Tuesday, 14 May 2019

linear algebra - AU=ULambda - single solution (by eigenvalue decomposition)?



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=VΛV
For any unitary V and Λ=VUΛUV.

An extreme example is
A=1Λ1


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...