Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If there are two matrixes that they have common eigenvectors for some eigenvalues that implies that those two matrixes are identical? What can we say for those two matrices?

share|improve this question
1  
Consider $\bigl(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\bigr)$ and $\bigl(\begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix}\bigr)$, all their eigenspaces are identical, but the matrices aren't identical. –  Najib Idrissi Sep 25 '12 at 9:36
    
What can we say about the geometry of those matrices if we represent them in the euclidean space? –  curious Sep 25 '12 at 9:46
    
You can't "represent" a matrix in a vector space. The fact that it is Euclidian or not does not matter... "Euclidian" is a topological property more used in metric spaces where you need to calculate a distance for example. In the case of two matrices that share the same set of eigenvectors you can think of this as the matrices "deforming" the vector space in the same way. You can see it as a combination of simultaneous dilatations in each direction defined by the eigenvectors. –  mak Nov 18 '13 at 2:31
add comment

2 Answers

up vote 3 down vote accepted

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. This similarity is a transitive property and then you can say that the two matrices you have represent the same endomorphism but in two different bases of the same vector space. The underlying endomorphism is the same but the matrices are definitely NOT identical. Remember that a matrix is a way to represent an endomorphism after one has chosen a vector base.

Edit:

Similar matrices $A$ and $B$ are such that there exists an invertible matrix $P$ such that: $A = PBP^{-1}$

share|improve this answer
    
What do you mean by similar matrices? –  curious Nov 4 '13 at 10:20
    
Edited the answer ;) –  mak Nov 18 '13 at 2:25
add comment

If two matrices have the same set of eigenvectors but different eigenvalues, then they can be simultaneously diagonalized, which means that the two matrices commute which each other, that is if the two matrices are A and B, AB = BA.

share|improve this answer
    
can you elaborate more on "simultaneously diagonalized"? –  curious Nov 4 '13 at 10:20
    
It means that they are similar to diagonal matrices written over the basis formed by the eigenvectors. –  mak Nov 18 '13 at 2:27
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.