Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}. [Graph: vertex 3 at top, connected to vertex 1 and vertex 4; vertex 1 also connected to vertex 2; vertex 4 also connected to vertex 5. One edge between vertex 1 and vertex 3 only visible from context.] Let lambda_1, lambda_2, lambda_3, lambda_4, and lambda_5 be the five eigenvalues of A. Note that these eigenvalues need not be distinct. The value of lambda_1^2 + lambda_2^2 + lambda_3^2 + lambda_4^2 + lambda_5^2 = ___

GATE 2023 · Engineering Mathematics · Eigen Value · medium

Answer: lambda_1^2 + lambda_2^2 + lambda_3^2 + lambda_4^2 + lambda_5^2 = 2

1. Identify edges in the graph: From the given graph with vertices {1,2,3,4,5}, count the number of undirected edges. The graph has exactly 1 edge: (1,3). Thus |E| = 1.
2. Apply eigenvalue sum-of-squares identity: lambda_1^2 + lambda_2^2 + lambda_3^2 + lambda_4^2 + lambda_5^2 = trace(A^2) = 2 * 1 = 2