Vectors and eigenvalues

Eigenvectors are vectors multiplied by an eigenvalue in the linear transformations of a matrix. The eigenvalues ​​are constants that multiply the eigenvectors in the linear transformations of a matrix.

In other words, eigenvectors translate the information from the original matrix into the multiplication of values ​​and a constant. The eigenvalues ​​are this constant that multiplies the eigenvectors and participates in the linear transformation of the original matrix.

Despite the fact that its name in Spanish is very descriptive, in English, the eigenvectors are called  eigenvectors and the eigenvalues ,  eigenvalues .

Recommended articles: matrix typologies, inverse matrix, determinant of a matrix.

Eigenvectors

Eigenvectors are sets of elements that, by multiplying any constant, are equivalent to multiplying the original matrix and the sets of elements.

Mathematically, an eigenvector  V = (v ₁ , …, v _n ) of a square matrix  Q is any vector  V that satisfies the following expression for any constant  h :

QV = hV

Own values

The constant h is the eigenvalue belonging to the eigenvector  V .

The eigenvalues ​​are the real roots (roots that have real numbers as a solution) that we find using the characteristic equation.

Characteristics of own values

• Each eigenvalue has infinite eigenvectors since there are infinite real numbers that can be part of each eigenvector.
• They are scalars, can be complex (non-real) numbers, and can be identical (more than one eigenvalue equal).
• There are as many eigenvalues ​​as the number of rows ( m) or columns ( n ) the original matrix has.

Vectors and eigenvalues

Between vectors and eigenvalues ​​there is a linear dependency relationship since the eigenvalues ​​multiply the eigenvectors.

Mathematically

If V is an eigenvector of matrix  Z and h is the eigenvalue of matrix Z , then  hV is a linear combination of vectors and eigenvalues.

Characteristic function

The characteristic function is used to find the eigenvalues ​​of a square Z matrix  .

Mathematically

(Z – hl) · V = 0

Where Z and  h are defined above and  I is the identity matrix.

Terms

To find vectors and eigenvalues ​​of a matrix, the following must be true:

• Square Zmatrix : the number of rows ( m ) is the same as the number of columns ( n ).
• Real Zmatrix . Most matrices used in finance have real roots. What advantage is there in using real roots? Well, the eigenvalues ​​of the matrix are never going to be complex numbers, and that friends, life solves us a lot.
• Non-invertible matrix ( Z–  hI ): determinant = 0. This condition helps us always find nonzero eigenvectors. If we found eigenvectors equal to 0, then the multiplication between values ​​and eigenvectors would be null.

Practical example

We assume that we want to find the vectors and eigenvalues ​​of a  Z matrix of dimension 2 × 2:

1. We substitute the matrix Zand  Iin the characteristic equation:
2. We fix the factors:
3. We multiply the elements as if we were looking for the determinant of the matrix.
4. The solution of this quadratic equation is h = 2 and h = 5. Two eigenvalues ​​because the number of rows or columns of the Zmatrix is 2. So, we have found the eigenvalues ​​of the Zmatrix that at the same time make the determinant 0.
5. To find the eigenvectors we will have to solve:
6. For example, (v ₁, v ₂) = (1,1) for h = 2 and (v ₁ , v ₂ ) = (- 1,2) for h = 5: