How to Find Eigenvalues and Eigenvectors of a Matrix in R - Step by Step Examples

How to Find Eigenvalues and Eigenvectors of a Matrix in R ?

Answer

To find eigenvalues and eigenvectors of a matrix in R, you can use the eigen() function. This function calculates the eigenvalues and eigenvectors of a square matrix.

✐ Examples

1 Finding Eigenvalues and Eigenvectors of a 3x3 Matrix

In this example,

We start by creating a 3x3 matrix named mat using the matrix() function. This matrix represents the coefficients of a system of linear equations.
Next, we use the eigen() function to calculate the eigenvalues and eigenvectors of the matrix mat. We assign the result to a variable named eigen_res.
We extract the eigenvalues from eigen_res using the $values attribute and assign them to a variable named eigenvalues.
We also extract the eigenvectors from eigen_res using the $vectors attribute and assign them to a variable named eigenvectors.
We print both the eigenvalues and eigenvectors to the console to see the results. This allows us to verify the calculated eigenvalues and corresponding eigenvectors.

R Program

mat <- matrix(c(1, 2, 3, 2, 4, 5, 3, 5, 6), nrow = 3, byrow = TRUE)
eigen_res <- eigen(mat)
eigenvalues <- eigen_res$values
eigenvectors <- eigen_res$vectors
print('Eigenvalues:')
print(eigenvalues)
print('Eigenvectors:')
print(eigenvectors)

Output

[1] "Eigenvalues:"
[1] 11.3448143  0.1709152 -0.5157295
[1] "Eigenvectors:"
           [,1]       [,2]       [,3]
[1,] -0.3279853  0.5910090  0.7369762
[2,] -0.5910090 -0.7369762  0.3279853
[3,] -0.7369762  0.3279853 -0.5910090

2 Finding Eigenvalues and Eigenvectors of a Symmetric Matrix

In this example,

We start by creating a symmetric matrix named sym_mat using the matrix() function. This matrix represents a symmetric system or set of equations.
Next, we use the eigen() function to calculate the eigenvalues and eigenvectors of the symmetric matrix sym_mat. We assign the result to a variable named eigen_res_sym.
We extract the eigenvalues from eigen_res_sym using the $values attribute and assign them to a variable named eigenvalues_sym.
We also extract the eigenvectors from eigen_res_sym using the $vectors attribute and assign them to a variable named eigenvectors_sym.
We print both the eigenvalues and eigenvectors to the console to see the results. This allows us to verify the calculated eigenvalues and corresponding eigenvectors.

R Program

sym_mat <- matrix(c(6, 2, 1, 2, 3, 2, 1, 2, 6), nrow = 3, byrow = TRUE)
eigen_res_sym <- eigen(sym_mat)
eigenvalues_sym <- eigen_res_sym$values
eigenvectors_sym <- eigen_res_sym$vectors
print('Eigenvalues (Symmetric Matrix):')
print(eigenvalues_sym)
print('Eigenvectors (Symmetric Matrix):')
print(eigenvectors_sym)

Output

[1] "Eigenvalues (Symmetric Matrix):"
[1] 8.464102 5.000000 1.535898
[1] "Eigenvectors (Symmetric Matrix):"
           [,1]          [,2]       [,3]
[1,] -0.6279630  7.071068e-01  0.3250576
[2,] -0.4597008  1.221245e-15 -0.8880738
[3,] -0.6279630 -7.071068e-01  0.3250576

Summary

In this tutorial, we learned How to Find Eigenvalues and Eigenvectors of a Matrix in R language with well detailed examples.

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How to Find Eigenvalues and Eigenvectors of a Matrix in R - Step by Step Examples

How to Find Eigenvalues and Eigenvectors of a Matrix in R ?

Answer

✐ Examples

1 Finding Eigenvalues and Eigenvectors of a 3x3 Matrix

R Program

Output

2 Finding Eigenvalues and Eigenvectors of a Symmetric Matrix

R Program

Output

Summary

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