# Characteristic equation formula for a nxn matrix ?

1. Apr 30, 2012

### sid9221

So I know that the characteristic equation for a 2x2 matrix can be given by

$$t^2 - traceA + |A|$$

So how would this be generalised for a 4x4 or higher matrix ?

2. Apr 30, 2012

### Office_Shredder

Staff Emeritus
You forgot some t's in your equation but it's clear what you are referring to.

The characteristic equation is a polynomial whose roots are the eigenvalues of the matrix. So if we have $$(x-\lambda_1)\times....\times (x-\lambda_n) = a_n x^n + a_{n-1}x^{n-1}.... + a_1x + a_0$$
then an-1 is the sum of the negatives of the eigenvalues, so an-1 is the negative trace. a0 is the product of the eigenvalues with minus signs, so a0 is plus or minus the determinant of the matrix (depending on whether n is even or odd). The other ones are constructed by adding up multiples of the eigenvalues, for example
$$a_{n-2} = \sum_{i<j} (-\lambda_i) (-\lambda_j)$$
an-3 requires adding every way to multiply three of the eigenvalues, etc. This just comes from expanding the multiplication on the right hand side of the original equation... to get an xn-2 power you have to use two of the eigenvalues and the rest x's