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Sequence of symmetric matrices

Source: CIIM 2023 - Problem 3

September 19, 2023
linear algebramatrixSequencessymmetric matrixeigenvalue and eigenvector

Problem Statement

Given a 3×33 \times 3 symmetric real matrix AA, we define f(A)f(A) as a 3×33 \times 3 matrix with the same eigenvectors of AA such that if AA has eigenvalues aa, bb, cc, then f(A)f(A) has eigenvalues b+cb+c, c+ac+a, a+ba+b (in that order). We define a sequence of symmetric real 3×33\times3 matrices A0,A1,A2,…A_0, A_1, A_2, \ldots such that An+1=f(An)A_{n+1} = f(A_n) for n≥0n \geq 0. If the matrix A0A_0 has no zero entries, determine the maximum number of indices j≥0j \geq 0 for which the matrix AjA_j has any null entries.
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