Confused with Eigenvalues and Eigenvectors and Vector transformations

Hello fellow mathematicians, I am studding " Eigenvalues and Eigenvectors " at this point and I need to understand something here:

I am actually performing automatic operations on finding them, but I don't really understand what they are and what they are used for.

Those operations really look like transformations. What is the difference of transforming a matrix and finding its Eigenvectors and Eigenvalues anyway?

Thank you

• Those descriptions are not what eigenvectors and eigenvalues are, so that might explain it. – Tobias Kildetoft Jul 16 '13 at 8:50
• But, we actually perform a transformation over the matrix and then we get a new matrix from that tranformation? – themhz Jul 16 '13 at 8:52
• No, we do not do anything to a matrix when we discuss eigenvectors and eigenvalues. We are given a specific matrix (which is the same as a linear transformation), and we look to see if there are eigenvectors for that matrix. To each of these eigenvectors we then associate an eigenvalue. – Tobias Kildetoft Jul 16 '13 at 8:53
• @themhz it doesn't matter how you get the eigenvalues and eigenvectors. What matters is what they are. They are not what you describe in your question. – Ittay Weiss Jul 16 '13 at 8:53
• The way you have edited the question has made nonsense out of much of the comments and answer. Also, you have invented the word, "confuded". – Gerry Myerson Jul 16 '13 at 9:23

There seems to be some rather deep confussion here and I'm not sure from where to start...so just let's:

(1) Eigenvectors are not "produced" by a matrix. They are vectors that fulfill a very precise relation wrt an operator/square matrix .

(2) Eigenvalues don't "scale up" transformations/matrices, whether "transformed" or not (what this means in this context). Eigenvalues are scalars that saisfy a certain polynomial equation very closely related to a trasnformation/matrix

(3) Eigenvectors are not trasnformations. Read (1) above.

Eigenvalues/eigenvectors are names proceeding from the german "eigen", meaning (its) "own" or "self", "inherent or proper", etc.

• Thank you. "hey are vectors that fulfill a very precise relation wrt an operator/square matrix ." what relation? what does wrt mean and why does my teacher give me a matrix and then tells me to find the "Eigenvalues and Eigenvectors " in such way that makes me think that I am actually producing a new matrix? I think I understand that the first vector packet as it seems to be the matrix has information of some n vectors in them. Why do I perform these operations? – themhz Jul 16 '13 at 9:01
• The relation is : $$Tv=\lambda v$$ with $\,T\,$ an operator/matrix, $\,v\neq 0\,$ an eigenvector and $\,\lambda\,$ this eigenvector's associated eigenvalue. – DonAntonio Jul 16 '13 at 9:03
• just to drive it home: What eigenvalues and eigenvectors are: certain scalars and certain vectors that are obtained from the matrix and which give a lot of information about the matrix. They are completely are self-values and self-vectors in the sense that they are derived from within the matrix and portray back on the matrix as an object. Self = eigen. – Ittay Weiss Jul 16 '13 at 9:03
• No, a matrix is not a "package of vectors"...In some circunstances and/or for some rather precise and important applications, you can think of the matrix's rows or columns as vectors in some vector space $\,\Bbb F^n\,\;,\;\;\Bbb F\;$ a field . When it is possible to form a new matrix out of the eigenvectors of another matrix (and not always is this possible!), it is for some precise and clear reason, say: to diagonalize the original matrix. I sincerely think you should read this stuff from scratch because I think you're confusing stuff big time here. – DonAntonio Jul 16 '13 at 9:24
• Yep, indeed. I will go back and read some things and use what you told me. Thank you also for telling me other uses of matrices. – themhz Jul 16 '13 at 9:31

I like to imagine a plane of arrows (vectors), which all are mulptiplied by a matrix. All the arrows will tranform and become new arrows (vectors) on the plane. Those arrows which keep their initial direction are eigenarrows and eigenvalues determine how the arrows scale (shrink or strech). A negative eigenvalue would mean the arrow changes its direction to the opposite it had. All the other arrows on the plane tranform differently: they are skewed and scaled.

• So basically the initial vectors create a new set of vectors and those are a totally new set on their own right? I mean they are not a transformation (rotation or scale) of the mother vectors. They just produce a new vector set and those eigenvalues scale those eigenvectors? – themhz Jul 16 '13 at 9:42