Spectral theorem for self-adjoint linear transformations

In summary, the conversation discusses the existence of scalars and linearly independent vectors that satisfy certain conditions for self-adjoint linear transformations P and Q. The use of the spectral theorem is mentioned to find an orthonormal basis for P and Q separately, but the question of how to connect them arises. The hint given is to use the linear transformations Ei to project into the subspace spanned by ei, which connects P and Q together.
  • #1
rainwyz0706
36
0
Let P,Q be self-adjoint linear transformations from V to V, Q is also positive-definite. Deduce that there exist scalars λ1 , . . . , λn and linearly independent vectors e1 , . . . , en in V such that, for i, j = 1, 2, . . . , n:
(i) P ei = λi Qei ;
(ii) <P ei , ej > = δi j λi ;
(iii) <Qei , ej > = δi j .
I could use the spectral theorem to find an orthonormal basis ei for P and Q separately, but how can I connect them together? Could anyone give me some hint? Any help is greatly appreciated!
 
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  • #2
Let Ei be the linear transformations that project into the subspace spanned by ei. Then (i) is essentially saying that

[tex]P = \sum \lambda_i QE_i = Q \sum \lambda_i E_i[/tex].

I hope I'm not giving away too much here.
 

What is the spectral theorem for self-adjoint linear transformations?

The spectral theorem for self-adjoint linear transformations is a mathematical theorem that states that every self-adjoint linear transformation on a finite-dimensional complex inner product space can be diagonalized by a unitary transformation. This means that the transformation can be represented as a diagonal matrix with real eigenvalues.

What is a self-adjoint linear transformation?

A self-adjoint linear transformation is a linear transformation on a vector space that satisfies the property of being equal to its own adjoint. In other words, the transformation and its adjoint have the same matrix representation with respect to an orthonormal basis.

What is the significance of the spectral theorem for self-adjoint linear transformations?

The spectral theorem for self-adjoint linear transformations is significant because it provides a powerful tool for analyzing and solving problems in linear algebra, quantum mechanics, and other areas of mathematics. It allows us to easily find the eigenvalues and eigenvectors of a self-adjoint transformation, which can be used to solve systems of linear equations and understand the behavior of linear systems.

Can the spectral theorem be applied to infinite-dimensional spaces?

Yes, the spectral theorem for self-adjoint linear transformations can be extended to infinite-dimensional spaces. However, the theorem only holds for certain types of infinite-dimensional spaces, such as Hilbert spaces. In these cases, the diagonalization of the transformation results in a countable set of eigenvalues and eigenvectors.

Are there any limitations to the spectral theorem for self-adjoint linear transformations?

Yes, the spectral theorem has limitations. It only applies to self-adjoint linear transformations, which are a special class of linear transformations. Additionally, the theorem is specific to finite-dimensional complex inner product spaces or certain types of infinite-dimensional spaces. It cannot be applied to other types of vector spaces, such as real vector spaces or spaces with non-standard inner products.

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