2021-11-13

Pinzheng Furniture

48

Hint: If a bounded linear operator on a Hilbert space is self-adjoint w.r.t. some inner product, its eigenvalues must be real (see the proof on ProofWiki). Now the matrix representation of the linear operator $A$ w.r.t. the standard basis of $M_2(mathbbC)$ is

$$

pmatrix1&0&-1&0

0&0&0&1

-1&0&1&0

0&-1&0&0.

$$

What are the eigenvalues of $A$?

Given the linear operator

$A in L(M_2(mathbbC))$

$A beginbmatrixa & b c & d endbmatrixbeginbmatrixa-b & -ab d & -c endbmatrix$

Is there a dot product where the operator becomes Hermitian? ($A^*A$)

I don't know how to prove that there exists one, but I don't know how to construct a dot product either. Any help is appreciated.

Â·OTHER ANSWER:

Given the linear operator

$A in L(M_2(mathbbC))$

$A beginbmatrixa & b c & d endbmatrixbeginbmatrixa-b & -ab d & -c endbmatrix$

Is there a dot product where the operator becomes Hermitian? ($A^*A$)

I don't know how to prove that there exists one, but I don't know how to construct a dot product either. Any help is appreciated.

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