Find the eigenvalues and a basis for each eigenspace of the linear operator defined by the state

Find the eigenvalues and a basis for each eigenspace of the...

Key Concepts

Eigenspaces

An eigenspace is the set of all eigenvectors corresponding to a particular eigenvalue, together with the zero vector. It forms a subspace of the vector space and illustrates the collection of directions that remain invariant under the transformation, aside from scaling.

Eigenvectors

Eigenvectors are nonzero vectors that only change by a scalar factor when a linear operator is applied, meaning they point in the same or exactly opposite direction after transformation. They reveal crucial insights into the invariant directions of the matrix or operator.

Characteristic Polynomial

The characteristic polynomial is formed by computing the determinant of the matrix of a linear operator minus ? times the identity matrix. The roots of this polynomial are the eigenvalues of the operator, and its factors can indicate the algebraic multiplicity and potential degeneracy of these eigenvalues.

Standard Matrix Representation

Any linear operator can be described by a matrix once a basis is fixed. The standard matrix provides a concrete way to perform calculations such as finding eigenvalues and eigenvectors, and it encapsulates the effect of the operator on the entire vector space.

Eigenvalues

Eigenvalues are scalars associated with a linear operator that indicate how much the corresponding eigenvectors are stretched or compressed during the transformation. They are found by solving the characteristic polynomial, which derives from setting the determinant of the operator’s matrix minus a scalar multiple of the identity to zero.

Linear Operator

A linear operator is a mapping between vector spaces (often from a vector space to itself) that preserves the operations of vector addition and scalar multiplication. This concept is central in linear algebra as it allows the study of invariant properties and the behavior of vectors under transformation.

Transcript

00:02 Hello there.

00:04 Okay, so for this exercise we have the following linear transformation and we need to find the eigen space and the eigen values for this linear transformation.

00:15 So it might seem to be complicated, but the best way to do this is to consider the matrix representation of this linear transformation.

00:25 So the matrix representation can be easily obtained by considering the coefficients of each variable at each position of the vector that we obtain here.

00:40 So this matrix is going to be 2, minus 1, minus 1, 1, 0, minus 1, and minus 1, 1, 2.

00:53 So now that we have a more familiar setting, we can work using this matrix to find the eigen values and the eigen space.

01:01 Case.

01:04 Okay, so how to, so the first thing we need to obtain is the eigenvalues and we know that for that we need to consider the determinant of lambda, in this case the identity 3 by 3, minus here the matrix that we're interested to obtain the eigen values and we need to equate this to 0.

01:30 This will return us the characteristic equation and and from that, so basically here we have these determinant of lambda minus 2, 1 minus 1 lambda, 1 and minus 1 and lambda minus 2.

01:55 So this determinant gives us the characteristic equation that in this case is equals to lambda so it is cube minus 4 lambda square plus 5 lambda and minus 2.

02:13 This should be equal to 0.

02:16 Ok, so we should have 3 roots for this 3 degree polynomial.

02:23 And happen that we can decompose this polynomial as lambda minus 2 times lambda minus 1 square.

02:33 Basically, what we have here is a double root and the root lambda equals to 2.

02:40 So we have the first eigenvalue that is 2 and we have our second eigenvalue 1, but in this case this has multiplicity equals to 2.

02:59 Great.

02:59 So we have the eigenvalues and now we need to obtain the eigenvectors.

03:05 So the second thing that we need to do is to find the eigen vectors...