Eigenvalues & eigenvectors of N x N matrix?

In summary: Or are you looking for a more general solution?If you are looking for a more general solution, then you might be interested in the Singular Value Decomposition.
  • #1
sapling_pk
3
0
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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  • #2
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?
 
  • #3
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?
 
  • #4
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.
 
  • #5
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?
 
  • #6
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
 
  • #7
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.
 
  • #9
Have you thought about using a standard eigensolver package, like LAPACK?
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to analyze the behavior of a linear transformation, represented by a square matrix. Eigenvalues are the scalar values that represent how the linear transformation affects the corresponding eigenvectors, which are the non-zero vectors that remain in the same direction after the transformation.

2. How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors can be calculated by finding the roots of the characteristic equation of a matrix. This equation is obtained by subtracting the identity matrix multiplied by a scalar value from the original matrix, and then solving for the determinant of the resulting matrix.

3. Why are eigenvalues and eigenvectors important?

Eigenvalues and eigenvectors are important because they provide insights into the behavior of a linear transformation. They can be used to determine the stability of a system, to identify the principal components of a dataset, and to simplify complex calculations by reducing the dimensionality of a matrix.

4. Can a matrix have multiple eigenvalues and eigenvectors?

Yes, a matrix can have multiple eigenvalues and corresponding eigenvectors. However, the number of eigenvalues and eigenvectors is limited by the size of the matrix. For an N x N matrix, there can be a maximum of N eigenvalues and N corresponding eigenvectors.

5. How are eigenvalues and eigenvectors used in real-world applications?

Eigenvalues and eigenvectors have various applications in fields such as physics, engineering, and data analysis. They are used in quantum mechanics to describe the energy levels of a system, in image processing to compress and classify images, and in machine learning to reduce data dimensionality and improve the efficiency of algorithms.

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