Eigenvalues and eigenvectors: a full information guide [LA4]

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Edited Photo of Volodymyr Hryshchenko on Unsplash.
  • Eigenvalues and Eigenvectors
  • Diagonalization
  • Calculating eigenvalue and eigenvector in R

1. The determinant of a matrix

We use determinants a lot in Linear Algebra, especially in the calculation of eigenvalues and eigenvectors. What is a determinant then?

Properties of determinants

a) Scalar multiplication: if we multiply a column of a matrix by k the determinant is multiplied by k.

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Scalar multiplication property of determinants.
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Vector addition property of determinants.
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The determinant of an identity matrix property.
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Calculation of determinants

a) 2x2 matrices:

Consider a 2x2 matrix

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Generic 2x2 matrix.
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The determinant of a 2x2 matrix.

b) 3x3 matrices:

Consider a 3x3 matrix

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Generic 3x3 matrix.
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Image from Wikipedia.

c) n x n matrices:

We calculate the determinant of a n x n matrix A with

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Image from Wikipedia.

2. Eigenvalues and Eigenvectors

If A is a n x n matrix, then a non-null vector v in ℝ^n is called an eigenvector of A (or of the linear transformation T) if Av is a scalar multiple of v, that is: Av = λv
with some scalar λ. The scalar λ is called the eigenvalue of A (or of the linear transformation T), and we say that v is an eigenvector associated with λ.

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Matrix A expanded by a factor 3. Image from Anton & Rorren (2012).
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3. Diagonalization

In this section, we will find a basis of ℝ^n that is based on the eigenvectors of a given matrix A n x n.

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P.
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AP.
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4. Calculating eigenvalues and eigenvectors in R

Calculate eigenvalues and eigenvectors in R is such an easy thing to do. First, we have to create a square matrix. You could use a correlation/covariance matrix, for example. But we will use a matrix from a previous example instead:

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A <- matrix(c(1/2, 0, 0, -1, 2/3, 0, 5, -8, -1/4), nrow = 3
print(A)
##      [,1] [,2] [,3]
## [1,] 1/2 0 0
## [2,] -1 2/3 0
## [3,] 5 -8 -1/4
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Bachelor in Psychology from PUC-Rio. https://rafavsbastos.wixsite.com/website

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