MathDB

8

Part of 2017 IMC

Problems(1)

IMC 2017 Problem 8

Source:

8/3/2017
Define the sequence A1,A2,…A_1,A_2,\ldots of matrices by the following recurrence: A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix},   A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix}   (n=1,2,\ldots) where ImI_m is the m×mm\times m identity matrix.
Prove that AnA_n has n+1n+1 distinct integer eigenvalues λ0<λ1<…<λn\lambda_0< \lambda_1<\ldots <\lambda_n with multiplicities (n0),(n1),…,(nn)\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}, respectively.
college contestsimc 2017IMClinear algebra