Consider a matrix A = u * u^T where u = (1, 1, 1)^T. Note that u^T denotes the transpose of u. The largest eigenvalue of...

Question

Consider a matrix A = u * u^T where u = (1, 1, 1)^T. Note that u^T denotes the transpose of u. The largest eigenvalue of A is ____.

Accepted Answer

3 A = uu^T is a rank-1 matrix formed by the outer product of u with itself. For any rank-1 matrix uu^T, the only nonzero eigenvalue is u^Tu (the inner product, a scalar), and it has multiplicity 1. The remaining n-1 eigenvalues are all zero. Here u = (1,1,1)^T, so u^Tu = 1+1+1 = 3, making the largest eigenvalue 3.