Determinant of exponential matrix

[Pushoam]

Homework Statement

Homework Equations

The Attempt at a Solution

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Det( $e^A$ ) = $e^{(trace A)}$

$trace(A) = trace( SAS^{-1}) = 0$ as trace is similiarity invariant.

Det( $e^A$ ) = 1

The answer is option (a).

Is this correct?

But in the question, it is not given that S AS^-1 is a similarity matrix of A. I assumed it without checking.
So, should I check it by calculating eigen values of both S and $SAS^{-1}$ and their eigen vectors?

 

[Orodruin]

Yes.

 

[Pushoam]

Yes.

But in the question, it is not given that $S AS{-1}$ is a similarity matrix of A. I assumed it without checking.
So, should I check it by calculating eigen values of both S and $SAS^{-1}$and their eigen vectors?

 

[Orodruin]

$S$ is clearly a unitary matrix. Anyway, all you really need to know is that $S$ is invertible (otherwise $S^{-1}$ does not exist) and the cyclic property of the trace.

 

[Pushoam]

Anyway, all you really need to know is that SSS is invertible (otherwise S−1S−1S^{-1} does not exist) and the cyclic property of the trace.

If there exists a non – singular matrix P, then $P^{-1} A P$ is known as a similarity matrix of A.

Similarity matrix is the name of $P^{-1} A P$.

Why is $P^{-1} A P$ called a "similiarity matrix" of A? Is there any intuitive reason behind the word " similiarity"?

Now, A and its similiarity matrix have the following properties:

1) They have the same set of eigen values.

2) If x is eigen vector of A, then $P^{-1} x$ is the eigen vector of corresponding similiarity matrix $P^{-1} A P$.

3) Trace( A ) = Trace ($P^{-1} A P$ )

But in the question, it is not given that $SAS^{-1}$. is a similarity matrix of A. I assumed it without checking.
So, should I check it by calculating eigen values of both S and $SAS^{-1}$ and their eigen vectors?

$SAS^{-1}$ is a similarity matrix of A by definition. There is no need to check it.

Is this correct?

 

[Orodruin]

Yes, but the point is that you do not need all those properties to solve this problem. You just need the cyclicity of the trace.

 

[Pushoam]

Thanks, I got it.