Understanding Eigenvalues and Eigenvectors
The words eigenvalue and eigenvector come from the German word eigen, meaning 'own' or 'characteristic'. In linear algebra, when a matrix A multiplies a vector v, it typically rotates and scales the vector. However, for certain special vectors, the multiplication only scales the vector, leaving its direction unchanged (or completely reversed). These special vectors are eigenvectors, and the scaling factors are the corresponding eigenvalues.
Mathematically, we write this relation as:
A × v = λ × v
where A is a square matrix, v is the non-zero eigenvector, and λ (lambda) is the eigenvalue.
How to Calculate Eigenvalues and Eigenvectors Step-by-Step
To find the eigenvalues and eigenvectors of a matrix A, we use the following algebraic steps:
1. Find the Eigenvalues
Rearranging the eigenvector equation gives: (A - λI)v = 0. For a non-zero vector v to exist as a solution, the matrix (A - λI) must be singular (non-invertible). This means its determinant must be zero:
det(A - λI) = 0
Solving this determinant yields a polynomial equation in λ, called the characteristic polynomial. The roots of this polynomial are the eigenvalues of the matrix.
2. Find the Eigenvectors
For each eigenvalue λ calculated in the first step, we substitute it back into the equation (A - λI)x = 0 and solve for the null space of the resulting matrix. The basis vectors of this null space represent the eigenvectors corresponding to that eigenvalue.
For example, if A is a 2x2 matrix, the characteristic equation is a quadratic polynomial: λ² - trace(A)λ + det(A) = 0. Solving this quadratic equation gives the two eigenvalues.