A203004 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A203003; by antidiagonals.

1, -1, 1, -18, 1, 1, -84, 116, -1, 1, -439, 1221, -839, 1, 1, -2475, 10435, -13855, 5658, -1, 1, -14312, 81690, -165715, 138669, -39038, 1, 1, -83270, 601411, -1661956, 2164099, -1292751, 266899, -1, 1, -485157

Offset: 1

Author

Clark Kimberling, Dec 27 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...-18....1
1...-84....116....-1
1...-439...1221...-839...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. A203003, A202605.

Programs

Mathematica

f[k_] := Fibonacci[k + 1]^2;
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}]]