# All eigenvalues of a Hermitian matrix are real

• Isaac.Wang88
In summary, all eigenvalues of a Hermitian matrix are real. This is explained from the physics point of view as the math model does not match the real world.
Isaac.Wang88
We know that all eigenvalues of a Hermitian matrix are real. How to explain this from the physics point of view?

Explain that none of our instruments are capable of measuring imaginary quantities? Seriously? You imply that physics depends on the matrix, rather than the matrix is used to describe the physics. There are all sorts of mathematical objects and concepts, none of which need be "explained" from a physics point of view.
Let me know when you measure a length, area, or count with a value of a+bi where a and b are real numbers and i = √-1. I am not familiar with any SI units using complex numbers, are you?
One ought not to confuse different levels of abstraction, if one can avoid it...

abitslow said:
Let me know when you measure a length, area, or count with a value of a+bi where a and b are real numbers and i = √-1.

Hmm... we measure quantities that we represent as complex numbers all the time, including distances, velocities, and accelerations.

But then i do engineering. Maybe we have different thresholds of confusion and/or abstraction from physicists.

I am not familiar with any SI units using complex numbers, are you?
You don't need any new units. if a+bi is a length, a and b both have units of meters. Otherwise, scaling a complex length by an arbitrary complex number wouldn't make any sense.

But like you, I'm not sure exactly what the OP wants "explained" about this. If the eigenvalues of the math are always real numbers and the corresponding quantities in the physics are not, the explanation is that the math model doesn't match the real world.

FWIW there are useful engineering math models where the matrices are not Hermitian, and the eigenvalues are physically meaningful but not real.

The obvious answer that we measure only real quantities is in fact quite superficial.

Measurement happens in an separate framework from the actual physical time evolution and encoding the possible results with hermitian operators is pure convenience. We could as well use antihermitian operators and take the imaginary eigenvalues as possible measurement outcomes. This would not change any of the physics involved.

However, the much more important use of hermitian operators is as generators of unitary representations of Lie groups. Interestingly, antihermitian operators can be used too and would simplify the notation somewhat and are in fact the choice of many mathematicians. But because it is nice to be able to directly identify generators of a symmetry with possible observables, the convention is to use hermitian operators for group generation and for encoding of possible measurement outcomes with real measured quantities.

Cheers,

Jazz

## 1. What is a Hermitian matrix?

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose.

## 2. What does it mean for a matrix to have eigenvalues?

Eigenvalues are a set of numbers associated with a matrix that represent the scaling factor for certain eigenvectors of the matrix.

## 3. Why are all eigenvalues of a Hermitian matrix real?

This is because Hermitian matrices are symmetric, meaning they have the same set of eigenvalues and eigenvectors as their transpose. Since all real numbers are their own conjugate, the eigenvalues of a Hermitian matrix must also be real.

## 4. Can a Hermitian matrix have complex eigenvalues?

No, a Hermitian matrix cannot have complex eigenvalues. This is because the conjugate of a real number is itself, meaning the eigenvalues of a Hermitian matrix must be real.

## 5. How is the realness of eigenvalues of a Hermitian matrix useful?

The realness of eigenvalues allows for simplification and easier computation in various applications, such as in quantum mechanics and signal processing. It also provides important insights into the properties of the matrix.

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