characteristic equation

 

The characteristic equation of a linear system is an equation that is derived from the system's matrix of coefficients. It is used to find the values of the variables that satisfy the system, known as the eigenvalues.

The characteristic equation is typically written in the form |A - λI| = 0, where A is the matrix of coefficients, λ is a scalar value, and I is the identity matrix. To find the eigenvalues of the system, you solve this equation for λ.

For example, consider the following system of linear equations:

2x - y = 0 -x + 2y = 0

The matrix of coefficients is:

[2 -1] [-1 2]

The characteristic equation for this system is:

|[2 -1] - λ[1 0]| = 0 |[-1 2] [0 1]|

This simplifies to:

(2 - λ)(2 - λ) - (-1)(-1) = 0

Which gives us the characteristic equation:

λ^2 - 4λ + 3 = 0

To find the eigenvalues of the system, we solve this equation for λ:

λ = (4 ± √(4^2 - 4(3))) / 2 = (4 ± √(16 - 12)) / 2 = (4 ± √(4)) / 2 = (4 ± 2) / 2 = {2, 6}

Therefore, the eigenvalues of the system are 2 and 6