Consider the following matrix:
M = [[0,1,1,1],[1,0,1,1],[1,1,0,1],[1,1,1,0]]
The largest eigenvalue of the above matrix ...

Question

Consider the following matrix:
M = [[0,1,1,1],[1,0,1,1],[1,1,0,1],[1,1,1,0]]
The largest eigenvalue of the above matrix is __________

Accepted Answer

3 Matrix M has 0 on the diagonal and 1 everywhere else. This is J - I where J is the 4x4 all-ones matrix and I is the identity. The all-ones matrix J has eigenvalues n (with eigenvector [1,1,...,1]^T) and 0 (with multiplicity n-1). So M = J - I has eigenvalues (n-1) and (-1). For n=4, the eigenvalues are 3 (once) and -1 (three times). The largest eigenvalue is 3.

Consider the following matrix: M = [[0,1,1,1],[1,0,1,1],[1,1,0,1],[1,1,1,0]] The largest eigenvalue of the above matrix is __________