Characteristic equation formula for a nxn matrix ?

sid9221

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

[tex] t^2 - traceA + |A| [/tex]

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

Office_Shredder

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 [tex](x-\lambda_1)\times....\times (x-\lambda_n) = a_n x^n + a_{n-1}x^{n-1}.... + a_1x + a_0 [/tex]
then a_n-1 is the sum of the negatives of the eigenvalues, so a_n-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
[tex] a_{n-2} = \sum_{i<j} (-\lambda_i) (-\lambda_j) [/tex]
a_n-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 x^n-2 power you have to use two of the eigenvalues and the rest x's