Let A be a symmetric, real matrix.
We want to find the matrices U and Λ such that AU=UΛ.
Obviously, a solution is given by the eigenvalue decomposition, where Λ is diagonal. But is there any other solution?
In other words: if AU=UΛ, where A is a known real symmetric matrix, then must Λ be diagonal?
Observation: In my context, I also know that Λ is symmetric. Maybe it helps.
Answer
Definitely No, In fact if A=UΛU† is an answer then so is
A=VΛ′V†
For any unitary V and Λ′=V†UΛU†V.
An extreme example is
A=1Λ1
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