How are the eigenvalues of A and B related If A and B are similar?

How are the eigenvalues of A and B related If A and B are similar?

Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is non-zero since P is invertible) is an eigenvector for B with eigenvalue λ.

Can similar matrices have different eigenvalues?

We see right away that if two matrices have different eigenvalues then they are not similar. Also, if two matrices have the same distinct eigen values then they are similar. Suppose A and B have the same distinct eigenvalues.

Can a matrix have different eigenvectors?

Matrices can have more than one eigenvector sharing the same eigenvalue. The converse statement, that an eigenvector can have more than one eigenvalue, is not true, which you can see from the definition of an eigenvector.

What does it mean when two matrices have the same eigenvalues?

If the two matrices have the same eigenvalues with the same multiplicity they have the same characteristic polynomial. If the multiplicity of these eigenvalues equal the dimension of the eigenspaces (vector sub-spaces) then the two matrices are similar to a diagonal matrix up to a change of base.

Can a matrix have multiple eigenvalues?

It may very well happen that a matrix has some “repeated” eigenvalues. That is, the characteristic equation det(A−λI)=0 may have repeated roots.

What is meant by saying two matrices A and B are similar matrices?

Similar Matrices The notion of matrices being “similar” is a lot like saying two matrices are row-equivalent. Definition (Similar Matrices) Suppose A and B are two square matrices of size n . Then A and B are similar if there exists a nonsingular matrix of size n , S , such that A=S−1BS A = S − 1 B S .

Can two different matrices have the same eigenvectors and eigenvalues?

Yes. Since there are n distinct eigenvalues the corresponding eigenvectors form a basis for Rn.

Are similar matrices both invertible?

Suppose that A and B are similar, i.e. that B = P–1AP for some matrix P. so the matrices have the same determinant, and one is invertible if the other is. so the matrices have the same characteristic polynomial and hence the same eigenvalues.

Is a similar to B?

Thus, “A is similar to B” is an equivalence relation. If A is similar to B, then A and B have the same eigenvalues. Proof Since A is similar to B, there exists an invertible matrix P so that .

Can We express linear transformations as matrices as well?

We will now see that we can express linear transformations as matrices as well. Hence, one can simply focus on studying linear transformations of the form \\(T(x) = Ax\\) where \\(A\\) is a matrix.

What is the difference between tuple and matrix representations of linear transformations?

As for tuple representations of vectors, matrix representations of a linear transformation will depend on the choice of the ordered basis for the domain and that for the codomain. Matrix representations of linear transformations Let \\(V\\) and \\(W\\) be vector spaces over some field \\(\\mathbb{F}\\).

How do you determine the value of a linear transformation?

In other words, a linear transformation is determined by specifying its values on a basis. Our first theorem formalizes this fundamental observation. Theorem 5.1Let U and V be finite-dimensional vector spaces over F, and let {eè, . . . , eñ} be a basis for U.

Is [Ta]Emen=a] a linear transformation?

TA⁢(v):=A⁢v. It is easy to see that TAis indeed a linear transformation. Furthermore, [TA]=[TA]EmEn=A, since the representationof vectors as n-tuples of elements in kis the same as expressing each vector under the standard basis(ordered) in the vector space kn.