Find matrix of linear transformation and show it's diagonalizable

In summary, a linear transformation is a mathematical function that preserves the structure of a vector space and can be represented by a matrix. To find the matrix of a linear transformation, the transformation is applied to a set of basis vectors and recorded in a matrix. A matrix is considered diagonalizable if it can be converted into a diagonal matrix through a similarity transformation. Showing that a matrix is diagonalizable can provide insight into the properties of a linear transformation and the process involves finding eigenvalues and eigenvectors and constructing a diagonal matrix. Techniques like the Jordan canonical form can simplify this process depending on the properties of the matrix.
  • #1
schniefen
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TL;DR Summary
Let ##V## be an ##n##-dimensional inner product space, where ##n>0##, and let ##F## be the linear transformation on ##V## defined by ##F(\textbf{u})=\langle \textbf{u},\textbf{c} \rangle \textbf{b}-\langle \textbf{b},\textbf{c} \rangle \textbf{u} ##, where ##\textbf{b},\textbf{c} \in V## and ## \langle \textbf{b},\textbf{c} \rangle \neq 0 ##. Show that ##V## has a basis consisting of eigenvectors of ##F## and find the matrix of ##F## with respect to some such basis.
The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...
 
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  • #2
I think it would be simpler to just figure out what an eigenvector of ##F(.)## looks like. Under what conditions does ##F(\mathbf u) = \lambda \mathbf u## for some ##\lambda## and ##\mathbf u##?
 
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Likes HallsofIvy and schniefen
  • #3
This is possible when the arithmetic and geometric multiplicities of each eigenvalue agree.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the structure of the vector space. In simpler terms, it is a function that takes in a vector and outputs a new vector, while maintaining the same properties such as linearity and proportionality.

2. How do you find the matrix of a linear transformation?

To find the matrix of a linear transformation, you first need to determine the basis vectors of the input and output vector spaces. Then, you apply the linear transformation to each basis vector and record the resulting vectors as columns of the matrix. The matrix will have the same number of columns as the dimension of the input vector space and the same number of rows as the dimension of the output vector space.

3. What does it mean for a matrix to be diagonalizable?

A matrix is diagonalizable if it can be expressed as a diagonal matrix, where all the entries off the main diagonal are zero. This means that the matrix can be simplified and is easier to work with, as it is essentially a collection of scalars and can be easily manipulated using basic matrix operations.

4. How do you show that a matrix is diagonalizable?

To show that a matrix is diagonalizable, you need to find a diagonal matrix D and an invertible matrix P such that A = PDP^-1, where A is the original matrix. This can be done by finding the eigenvalues and eigenvectors of the matrix, and using them to construct the diagonal matrix D and the invertible matrix P.

5. Why is it important to find the matrix of a linear transformation and show its diagonalizability?

Finding the matrix of a linear transformation allows us to represent the transformation in a more concrete and easily manipulable form. Showing its diagonalizability is important because it simplifies the matrix and makes it easier to perform calculations and solve problems involving the transformation. It also provides valuable insights into the behavior of the transformation and can help in understanding its properties and applications.

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