Linear algebra+ linear operators

In summary: This is the norm that is induced by the definition on the vector space of the linear operators on that vector space. In summary, the coeffisients of the matrix/operator A don't seem to matter for the norm of A.
  • #1

Homework Statement



In [tex]R^{3}[/tex] ||x||= [tex]a_{1}[/tex]*|[tex]x_{1}[/tex]|+ [tex]a_{2}[/tex]*|[tex]x_{2}[/tex]|+ [tex]a_{3}[/tex]*|[tex]x_{3}[/tex]|. where [tex]a_{i}[/tex]>0

What is ||A||(indused norm = sup||Ax|| as ||x||=1). (Suppose we know the coeffisients of the matrix/operator A)??


Homework Equations





The Attempt at a Solution


 
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  • #2
If A is given by a matrix, it's the square root of the largest eigenvalue of AA* (*=hermitian conjugate or transpose in the real case).
 
  • #3
Thanks, but do you know how to proove it?
 
  • #4
By the way, do you want to say that it does nod depend on a_i??
 
  • #5
(i just need the real case)
 
  • #6
Mechmathian said:
Thanks, but do you know how to proove it?

|Av|^2=v^(T)*A^T*A*v. A^T*A is a symmetric matrix, so it has a complete orthogonal set of eigenvectors. Can you show the eigenvalues are nonnegative? Now expand a general v in terms of those eigenvectors and go from there...
 
  • #7
Mechmathian said:

Homework Statement



In [tex]R^{3}[/tex] ||x||= [tex]a_{1}[/tex]*|[tex]x_{1}[/tex]|+ [tex]a_{2}[/tex]*|[tex]x_{2}[/tex]|+ [tex]a_{3}[/tex]*|[tex]x_{3}[/tex]|. where [tex]a_{i}[/tex]>0

What is ||A||(indused norm = sup||Ax|| as ||x||=1). (Suppose we know the coeffisients of the matrix/operator A)??


Homework Equations





The Attempt at a Solution


Hey, that's not a matrix operator at all, is it? You just want to maximize the inner product of (a1,a2,a3) and x. That's much easier. Think Cauchy-Schwarz.
 
  • #8
Unfortunately I do not understand why

|Av|^2=v^(T)*A^T*A*v is true.. Why does it not depend on a_i??
2. I don't think I know how to proove that the eigenvalues are nonnegative or where to go from there..
 
  • #9
I think that it is a matrix operator..
 
  • #10
Mechmathian said:
I think that it is a matrix operator..

What you wrote is the dot product of (a1,a2,a3) with (x1,x2,x3). That's a matrix operator only in the sense it's a 1x3 matrix. You want to maximize it. Will you look up the Cauchy-Schwarz inequality. Please?
 
  • #11
The thing is that the dot product does not have the modules..
Even if I do maximize it.. I do not see where to go from there..
 
  • #12
Cauchy Shwartz: (x,y)<=||x||*||y||
 
  • #13
Mechmathian said:
Cauchy Shwartz: (x,y)<=||x||*||y||

Yes, and add "with equality holding only when y is a multiple of x". That's the part you want.
 
  • #14
Dick said:
Hey, that's not a matrix operator at all, is it? You just want to maximize the inner product of (a1,a2,a3) and x. That's much easier. Think Cauchy-Schwarz.

Dick said:
What you wrote is the dot product of (a1,a2,a3) with (x1,x2,x3). That's a matrix operator only in the sense it's a 1x3 matrix. You want to maximize it. Will you look up the Cauchy-Schwarz inequality. Please?

Dick, I think you have misinterpreted the question. What he wrote was the definition of norm (not an inner product between (a1, a2, a3) and (x1, x2, x3)- a1, a2, and a3 are fixed for all x) THEN he asked
"What is ||A||(indused norm = sup||Ax|| as ||x||=1). (Suppose we know the coeffisients of the matrix/operator A)??"

In other words, what norm does that defined norm on the vector space induce on the linear operators.

The definition of "induced norm" of A is ||A||= lub ||Ax|| for all x with norm 1.
 

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other algebraic structures to solve problems related to systems of linear equations.

2. What are linear operators?

Linear operators are functions that map a vector space to itself in a linear manner. They are represented by matrices and can be used to perform operations such as scaling, rotation, and reflection on vectors.

3. What is the significance of linear algebra in science?

Linear algebra has numerous applications in science, including physics, engineering, computer science, and economics. It is used to model and solve complex systems, analyze data, and make predictions or optimizations.

4. How are linear algebra and linear operators used in machine learning?

In machine learning, linear algebra and linear operators are used to represent and manipulate data, perform dimensionality reduction, and build predictive models. They are also essential in algorithms such as principal component analysis and linear regression.

5. What are some real-life examples of linear algebra and linear operators?

Some real-life examples of linear algebra and linear operators include image processing in digital cameras, image compression algorithms, and navigation systems in airplanes. They are also used in financial models to analyze stock prices and make investment decisions.

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