Proving properties of a 2x2 complex positive matrix

In summary: And whoops, I accidentally replied to the summary instead of the conversation, my bad!In summary, the conversation discusses proving that a 2x2 complex matrix is positive if and only if it satisfies the conditions (i) A=A* and (ii) a, d, and |A| = ad-bc are real and positive. The first part of the proof is discussed and it is shown that if A is positive, then A=A* and vice versa. The second part involves proving that a, d, and ad-bc are nonnegative, but the poster is struggling with this part. Suggestions are made to think about the product of eigenvalues and the trace of A to help with the proof.
  • #1
Adgorn
130
18

Homework Statement


Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\
c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc##

Homework Equations


N/A

The Attempt at a Solution


I got stuck at the first part. if ##A## is positive then by definition ##A=S^*S## for some matrix ##S## and thus ##A^*=(S^*S)^*=S^*S=A## and so ##A=A^*##.
However I cannot prove the opposite, that if ##A=A^*## then A is positive. Since ##A=A^*## A is normal and thus there exists an orthonormal basis of ##C^2## comprised of eigenvectors of A, say, {##v_1.v_2##}. Furthermore since ##A## is self-adjoint all eigenvalues of ##A## are real. If I could prove the eigenvalues of ##A## are also nonnegative the proof will be complete, but I cannot seem to manage to prove that and it might not even be true.

Regarding the second part, assuming I have proven the first part it is easy to show how ##a##, ##d## and ##ad-bc## must be real. However I can't seem to figure out how to prove they are nonnegative.
Help would be appriciated.
 
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  • #2
Adgorn said:

Homework Statement


Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\
c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc##
Is the last part missing "are real"?
That seems to be the case based on what you wrote in your attempt.
 
  • #3
Adgorn said:

Homework Statement


Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\
c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc##

...

Regarding the second part, assuming I have proven the first part it is easy to show how ##a##, ##d## and ##ad-bc## must be real. However I can't seem to figure out how to prove they are nonnegative.
Help would be appriciated.
I would spend some time thinking about the second part. notice that ad-bc is the determinant. If the product of two eigenvalues is negative, and both eigs are real (as they are in Hermitian operator) then what does that tell you about the eigenvalues. What if the product is positive?

Now let's think about the trace for a minute...

you said if Hermitian positive (semi) definite, then ##A=S^* S##

I claim that I can get the squared Frobenius norm of ##S##, by taking the ##\big \Vert S \big \Vert_F^2= trace(A) = trace(S^* S)##? Why is this true? What do the diagonal entries of ##A## look like if they come from ##S^* S##?
 
Last edited:
  • #4
Mark44 said:
Is the last part missing "are real"?
That seems to be the case based on what you wrote in your attempt.
Are real and positive yes, forgot to write that for some reason.
 

Related to Proving properties of a 2x2 complex positive matrix

1. How do you define a 2x2 complex positive matrix?

A 2x2 complex positive matrix is a square matrix with complex entries that satisfies the following conditions: all its eigenvalues have positive real parts, its diagonal entries are positive, and its off-diagonal entries are nonnegative.

2. What properties can be proven about a 2x2 complex positive matrix?

Some properties that can be proven about a 2x2 complex positive matrix include: it has a positive determinant, it is invertible, and it has a unique positive definite square root.

3. How can one prove the positive definiteness of a 2x2 complex positive matrix?

The positive definiteness of a 2x2 complex positive matrix can be proven by showing that all its eigenvalues have positive real parts. This can be done by calculating the eigenvalues using the characteristic polynomial or by using the Gershgorin circle theorem.

4. Can a 2x2 complex positive matrix have negative or zero entries?

No, a 2x2 complex positive matrix must have all positive entries on its diagonal and nonnegative entries on its off-diagonal. If any of its entries are negative or zero, it would not satisfy the conditions for being a complex positive matrix.

5. How is a 2x2 complex positive matrix useful in scientific research?

A 2x2 complex positive matrix is useful in various fields of science such as physics, engineering, and economics. It is used in analyzing systems with multiple variables and in solving systems of linear equations. It can also be used in studying the stability of a system and in understanding complex dynamical systems.

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