A202971 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A202970; by antidiagonals.

1, -1, 1, -11, 1, 1, -30, 57, -1, 1, -53, 338, -224, 1, 1, -80, 992, -2600, 752, -1, 1, -111, 2171, -11803, 15614, -2304, 1, 1, -146, 4039, -35908, 105335, -79786, 6665, -1, 1, -185, 6776, -87154, 434244, -770624, 362449, -18595

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...-11...1
1...-30...57....-1
1...-53...338...-224...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. A202970, A202605.

Programs

Mathematica

f[k_] := -2 + Fibonacci[k + 3]
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}]]