Relationship between the eigenvalues of a matrix acting on different spaces.

In summary, when treating an nxn matrix as a linear operator over ℝn, it will have n distinct eigenvectors vi. However, when treating it as a linear operator over ℝnxn, the space of nxn matrices, the eigenvectors will have all columns vi or the zero vector for any given i. This results in n2n eigenvectors, which cannot be independent. Further clarification is needed on the action of the matrix on ℝ^n^2.
  • #1
Daron
15
0
Suppose an nxn matrix has n distinct eigenvectors vi when treated as a linear operator over ℝn. What is the relationship between these and the eigenvectors of the matrix when treated as a linear operator over ℝnxn, the space of nxn matrices?

Since a matrix L acting on one with columns a1, a2, ... an returns one with columns La1, La2, ... Lan, my initial assumption is that the eigenvectors in ℝnxn have all columns vi or the zero vector for any given i. This gives n2n eigenvectors, which can obviously not be independent.

Can anyone tell me more about this?
 
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  • #2
Not sure what you mean by he action on R^n^2

Do you mean point wise multiplication of the coordinates?
 

Related to Relationship between the eigenvalues of a matrix acting on different spaces.

What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is commonly used in mathematics and science to represent linear transformations and systems of equations.

What are eigenvalues?

Eigenvalues are a set of numbers associated with a matrix that represent the amount by which a linear transformation changes the length of a vector in a specific direction. They are important in understanding the behavior of a matrix and its corresponding transformation.

How are eigenvalues calculated?

To calculate the eigenvalues of a matrix, you need to solve the characteristic equation, which is det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. This will give you a set of values that represent the eigenvalues of the matrix.

What is the relationship between eigenvalues and eigenvectors?

Eigenvectors are the corresponding vectors to eigenvalues and represent the direction of the linear transformation. The eigenvalue determines the magnitude of the transformation in that direction. The relationship between eigenvalues and eigenvectors is vital in understanding the behavior of a matrix and its transformation.

How does a matrix act on different spaces?

A matrix can act on different spaces by transforming vectors in those spaces. Each space has its own set of eigenvectors and eigenvalues, which determine how the matrix will transform vectors in that space. The relationship between the eigenvalues of a matrix in different spaces can provide insight into the transformation behavior between those spaces.

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