Consider the following matrix. \(\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\…
2021
Consider the following matrix.
\(\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}\)
The largest eigenvalue of the above matrix is __________.
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Correct answer: 3
Key idea: write the matrix as J - I, where J is the 4x4 all-ones matrix and I is the 4x4 identity matrix.
Let A denote the given matrix. Then A = J - I.
The all-ones vector v = (1,1,1,1)^T satisfies Jv = 4v, so Av = (J - I)v = 4v - v = 3v. Hence 3 is an eigenvalue.
Any vector w whose entries sum to zero satisfies Jw = 0, so Aw = (J - I)w = -w. Therefore -1 is an eigenvalue with multiplicity 3.
Thus the eigenvalues are 3 and -1 (three times), and the largest eigenvalue is 3.
Shortcut:
When each row sum is equal that would be the maximum Eigen value of the matrix. In this case sum is 3 so answer will be 3.