- #1

boo

- 26

- 8

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

In summary, the complex spectral theorem for operators on vector space V states that T is normal if and only if V has an orthonormal basis consisting of eigenvectors of T and the matrix representation of T is diagonal with respect to some orthonormal basis of V. It is not possible for T to be diagonalizable with non-orthogonal eigenvectors if it is not normal, both in a complex and real inner product space. Additionally, in a complex inner product space, every normal operator is guaranteed to have a full orthonormal set of basis vectors, while in a real inner product space, every self-adjoint operator does as well.

- #1

boo

- 26

- 8

Physics news on Phys.org

- #2

mathwonk

Science Advisor

Homework Helper

- 11,774

- 2,041

- #3

boo

- 26

- 8

- #4

boo

- 26

- 8

Axler's Spectral Theorem is a theorem in linear algebra that states that any normal matrix can be diagonalized by a unitary matrix. This means that a normal matrix can be expressed as a diagonal matrix with its eigenvalues on the diagonal.

Axler's Spectral Theorem is specifically applicable to normal matrices. It states that any normal matrix can be diagonalized, which means that the matrix can be expressed in terms of its eigenvalues and eigenvectors.

Normal matrices are square matrices that satisfy the condition that the matrix multiplied by its conjugate transpose is equal to the conjugate transpose multiplied by the matrix. In simpler terms, a normal matrix is a matrix that commutes with its conjugate transpose.

Yes, Axler's Spectral Theorem is a general theorem that applies to all normal matrices. It is a fundamental result in linear algebra and has been proven to hold true for all normal matrices.

Axler's Spectral Theorem is important because it provides a way to easily diagonalize normal matrices, which simplifies calculations and makes it easier to understand the properties of these matrices. It is also a crucial result in quantum mechanics, where normal matrices are used to represent observables.

- Replies
- 2

- Views
- 1K

- Replies
- 3

- Views
- 3K

- Replies
- 3

- Views
- 2K

- Replies
- 3

- Views
- 2K

- Replies
- 4

- Views
- 4K

- Replies
- 4

- Views
- 359

- Replies
- 2

- Views
- 1K

- Replies
- 4

- Views
- 2K

- Replies
- 1

- Views
- 1K

- Replies
- 4

- Views
- 1K

Share: