Geometric and Physical Interpretation of Diagonalization

• hliu8
In summary, simultaneous diagonalization of matrices is important in quantum mechanics because it allows one to measure observables that are independent of each other.
hliu8
I am a second year student in quantum mechanics. I heard in lecture that simultaneous diagonalization of matrices is important in quantum mechanics. I would like to know why is it significant when two matrices can be simultaneous diagonalized, and what is the geometric and physical interpretation of simultaneous diagonalization of matrices in quantum mechanics and in physics in general.

Thank you for everyone's help.

The physical interpretation of this result is that if you can simultaneously diagonalise two matrices then the corresponding operators (if the matrices in question are Hermitian) are compatiable. This means that they commute since it is possible to find a commom eigenvector basis for both of the matrices. Ultimately, this means that one can simultaneously measure both of those observables on the quantum system of interest. Thus, in this case there should be no uncertainty relation between the operators and one can be expressed as a function of the other.

Matiasss
To put it a little more physically, (hermitian) matrices represent observable physical quantities. If you have a state which, when acted upon by a matrix, returns itself multiplied by a constant, that state has a definite value of the observable associated with the matrix. A compete set of such states will diagonalize the matrix.

If a common set of states can diagonalize two matrices simultaneously, it means that those states have definite values of both observables. So, the statement that there is no uncertainty relation between the observables is quite correct. However, it is not the case that one can be expressed as a function of the other. In general, the observables are independent (like the energy and angular momentum of the electron in a hydrogen atom).

Matiasss
I would like to know why is it significant when two matrices can be simultaneous diagonalized, and what is the geometric and physical interpretation of simultaneous diagonalization of matrices in quantum mechanics and in physics in general.
It's often useful, when studying a matrix, to look at the eigenvectors. Even better if you can choose a basis whose vectors are all eigenvectors.

So, if two matrices can be simultaneously diagonalized...

while diagonalizing an operator A with a matrix S (formed from eigenvectors of A), does S need to be unitary (i.e., whether I have to form it with the NORMALIZED eigenvectors)? Even if S is not unitary, A gets diagonalized.

1. What is diagonalization in mathematics?

Diagonalization is a mathematical process that involves transforming a matrix into a diagonal matrix through a series of operations. This process is used to simplify calculations and solve equations involving matrices.

2. What is the geometric interpretation of diagonalization?

The geometric interpretation of diagonalization is that it represents a change of basis in a vector space. This means that the original vectors are being transformed into a new set of vectors that are easier to work with and understand.

3. Can diagonalization be used to solve systems of linear equations?

Yes, diagonalization can be used to solve systems of linear equations by transforming the system into a diagonal form, which makes it easier to solve. This method is especially useful when dealing with large systems of equations.

4. What is the physical interpretation of diagonalization?

The physical interpretation of diagonalization refers to the application of diagonalization in real-world problems, such as in physics and engineering. It allows for the simplification of complex systems and helps in understanding the underlying principles and relationships between variables.

5. How does diagonalization relate to eigenvalues and eigenvectors?

Diagonalization is closely related to eigenvalues and eigenvectors, as the process involves finding these values and vectors for a given matrix. The eigenvalues and eigenvectors are used to create the diagonal matrix, which is the end result of diagonalization.

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