Eigen values and Eigenvectors for a special case of a symmetric matrix

In summary, the conversation discusses the calculation of eigenvectors and eigenvalues for a vector x and a matrix X^T*X. The solution involves finding the dot product of x and v, which results in one eigenvector with a nonzero eigenvalue. The rank of the matrix is always one.
  • #1
mihalisla
15
0
Hey guys if i have a vector x=[x1,x2, ... xn]
what are the eigenvectors and eigenvalues of X^T*X ?
I know that i get a n by n symmetric matrix with it's diagonal entries in
the form of Ʃ xii^2 for i=1,2,3,. . . ,n

Thank you in advance once again!
 
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  • #2
You can calculate them fairly directly. Let [itex] A = x^t x [/itex] Then for a vector [itex] v = (v_1,...,v_n) [/itex]
[tex] Av^t = (x^t x) v^t = x^t (x v^t) = x^t \left( x\cdot v \right) [/tex]
where [itex] x\cdot v [/itex] is the dot product of x and v. Based on this you should be able to spot that there can only be one eigenvector with a nonzero eigenvalue (and what the eigenvectors and eigenvalues are)
 
  • #3
I ll try it . thank you very much!
 
  • #4
I tried it but i get a nx1 in size matrix. Aren't eigenvalues and eigenvectors for nxn matrices . . . ?
How can I get the values ?
Thank you .
 
  • #5
Yes, Office Shredder said that would give the eigenvectors, not a matrix.
 
  • #6
Sorry but i still don't get it. What is the eigenvector in the second part of the equation? Could you provide the solution beyond that first step! Thank you !
 
  • #8
Solved

Solved ! lamda=x*x' and the corresponding eigenvector is x
 

FAQ: Eigen values and Eigenvectors for a special case of a symmetric matrix

What are eigenvalues and eigenvectors for a symmetric matrix?

Eigenvalues and eigenvectors are special values and corresponding vectors that can be found for a given matrix. In a symmetric matrix, the eigenvectors are perpendicular to each other and the eigenvalues are real numbers.

What is the significance of finding eigenvalues and eigenvectors for a symmetric matrix?

Finding eigenvalues and eigenvectors for a symmetric matrix can help in understanding the behavior of the matrix and its corresponding linear transformation. It can also aid in solving systems of linear equations and finding important properties of the matrix, such as determinants and inverses.

How can eigenvalues and eigenvectors be calculated for a symmetric matrix?

Eigenvalues and eigenvectors for a symmetric matrix can be calculated using various methods such as the characteristic polynomial, power iteration method, and Jacobi method. These methods involve finding the roots of the characteristic polynomial or iteratively solving for the eigenvectors using a starting vector.

Can a symmetric matrix have complex eigenvalues and eigenvectors?

No, a symmetric matrix can only have real eigenvalues and eigenvectors. This is because the eigenvectors are perpendicular to each other and the eigenvalues are real numbers.

What are some real-world applications of eigenvalues and eigenvectors for a symmetric matrix?

Eigenvalues and eigenvectors for a symmetric matrix have various applications in fields such as physics, engineering, and economics. They are used in analyzing vibrations and oscillations, predicting stock market trends, and understanding the stability of structures and systems.

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