how to get basis vector from eigenvalues

Question

Laura T 2021-12-8

0
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1605980-how-to-get-basis-vector-from-eigenvalues

回答： sai charan sampara 2024-5-9

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?

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Answer 1

sai charan sampara 2024-5-9

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1605980-how-to-get-basis-vector-from-eigenvalues#answer_1454916

在 MATLAB Online 中打开

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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<mw-icon class=""></mw-icon>
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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<mw-icon class=""></mw-icon>
basis_vector_2 = U(:, 2)
basis_vector_2 = 5x1
    0.7121
    0.2109
   -0.4820
    0.3589
   -0.2954
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