Maths - Matrix algebra - Matrix algebra - Eigenvectors and Eigenvalues

Maths - Eigenvectors and Eigenvalues of 2×2 Matrix

The method for symbolic computation of eigenvectors and eigenvalues involves first finding the eigenvalues from the characteristic polynomial:

det(M - λ I) = 0

where, in this case for two dimensional matrix M =

m00	m01
m10	m11

which gives the characteristic equation:

(m00 - λ)(m11 - λ) - m01 m10 = 0

expanding out gives:

λ² - (m00 + m11)λ + (m00 m11 - m01 m10) = 0

where:

(m00 + m11) = trace
(m00 m11 - m01 m10) = determinant

solving the quadratic equation gives:

λ = (m00 + m11)/2 ± (√(4*m01*m10 + (m00 - m11)²))/2

We can then substitute these into

m00 - λ	m01
m10	m11 - λ

which is equivalent to solving 2 simultaneous equations.

Special Cases

If the matrix is symmetrical around the leading diagonal:

m01=m10

then:

λ = (m00 + m11)/2 ± (√(2*m01)² + (m00 - m11)²))/2

Code

Here is a Java function to return the eigenvalues:

/**
code to return eigenvalues of a 2x2 matrix
result we be returned in resultReal[0],resultImaginary[0],resultReal[1] and resultImaginary[1]
expects arrays to be setup before being called, like this:
double[] resultReal= new double[2];
double[] resultImaginary = new double[2];
*/
public void eigenValue(double[][] m, double[] resultReal , double[] resultImaginary) {
  quadratic(1.0,m[0][0] + m[1][1],m[0][0]*m[1][1] - m[0][1]*m[1][0],resultReal,resultImaginary);
    // the quadratic function is defined on this page:
    // http://www.euclideanspace.com/maths/algebra/equations/polynomial/quadratic/
	return;
}

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