Chewybakas said:Homework Statement
Find the eigenvalues and eigenspace of the given vector.
Homework Equations
Matrix = (3,0)
(8,-1)
The Attempt at a Solution
I have determined the eigenvalues to be -1 and 3, but when I try compute the eigenspace when lambda = -1 I constantly get confused and end up with the space equal to span(t[0,0]) tεℝ. Can anyone help or confirm that answer!
Chewybakas said:I reduced the original matrix to reduced row echelon form which gave me the identity 2x2 matrix, which when finding two values v1,v2 where when multiplied by the identity matrix gives zero, I get the answer stated above but when i tried a different way I get the eigenspace 2v1 and 9v1 which again confuses me.
A matrix eigenspace is the set of all eigenvectors corresponding to a particular eigenvalue of a matrix. In other words, it is the subspace of the original vector space that is transformed only by a scalar factor when multiplied by the matrix.
To find the eigenspace of a matrix, you first need to find the eigenvalues of the matrix. Then, for each eigenvalue, you can find the corresponding eigenvectors by solving the equation (A - λI)x = 0, where A is the original matrix, λ is the eigenvalue, and x is the eigenvector. The set of all these eigenvectors will make up the eigenspace.
The eigenspace is important because it provides a way to break down a matrix into simpler components. This can be useful in solving systems of linear equations, diagonalizing matrices, and understanding the behavior of linear transformations.
Yes, a matrix can have multiple eigenspaces. Each eigenvalue will have its own corresponding eigenspace, and it is possible for multiple eigenvalues to have the same eigenspace.
The dimension of the eigenspace is equal to the multiplicity of the eigenvalue. In other words, the number of linearly independent eigenvectors corresponding to an eigenvalue is equal to its multiplicity.