Is the mentioned matrix possible? If so, what is an example of one?
2026-03-29 18:33:08.1774809188
Example of 4x4 symmetric matrix that is not diagonalizable
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It suffices to select a colum-vector $u$ such that $u^Tu = 0$, then take $A = uu^T$. In particular, if $u = (1,i,0,0),$ we find $$ A = \pmatrix{1&i&0&0\\i&-1&0&0\\ 0&0&0&0\\0&0&0&0} $$