Eigenvalues & eigenvectors of N x N matrix?

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SUMMARY

The discussion focuses on computing eigenvalues and eigenvectors of large matrices, specifically a 12 x 70,000 matrix, which poses significant computational challenges. Key methods mentioned include using the determinant condition for eigenvalues and Gaussian elimination for finding eigenvectors. The conversation highlights the importance of linear algebra textbooks for foundational algorithms and suggests utilizing established libraries like LAPACK for efficient computation. Additionally, the Rayleigh quotient method is discussed as a means to approximate the dominant eigenvalue through iterative calculations.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly eigenvalues and eigenvectors.
  • Familiarity with Gaussian elimination techniques.
  • Knowledge of determinant calculations and polynomial roots.
  • Experience with numerical libraries, specifically LAPACK for eigensolver functions.
NEXT STEPS
  • Research the implementation of LAPACK for eigenvalue problems.
  • Study the Rayleigh quotient method for approximating dominant eigenvalues.
  • Explore algorithms for handling large matrices in numerical linear algebra.
  • Learn about diagonalization and its implications for eigenvalue computation.
USEFUL FOR

Mathematicians, data scientists, and software engineers involved in numerical analysis, particularly those working with large datasets in applications like face recognition algorithms.

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How to get eigenvalues & eigenvectors of N x N matrix?
Please can anyone help me out i have searched a lot but not able to find the solution.

Regards
 
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Your textbook should present a complete algorithm for computing them; have you looked there? If you've already looked at it, then in what way are you having trouble using it?
 
My goodness! This is one of the major problems of Linear Algebra and, indeed, of mathematics in general! Surely, as Hurkyl suggests, any textbook on Linear Algebra will devote one or more chapters to this!

This is much too general a question for a forum like this. Can you post specific problems?
 
c is an eigenvalue of A iff A-c fails to be invertible iff det(A-c) = 0. so compute det(A-c) considering c as a variable and set this polynomial equal to zero. if c is a root of it, then compute a basis for the kernel of A-c by gaussian elimination.

doing this for all roots c of det(A-c) gives a maximal independent set of eigenvectors, hence basis of them if one exists.
 
well actually i want find eigenvalues of huge matrix i.e 12 x 70000 so hope you have understood my problem.
thanks to all for replying.
Regards

HallsofIvy said:
My goodness! This is one of the major problems of Linear Algebra and, indeed, of mathematics in general! Surely, as Hurkyl suggests, any textbook on Linear Algebra will devote one or more chapters to this!

This is much too general a question for a forum like this. Can you post specific problems?
 
And not only eigenvalues but also the eigenvectors.Because i am implementing a face recognition algorithm if someone give me any idea with respect to programming that will be appreciated.Thanks
 
apparently you knlow more than i do, but here is what my old linear aklgebra book says:assuming your matrix A is diagonalizable, and the largest eigenvalue is unique and much larger than the other eigenvalues, then for any vector u which has a non zero coefficient with respect to the corresponding "largest" eigenvector, Au has a large component of that eigenvector.

then (Au.u)/(u.u) is an approximation to the dominant eigenvalue.

iterating A makes the dominance more pronounced, so (Au.u)/u.u) will hopefully converge to the dominant eigenvalue if we repeat the calculation with Au in place of u, and continue many times.

these are called rayleigh quotients.
 
Have you thought about using a standard eigensolver package, like LAPACK?
 

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