A strange definition for Hermitian operator

In summary, a Hermitian operator is defined as having real eigenvalues and being diagonalizable by a unitary transformation. However, this definition is not entirely accurate as there can be non-Hermitian operators with real eigenvalues, as shown by the counterexample provided. Therefore, the statement given in the lecture notes is incorrect and should be revised.
  • #1
struggling_student
9
1
In lecture notes at a university (I'd rather not say which university) the following definition for Hermitian is given:

An operator is Hermitian if and only if it has real eigenvalues.


I find it questionable because I thought that non-Hermitian operators can sometimes have real eigenvalues. We can correctly say that Hermitian operators can only have real eigenvalues but that does not define the operator, right? Is it some kind of convention or is it just plain wrong? Alas the physicists often don't understand the difference between an implication and equivalence.
 
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  • #2
The statement which was give to you is wrong. One can find a non-hermitean matrix with real eigenvalues.
 
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  • #3
Counterexample: $$
\begin{pmatrix}
1 & 1 \\
0 & 1
\end{pmatrix} $$
has eigenvalue 1 with multiplicty 2. It's not Hermitian.
 
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  • #4
A matrix is hermitian if it has real eigenvalues and you can diagonalize it with a unitary transformation. This means that if and only if matrix ##A## is hermitian, there exists a matrix ##U## such that ##U^\dagger U = UU^\dagger = 1## and ##U^\dagger A U## is a diagonal matrix with real numbers on the diagonal.
 
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Related to A strange definition for Hermitian operator

1. What is a Hermitian operator?

A Hermitian operator is a type of linear operator in mathematics that is associated with a complex inner product space. It is defined as an operator that is equal to its own conjugate transpose, meaning that its matrix representation is equal to the conjugate transpose of its own matrix.

2. How is a Hermitian operator different from a normal operator?

A Hermitian operator is different from a normal operator in that it satisfies the additional condition of being self-adjoint, meaning that it is equal to its own adjoint. This implies that its eigenvalues are all real numbers, whereas a normal operator's eigenvalues can be complex.

3. What are some examples of Hermitian operators?

Some common examples of Hermitian operators include the position and momentum operators in quantum mechanics, the Laplace operator in differential equations, and the covariance matrix in statistics.

4. What is the significance of Hermitian operators in physics?

Hermitian operators play a crucial role in quantum mechanics, where they represent physical observables such as position, momentum, and energy. They also have important applications in other areas of physics, such as statistical mechanics and signal processing.

5. How are Hermitian operators related to the concept of symmetry?

In physics, Hermitian operators are closely related to the concept of symmetry. This is because a Hermitian operator represents a physical observable that remains unchanged under certain transformations, such as rotations and reflections. This symmetry property is essential in understanding the behavior of physical systems.

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