Linear Transformation and Diagonalization Problem

In summary, linear transformations are mathematical functions that preserve the linear structure of a vector space. Diagonalization is a technique used in linear algebra to simplify and better understand these transformations by finding a new basis with a diagonal matrix representation. It is closely related to eigenvalues and eigenvectors, and a transformation can be either diagonalizable or non-diagonalizable. Diagonalization has various real-world applications, including in fields such as physics, engineering, and economics, where it is used to simplify complex systems and solve optimization problems.
  • #1
Stapler2000
2
0
Nevermind -- Polygons and Polywags.
 
Last edited:
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  • #2
matrix in standard basis looks good, but i would check your characteristic polynomial
 
  • #3
i got x^2 - x + 6 = 0
 
  • #4
however as you mention it is correct that in the case T has complex eigenvalues it is not diagonalisable over the reals
 
  • #5
Agreed -- it is x^2-x+6.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps a vector space to itself, preserving its linear structure. In simpler terms, it is a transformation that does not change the shape of an object, only its size and orientation.

2. What is the purpose of diagonalization in linear algebra?

Diagonalization is a technique used in linear algebra to simplify and better understand a linear transformation. It involves finding a new basis for the vector space in which the transformation has a diagonal matrix representation, making it easier to perform calculations and analyze the properties of the transformation.

3. How is diagonalization related to eigenvalues and eigenvectors?

Diagonalization is closely related to eigenvalues and eigenvectors, as the process of diagonalization involves finding a basis of eigenvectors for the given linear transformation. The eigenvalues are then used to construct the diagonal matrix representation of the transformation.

4. What is the difference between a diagonalizable and non-diagonalizable linear transformation?

A linear transformation is diagonalizable if it can be represented by a diagonal matrix, while a non-diagonalizable transformation cannot be represented in this form. In other words, a diagonalizable transformation has a full set of linearly independent eigenvectors, while a non-diagonalizable transformation may have repeated eigenvalues or a lack of eigenvectors.

5. How is diagonalization used in real-world applications?

Diagonalization has many applications in various fields such as physics, engineering, and economics. It is used to simplify and better understand complex systems, such as circuits, differential equations, and financial models. It also plays a crucial role in solving optimization problems and in data analysis and machine learning techniques.

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