newgrad said:Homework Statement
I am working on a problem where I made a matrix representation of a linear transformation and I am asked what is the eigenspace for a particular eigenvalue.
Homework Equations
The Attempt at a Solution
The problem for me is, I came out with a diagonal matrix. what is the null space of a diagonal matrix? Is it just the 0 vector? If so, can I have an eigenspace?
newgrad said:thanks. I think I am seeing the light. So when I subtract off the eigenvalue times I, then i get a 0 in that spot, so the eigenvector can have a 1, or anything, in that corresponding spot. so the eigenspace would be all the vectors with the basis of all O's and a 1 in a column, and the one is where that particular eigenvalue was. did that make sense?
The null space of a diagonal matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the diagonal matrix.
The null space of a diagonal matrix is closely related to its eigenvalues. The eigenvalues of a diagonal matrix are the entries along the main diagonal, and any vector in the null space of the matrix must contain a 0 in the corresponding position to each eigenvalue. This means that the eigenvalues of a diagonal matrix are also the roots of the characteristic polynomial of the matrix.
Yes, a diagonal matrix can have a non-zero null space. This occurs when the matrix contains at least one 0 along the main diagonal, which means that there is at least one eigenvalue of 0. This results in a null space that contains all vectors with a 0 in the corresponding position.
An eigenspace of a diagonal matrix is a set of all vectors that, when multiplied by the matrix, result in a scalar multiple of the original vector. In other words, it is the set of all solutions to the equation Ax = λx, where A is the diagonal matrix and λ is an eigenvalue of the matrix.
To find the null space of a diagonal matrix, simply set up and solve the equation Ax = 0, where A is the matrix. This will result in a set of vectors that make up the null space. To find the eigenspace, set up and solve the equation Ax = λx, where A is the matrix and λ is an eigenvalue. This will result in a set of vectors that make up the eigenspace.