M_n)^+ M_n is also sqrt(n), for n >= 1. See the conjecture in the first comment above.

The square of the Frobenius norm (aka Hilbert-Schmidt norm) of M_n is max_{i,j=1..n} |M_n(i,j)| ^2 = 3*n - 2 = A016777(n-1), for n >= 1.

approved

editing

proposed

approved

editing

proposed

The norm of the matrix M appears to be sqrt(n, ), where with norm is meant the eigenvalue of the largest magnitude, negative or positive. Row sums appear to be A085750 [see below for the proof].

The eigenvalues of M_n are +1 for n = 1 and for n >= 2 they are +sqrt(n, ), -sqrt(n, ), and n-2 times -1.

Therefore the spectral radius (absolute value of the maximal eigenvector) is rho_n = sqrt(n, ), and the spectral norm of M_n (square root of the maximal eigenvalue of (M_n)^+ M_n) is also sqrt(n, ), for n >= 1. See the conjecture in the first comment above.

From Wolfdieter Lang, Feb 02 32 2018 : (Start)

The norm of the matrix M appears to be sqrt(n), , where with norm is meant the eigenvalue of the largest magnitude, negative or positive. Row sums appear to be A085750 [see below for the proof].

The eigenvalues of M_n are +1 for n = 1 and for n >= 2 they are +n, -n, and n-2 times -1.

Therefore the spectral radius (absolute value of the maximal eigenvector) is rho_n = n, and the spectral norm of M_n (square root of the maximal eigenvalue of M_n^+ M_n) is also n, for n >= 1. See the conjecture in the first comment above.

The square of the Frobenius norm (aka Hilbert-Schmidt norm) of M_n is max_{i,j=1..n} |M_n(i,j)| = 3*n - 2 = A016777(n-1), for n >= 1.

Cf. A016777, A085750 (row sums), A067998, A110427 (column k=2), -A131386 (column k=1), A181983 (Det M_n), A191898.

proposed

editing

editing

proposed