Applied linear operators and spectral methods/Lecture 3

Review

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In the last lecture we talked about norms in inner product spaces. The induced norm was defined as

 

We also talked about orthonomal bases and biorthonormal bases. The biorthonormal bases may be thought of as dual bases in the sense that covariant and contravariant vector bases are dual.

The last thing we talked about was the idea of a linear operator. Recall that

 

where the summation is on the first index.

In this lecture we will learn about adjoint operators, Jacobi tridiagonalization, and a bit about the spectral theory of matrices.

Adjoint operator

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Assume that we have a vector space with an orthonormal basis. Then

 

One specific matrix connected with   is the Hermitian conjugate matrix. This matrix is defined as

 

The linear operator   connected with the Hermitian matrix is called the adjoint operator and is defined as

 

Therefore,

 

and

 

More generally, if

 

then

 

Since the above relation does not involve the basis we see that the adjoint operator is also basis independent.

Self-adjoint/Hermitian matrices

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If   we say that   is self-adjoint, i.e.,   in any orthonomal basis, and the matrix   is said to be Hermitian.

Anti-Hermitian matrices

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A matrix   is anti-Hermitian if

 

There is a close connection between Hermitian and anti-Hermitian matrices. Consider a matrix  . Then

 

Jacobi Tridiagonalization

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Let   be self-adjoint and suppose that we want to solve

 

where   is constant. We expect that

 

If   is "sufficiently" small, then

 

This suggest that the solution should be in the subspace spanned by  .

Let us apply the Gram-Schmidt orthogonalization procedure where

 

Then we have

 

This is clearly a linear combination of  . Therefore,   is a linear combination of  . This is the same as saying that   is a linear combination of  .

Therefore,

 

Now,

 

But the self-adjointeness of   implies that

 

So   is   or  . This is equivalent to expressing the operator   as a tridiagonal matrix   which has the form

 

In general, the matrix can be represented in block tridiagonal form.

Another consequence of the Gram-Schmidt orthogonalization is that

Lemma:

Every finite dimensional inner-product space has an orthonormal basis.

Proof:

The proof is trivial. Just use Gram-Schmidt on any basis for that space and normalize.  

A corollary of this is the following theorem.

Theorem:

Every finite dimensional inner product space is complete.

Recall that a space is complete is the limit of any Cauchy sequence from a subspace of that space must lie within that subspace.

Proof:

Let   be a Cauchy sequence of elements in the subspace   with  . Also let   be an orthonormal basis for the subspace  .

Then

 

where

 

By the Schwarz inequality

 

Therefore,

 

But the  ~s are just numbers. So, for fixed  ,   is a Cauchy sequence in   (or  ) and so converges to a number   as  , i.e.,

 

which is is the subspace  .  

Spectral theory for matrices

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Suppose   is expressed in coordinates relative to some basis  , i.e.,

 

Then

 

So   implies that

 

Now let us try to see the effect of a change to basis to a new basis   with

 

For the new basis to be linearly independent,   should be invertible so that

 

Now,

 

Hence

 

Similarly,

 

Therefore

 

So we have

 

In matrix form,

 

where the objects here are not operators or vectors but rather the matrices and vectors representing them. They are therefore basis dependent.

In other words, the matrix equation  


Similarity transformation

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The transformation

 

is called a similarity transformation. Two matrices are equivalent or similar is there is a similarity transformation between them.

Diagonalizing a matrix

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Suppose we want to find a similarity transformation which makes   diagonal, i.e.,

 

Then,

 

Let us write   (which is a   matrix) in terms of its columns

 

Then,

 

i.e.,

 

The pair   is said to be an eigenvalue pair if   where   is an eigenvector and   is an eigenvalue.

Since   this means that   is an eigenvalue if and only if

 

The quantity on the left hand side is called the characteristic polynomial and has   roots (counting multiplicities).

In   there is always one root. For that root   is singular, i.e., there always exists at least one eigenvector.

We will delve a bit more into the spectral theory of matrices in the next lecture.

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