This is simpler than you seem to be making it out to be. If 0 is an eigenvalue, then det(A) = 0, and Ax = 0x for any eigenvector of 0.flyingpig said:Homework Statement
http://img820.imageshack.us/img820/4874/cah.th.png
Uploaded with ImageShack.us
The Attempt at a Solution
a) Did it already, 3 is the eigenvalue
b) This is just finding the nullspace and the basis of the nullspace are my eigenvectors right?
flyingpig said:c) ignore this one, we cover this next term
You can't just move a bunch of symbols around. You need to do something with your matrix A.flyingpig said:Wait for b)
Ax = λx = 0x = 0
Ax = 0 <=== not nullspace?
Are you implying that
det(Ax) = det(0I) = 0
det(Ax) = 0
How do you pull the x out?
Eigenvectors and eigenvalues are concepts in linear algebra that are used to describe the behavior of a linear transformation or a matrix. An eigenvector is a vector that, when multiplied by a matrix, results in a scalar multiple of itself. The scalar multiple is called the eigenvalue. In other words, an eigenvector is a special vector that does not change direction when multiplied by a matrix.
Eigenvectors and eigenvalues are used to understand the properties of linear transformations and matrices. They are particularly useful in solving systems of differential equations, finding equilibrium points in dynamical systems, and in the analysis of data sets in fields such as data science and machine learning.
Eigenvectors and eigenvalues are two different mathematical concepts, but they are closely related. An eigenvector is a vector, whereas an eigenvalue is a scalar. An eigenvector represents a direction, while an eigenvalue represents a magnitude. In other words, an eigenvector tells us which direction a linear transformation or a matrix stretches or shrinks, while an eigenvalue tells us by how much.
To find eigenvectors and eigenvalues, we need to solve the characteristic equation for a given matrix. The characteristic equation is obtained by setting the determinant of the matrix minus a scalar multiple of the identity matrix equal to zero. The solutions to this equation are the eigenvalues, and the corresponding eigenvectors can be found by plugging in the eigenvalues into the equation (A - cI)x = 0 and solving for x.
Yes, a matrix can have multiple eigenvectors and eigenvalues. In fact, most matrices have multiple eigenvectors and eigenvalues. The number of eigenvectors and eigenvalues a matrix has is equal to its dimension. However, not all matrices have distinct eigenvectors and eigenvalues. In some cases, there may be repeated eigenvalues or linearly dependent eigenvectors.