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Isaac.Wang88
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We know that all eigenvalues of a Hermitian matrix are real. How to explain this from the physics point of view?
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.
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.I am not familiar with any SI units using complex numbers, are you?
A Hermitian matrix is a square matrix that is equal to its own conjugate transpose.
Eigenvalues are a set of numbers associated with a matrix that represent the scaling factor for certain eigenvectors of the matrix.
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.
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.
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.