Real Vector Spaces and the Real Spectral Theorem

In summary, the Real Spectral Theorem states that a real-inner product space V has an orthonormal basis consisting of eigenvectors of T if and only if T is self-adjoint. To prove this, one must come up with an inner product that transforms the original basis into an orthonormal one. This can be achieved by defining a new scalar product using the original scalar product and the eigenvectors of T.
  • #1
jesusfreak324
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0

Homework Statement


Proof:Suppose that V is a real inner product space and T[tex]\in \wp[/tex](V). If (v1... vn) is a basis for V consisting of eigenvectors for T, then there exists an inner product for V such that T is self-adjoint.


Homework Equations


The Real Spectral Theorem: Suppose that V is a real-inner product space and T [tex]\in \wp[/tex](V). Then V has an orthonormal basis consisting of eigenvectors of T if.f. T is self-adjoint.

Eigenvalue / Eigenvector Problem: T(v) = [tex]\lambda[/tex]v


The Attempt at a Solution


Ummm... I have spent days on trying to figure this out, and the only advice my professor gave to me was to use the real spectral theorem. But the only way to do this is by, as my professor goes on, to come up with an inner product that "turns" the basis list into an orthonormal one. But this seems like a complete contradiction... idk ~~~~
 
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  • #2
I guess your n is the same as the dimension of V? Suppose it is so. Let [itex](v,w)[/tex] be the original scalar product. Then you define new scalar product of two vectors v,w by

[tex]\langle v,w\rangle=\sum_{ij}\delta_{ij} (v,v_i)(w,v_j)[/tex]

Your basis is now orthonormal - check it. Then play with it.
 

1. What is a real vector space?

A real vector space is a collection of elements called vectors, which can be added together and multiplied by real numbers. This space follows specific properties such as closure under addition and scalar multiplication, and it must contain a zero vector and have an inverse for each vector. Examples of real vector spaces include the set of all n-dimensional real vectors and the set of all polynomials with real coefficients.

2. What is the Real Spectral Theorem?

The Real Spectral Theorem, also known as the Spectral Decomposition Theorem, states that any symmetric matrix can be diagonalized by an orthogonal matrix. This means that the matrix can be expressed as a product of its eigenvalues and eigenvectors. This theorem is useful in various areas of mathematics and physics, including linear algebra and quantum mechanics.

3. What is the difference between a real vector space and a complex vector space?

The main difference between a real vector space and a complex vector space is the type of numbers used. Real vector spaces use real numbers, while complex vector spaces use complex numbers. Both types of vector spaces follow similar properties, but complex vector spaces allow for more operations, such as multiplication by complex numbers and the existence of complex eigenvalues and eigenvectors.

4. How is the Real Spectral Theorem used in applications?

The Real Spectral Theorem has various applications in mathematics and physics. In linear algebra, it is used to diagonalize symmetric matrices, making it easier to solve systems of linear equations and perform other operations. In quantum mechanics, it is used to describe the state of a quantum system and analyze its behavior. This theorem also has applications in signal processing, control theory, and statistics.

5. Are there any extensions or generalizations of the Real Spectral Theorem?

Yes, there are extensions and generalizations of the Real Spectral Theorem. One example is the Complex Spectral Theorem, which states that any Hermitian matrix can be diagonalized by a unitary matrix. There are also generalizations to infinite-dimensional vector spaces, such as the Spectral Theorem for compact self-adjoint operators. These extensions and generalizations allow the theorem to be applied to a wider range of mathematical and physical problems.

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