A202875 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202874; by antidiagonals.

1, -1, 1, -6, 1, 1, -12, 20, -1, 1, -19, 69, -59, 1, 1, -27, 159, -303, 162, -1, 1, -36, 302, -943, 1149, -434, 1, 1, -46, 511, -2284, 4599, -3991, 1147, -1, 1, -57, 800, -4743, 13733, -19785, 13090, -3016, 1, 1, -69, 1184, -8867, 34141, -70945

Offset: 1

Author

Clark Kimberling, Dec 26 2011

Keywords

• tabl
• sign

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

Examples

			Top of the array:
1...-1
1...-6....1
1...-12...20...-1
1...-19...69...-59...1
		

Links

• S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
• A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Crossrefs

Cf. A202874, A202605.

Programs

Mathematica

f[k_] := Fibonacci[k + 1]
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]