Eigenvalues & eigenvectors of N x N matrix?

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Discussion Overview

The discussion revolves around the methods for calculating eigenvalues and eigenvectors of an N x N matrix, particularly focusing on challenges faced when dealing with large matrices. The scope includes theoretical aspects of linear algebra, practical computational methods, and programming applications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related
  • Mathematical reasoning
  • Experimental/applied

Main Points Raised

  • One participant requests help in finding eigenvalues and eigenvectors of an N x N matrix, indicating difficulty in locating a solution.
  • Another participant suggests consulting a textbook for a complete algorithm and asks for clarification on the specific troubles faced.
  • A different participant emphasizes the significance of the topic in linear algebra and requests more specific problems to address.
  • A participant describes a method involving the determinant of the matrix and Gaussian elimination to find eigenvalues and eigenvectors.
  • One participant specifies the challenge of finding eigenvalues for a large matrix (12 x 70000) and expresses gratitude for responses.
  • Another participant mentions the application of eigenvalues and eigenvectors in a face recognition algorithm and seeks programming-related advice.
  • A participant discusses an iterative method for approximating the dominant eigenvalue using Rayleigh quotients, assuming the matrix is diagonalizable.
  • One participant references a previous post that may contain useful information.
  • A suggestion is made to consider using a standard eigensolver package, such as LAPACK.

Areas of Agreement / Disagreement

Participants express varying levels of familiarity with the topic, and while some suggest textbook methods, others highlight the complexity of the problem, especially in the context of large matrices. No consensus is reached on a specific method or solution.

Contextual Notes

The discussion does not resolve the challenges associated with large matrices or the specific implementation details for programming applications. Assumptions about the matrix properties, such as diagonalizability, are mentioned but not universally accepted.

sapling_pk
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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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