Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?
[Math] non-symmetric matrix with orthogonal eigenvectors
diagonalizationeigenvalues-eigenvectorslinear algebramatrices
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Best Answer
If a matrix $A$ has orthogonal eigenvectors, then there exists an orthogonal $P$ and a diagonal matrix $\Lambda$ such that $$ A=P\Lambda P^\top $$ It follows that $$ A^\top = (P\Lambda P^\top)^\top = (P^\top)^\top\Lambda^\top P^\top=P\Lambda P^\top=A $$ Hence $A$ is symmetric.