Okay, that's [itex]\sigma[/itex]. What is M? we can't write something "as a linear combination of the eigenvectors of M without knowing what M is!JordanGo said:Homework Statement
Write the eigenvector of [itex]\sigma[/itex]x with +1 eigenvalue as a linear combination of the eigenvectors of M.
Homework Equations
[itex]\sigma[/itex]x = (0,1),(1,0) (these are the columns)
The Attempt at a Solution
... Don't know what to do. Can someone show me how to do this using arbitrary eigenvectors, say (a,b) and (c,d)?
An eigenvector is a vector that does not change its direction when multiplied by a given matrix. Instead, it only changes in length, which is represented by a scalar value known as the eigenvalue.
Making an eigenvector a linear combination of other eigenvectors can help simplify and generalize a problem. It can also help in solving systems of linear equations, diagonalizing matrices, and finding dominant eigenvalues and eigenvectors.
To make an eigenvector a linear combination of other eigenvectors, you can use a method called diagonalization. This involves finding a matrix that is composed of the eigenvectors of the original matrix, as well as a diagonal matrix with the corresponding eigenvalues. Once these matrices are found, the original matrix can be written as a linear combination of the eigenvectors and eigenvalues.
Yes, any eigenvector can be written as a linear combination of other eigenvectors. This is because the eigenvectors of a matrix form a basis for the vector space, meaning that any vector in that space can be written as a linear combination of the eigenvectors.
One limitation is that the matrix must be diagonalizable, meaning that it must have a full set of linearly independent eigenvectors. Additionally, this method may not always be practical or useful in certain situations, as it may result in a large number of calculations and a more complex solution.