how to get basis vector from eigenvalues

Hi Guys,

I have i have just calculated the eigenvalue:

[V, D] = eig(B)

where previously B = cov(A) gave me a 5x5matrix.

How do i get the 2 basis vectors belonging to axes with the greatest variance? And how to get their corresponding variances?

Answer 1

[charan]

Hello,

You can get the 2 basis vectors by simply sorting the eigen values in decreasing order and then usig the sorted indexes to index through the eigen vectors and get the vectors with the 2 largest eigen values as the basis vectors. The same is done in singular value decomposition done by "svd" function in MATLAB. Here is an example code:

A=randi(5,5);
B=cov(A)

B = 5x5

2.3000 1.2500 -0.1000 -0.6500 -0.1000 1.2500 1.0000 0.5000 -0.7500 0.2500 -0.1000 0.5000 1.2000 -1.2000 0.7000 -0.6500 -0.7500 -1.2000 2.2000 -0.4500 -0.1000 0.2500 0.7000 -0.4500 0.7000

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[V,D]=eig(B)

V = 5x5

0.4082 0.2587 0.0518 -0.7121 0.5066 -0.6124 -0.5296 0.3352 -0.2109 0.4333 0.6124 -0.4114 0.3025 0.4820 0.3633 0.2041 -0.2089 0.6271 -0.3589 -0.6265 -0.2041 0.6632 0.6326 0.2954 0.1764

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D = 5x5

-0.0000 0 0 0 0 0 0.1688 0 0 0 0 0 0.7128 0 0 0 0 0 2.4516 0 0 0 0 0 4.0667

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[sorted_eigenvalues, idx] = sort(diag(D), 'descend')

sorted_eigenvalues = 5x1

4.0667 2.4516 0.7128 0.1688 -0.0000

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idx = 5x1

5 4 3 2 1

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sorted_eigenvectors = V(:, idx);
basis_vector_1 = sorted_eigenvectors(:, 1)

basis_vector_1 = 5x1

0.5066 0.4333 0.3633 -0.6265 0.1764

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basis_vector_2 = sorted_eigenvectors(:, 2)

basis_vector_2 = 5x1

-0.7121 -0.2109 0.4820 -0.3589 0.2954

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[U,S,V] = svd(B)

U = 5x5

-0.5066 0.7121 -0.0518 0.2587 -0.4082 -0.4333 0.2109 -0.3352 -0.5296 0.6124 -0.3633 -0.4820 -0.3025 -0.4114 -0.6124 0.6265 0.3589 -0.6271 -0.2089 -0.2041 -0.1764 -0.2954 -0.6326 0.6632 0.2041

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S = 5x5

4.0667 0 0 0 0 0 2.4516 0 0 0 0 0 0.7128 0 0 0 0 0 0.1688 0 0 0 0 0 0.0000

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V = 5x5

-0.5066 0.7121 -0.0518 0.2587 -0.4082 -0.4333 0.2109 -0.3352 -0.5296 0.6124 -0.3633 -0.4820 -0.3025 -0.4114 -0.6124 0.6265 0.3589 -0.6271 -0.2089 -0.2041 -0.1764 -0.2954 -0.6326 0.6632 0.2041

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basis_vector_1 = U(:, 1)

basis_vector_1 = 5x1

-0.5066 -0.4333 -0.3633 0.6265 -0.1764

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basis_vector_2 = U(:, 2)

basis_vector_2 = 5x1

0.7121 0.2109 -0.4820 0.3589 -0.2954

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