Table of Contents
Why do similar matrices have different eigenvectors?
If A and B are similar matrices, then they represent the same linear transformation T, albeit written in different bases. So really the two matrices have the same eigenvectors, they just look different because you’re expressing them in terms of a different basis.
Can eigenvectors be the same?
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.
Are similar matrices the same?
Two similar matrices are not equal, but they share many important properties. 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 .
If two matrices are similar, they have the same eigenvalues and the same number of independent eigenvectors (but probably not the same eigenvectors). If two matrices have the same n distinct eigenvalues, they’ll be similar to the same diagonal matrix.
Do similar matrices have the same determinant?
Similar matrices have the same rank, the same determinant, the same characteristic polynomial, and the same eigenvalues.
Can different matrices have the same eigenvalues and eigenvectors?
Yes. Since there are n distinct eigenvalues the corresponding eigenvectors form a basis for Rn. So knowing the eigenvectors and eigenvalues completely determines what the linear transformation corresponding to the matrix is.
Can two different matrices have the same eigenvalues and eigenvectors?
Two similar matrices have the same eigenvalues, even though they will usually have different eigenvectors. Said more precisely, if B = Ai’AJ. I and x is an eigenvector of A, then M’x is an eigenvector of B = M’AM. Also, if two matrices have the same distinct eigen values then they are similar.
How do you show that matrices are similar?
Also, if two matrices have the same distinct eigen values then they are similar. Suppose A and B have the same distinct eigenvalues. Then they are both diagonalizable with the same diagonal 2 Page 3 matrix A. So, both A and B are similar to A, and therefore A is similar to B.
How many matrices can have the same eigenvector?
(This is reasonable to assume because, strictly speaking, every square matrix has an infinite number of eigenvectors, since if is an eigenvector, so is any multiple of .) Since and are diagonal matrices, they commute, and hence so do and by the last two equations.
Do similar matrices have similar inverses?
If A and B are similar, then B = P–1AP. Since all the matrices are invertible, we can take the inverse of both sides: B–1 = (P–1AP)–1 = P–1A–1(P–1)–1 = P–1A–1P, so A–1 and B–1 are similar. If A and B are similar, so are Ak and Bk for any k = 1, 2, .
Are inverse matrices similar?
Diagonal matrices will do. So, A and inverse of A are similar, so their eigenvalues are same. if one of A’s eigenvalues is n, a eigenvalues of its inverse will be 1/n.
Are eigenvectors and eigenvalues same?
Eigenvalues and eigenvectors are only for square matrices. Eigenvectors are by definition nonzero. Eigenvalues may be equal to zero. We do not consider the zero vector to be an eigenvector: since A 0 = 0 = λ 0 for every scalar λ , the associated eigenvalue would be undefined.