Finding a Matrix that is Similar to A=[1 1;0 1]


by BrainHurts
Tags: matrix, similar
BrainHurts
BrainHurts is offline
#1
Nov9-12, 06:32 PM
P: 80
Is there one?

I know A=[1 1;0 1] and A-1=[1 -1;0 1]

So I know that A and A-1 have the same eigenvalues, I know that this is not sufficient to say that A and A-1 are similar (or maybe) but the dimension of the Eigenspace with eigen value 1 is 1.

In other words the geometric multiplicity does not equal the algebraic multiplicity.

So does this ultimately mean that A and A-1 are not similar?
Phys.Org News Partner Science news on Phys.org
Better thermal-imaging lens from waste sulfur
Hackathon team's GoogolPlex gives Siri extra powers
Bright points in Sun's atmosphere mark patterns deep in its interior
DonAntonio
DonAntonio is offline
#2
Nov9-12, 07:06 PM
P: 606
Quote Quote by BrainHurts View Post
Is there one?

I know A=[1 1;0 1] and A-1=[1 -1;0 1]

So I know that A and A-1 have the same eigenvalues, I know that this is not sufficient to say that A and A-1 are similar (or maybe) but the dimension of the Eigenspace with eigen value 1 is 1.

In other words the geometric multiplicity does not equal the algebraic multiplicity.

So does this ultimately mean that A and A-1 are not similar?

First, let us see if what I understand is what you meant:

$$A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\;\;\;,\;\;\;A^{-1}=\begin{pmatrix}1&\!\!-1\\0&1\end{pmatrix}$$

When we talking of square matrices [itex]\,n\times n\;\;,\;n\leq 3\,[/itex], similarity is determined by the characteristic and the minimal polynomials. Since

$$p_A(x)=p_{A^{-1}}(x)=(x-1)^2=m_A(x)=m_{A^{-1}}(x)$$

both matrices are similar.

I'll let you to find out what is the general form of the matrix [itex]\,P\,[/itex] that fulfills

$$P^{-1}AP=A^{-1}$$

DonAntonio
BrainHurts
BrainHurts is offline
#3
Nov9-12, 07:11 PM
P: 80
thank you very much DonAntonio that was quite helpful!

BrainHurts
BrainHurts is offline
#4
Nov10-12, 12:59 AM
P: 80

Finding a Matrix that is Similar to A=[1 1;0 1]


you know what, I just read this again, sorry for jumping the gun

I'm not assuming that A and A-1 are similar, and when I took a good look at this problem again there is only 1 eigenvector for A, so there aren't 2 linearly independent eigenvectors

so far i understand that if two matrices are similar then their characteristic polynomials are the same and as a consequence their eigenvalues.

in this case specifically we can say that (x-1)2 is the minimum polynomial. Is that enough to say that the two matricies are similar?

i guess the main concern of my question comes from the eigenvectors that make up the similarity matrix. in this problem i only have 1 eigenvector, so how can I find a P s.t. P is invertible?
DonAntonio
DonAntonio is offline
#5
Nov10-12, 01:11 PM
P: 606
Quote Quote by BrainHurts View Post
you know what, I just read this again, sorry for jumping the gun

I'm not assuming that A and A-1 are similar, and when I took a good look at this problem again there is only 1 eigenvector for A, so there aren't 2 linearly independent eigenvectors

so far i understand that if two matrices are similar then their characteristic polynomials are the same and as a consequence their eigenvalues.

in this case specifically we can say that (x-1)2 is the minimum polynomial. Is that enough to say that the two matricies are similar?

i guess the main concern of my question comes from the eigenvectors that make up the similarity matrix. in this problem i only have 1 eigenvector, so how can I find a P s.t. P is invertible?

In this very particular case it is enough that both the characteristic and the minimal polynomials of two matrices

are equal, just as I mentioned in my first post, since then they both have the very same Jordan Canonical Form (in fact, the

matrix A is already in JCF) , since we're talking of square matrices of order less than 4.

If these were matrices of order 4 or more then the above would not suffice.

And it is unimportant about the eigenvalues, though if there were two difrerent eigenvalues then the matrix would

be diagonal, which in this case is impossible.

About P: you don't need P to be constructed out of eigevectors of the matrix to show some matrix is similar to another one.

DonAntonio
BrainHurts
BrainHurts is offline
#6
Nov10-12, 07:57 PM
P: 80
Quote Quote by DonAntonio View Post
If these were matrices of order 4 or more then the above would not suffice.
what would?
DonAntonio
DonAntonio is offline
#7
Nov10-12, 09:56 PM
P: 606
Quote Quote by BrainHurts View Post
what would?

Well, the ultimative test: they both must have the very same and exact JCF, which means the very same eigenvalues with the

same algebraic and geometric multiplicities each one...and THEN one must also check the corresponding Jordan blocks are a match.

DonAntonio
BrainHurts
BrainHurts is offline
#8
Nov14-12, 04:57 PM
P: 80
Hmm I just thought of a question. Suppose A is nxn with eigenvalue 1 with algebraic multiplicity n, then A-1 would have eigenvalue 1 with the same algebraic multiplicity.

Can we always say that A and A-1 are similar?

We can't make any assumptions on the minimal polynomial, but if they have the same minimal polynomial then A and A-1 would be similar to the same jordan matrix.

So where can one go from here?


Register to reply

Related Discussions
Matrix similar to its transpose Linear & Abstract Algebra 2
Triangular Similar Matrix question Linear & Abstract Algebra 3
rank of a similar matrix Linear & Abstract Algebra 1
Help! Diagonal matrix similar to upper triangular matrix? Linear & Abstract Algebra 1
Similar matrix and characteristic polynomial Linear & Abstract Algebra 7