Transform a 2x2 matrix into an anti-symmetric matrix

In summary, to transform a 2x2 matrix into an anti-symmetric matrix, you need to take the transpose of the original matrix, multiply it by -1, and add it to the original matrix. An anti-symmetric matrix is defined as a square matrix with negative values below the main diagonal and zero values on the main diagonal. This transformation is important in physics, engineering, and linear algebra. However, not all 2x2 matrices can be transformed into anti-symmetric matrices. There is a formula for transforming a matrix into an anti-symmetric matrix without taking the transpose and multiplying by -1.
  • #1
dRic2
Gold Member
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Hi,
I have a 2x2 hermitian matrix like:
$$
A = \begin{bmatrix}
a && b \\
-b && -a
\end{bmatrix}
$$
(b is imaginary to ensure that it is hermitian). I would like to find an orthogonal transformation M that makes A skew-symmetric:
$$
\hat A = \begin{bmatrix}
0 && c \\
-c && 0
\end{bmatrix}
$$
Is it possible, or I need to constrain my problem more? I need M to be orthogonal and with det(M) = 1. I was thinking maybe there are some tricks involving Pauli matrices.Ric
 
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  • #2
This is possible since the minimal polynomials of [itex]A[/itex] and [itex]\hat A[/itex] are [itex]m_A(\lambda) = \lambda^2 + b^2 - a^2[/itex] and [itex]m_{\hat A}(\lambda) = \lambda^2 + c^2[/itex], so if [itex]c^2 = b^2 - a^2[/itex] they will have the same minimal polynomial and the same Jordan normal form (which in this case is diagonal).

However, I don't think [itex]M[/itex] will be orthogonal unless the eigenvectors of [itex]A[/itex] are orthogonal, which does not appear to be the case in general: the eigenvectors are [itex](b, a \mp \lambda)[/itex] and their inner product is [tex]|a|^2 + |b|^2 + 2\operatorname{Im}(a\bar{\lambda}) - |\lambda|^2[/tex] where [itex]\lambda^2 = a^2 - b^2[/itex].
 
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  • #3
Question: since A is hermitian ##\lambda## are reals, so if ##a## is real, then ##\text{Im}(a\lambda) = 0##
and
$$
|a|^2 - |b|^2 - (a^2 - b^2) = |b|^2 + b^2 = 0
$$
because ##b## is purely imaginary. Right?
 
  • #4
No, you have ##|b|^2## with the wrong sign on the left.
 
  • #5
Maarten Havinga said:
No, you have ##|b|^2## with the wrong sign on the left.
Sorry it was a typo. It should read ##+|b|^2## and it should be correct.
 
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  • #6
It is correct with that addition
 
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FAQ: Transform a 2x2 matrix into an anti-symmetric matrix

1. How do you transform a 2x2 matrix into an anti-symmetric matrix?

To transform a 2x2 matrix into an anti-symmetric matrix, you need to swap the elements in the first row with the elements in the second row and change their signs. For example, if the original matrix is [a b; c d], the anti-symmetric matrix would be [d -b; -c a].

2. What is the purpose of transforming a matrix into an anti-symmetric matrix?

The purpose of transforming a matrix into an anti-symmetric matrix is to create a matrix that is equal to its own negative transpose. This property is useful in various mathematical and scientific applications, such as in linear algebra and mechanics.

3. Can a 2x2 matrix always be transformed into an anti-symmetric matrix?

Yes, a 2x2 matrix can always be transformed into an anti-symmetric matrix. This is because a 2x2 matrix only has four elements, and swapping and changing the signs of these four elements will always result in an anti-symmetric matrix.

4. Is the process of transforming a matrix into an anti-symmetric matrix reversible?

Yes, the process of transforming a matrix into an anti-symmetric matrix is reversible. If you take the anti-symmetric matrix and perform the same steps of swapping and changing signs, you will get back the original matrix.

5. Are there any real-world applications of anti-symmetric matrices?

Yes, anti-symmetric matrices have many real-world applications. They are used in mechanics to describe the motion of rigid bodies, in physics to represent electromagnetic fields, and in control theory to model systems with oscillatory behavior.

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