Algebraic Difference
what is difference btwn algebraic and geometric multiplicity?
Algebraic multiplicity is the exponent related to the eingenvalue, for example if you have a charateristic equation for the egenvalue p
[(p-3)^(2)](p-2)=0
you realize that you have a root p=3 with algebraic multiplicity 3 and a root p=2 with algebraic multiplicity 1.
The geometric multiplicity is a bit more difficult. Consider the matrix A and the eigenvalue p. When you try to search the corrispondent eigenvector you have to resolve the system
A*x=p*x
This is a linear system that, as you know, can have no solution or 1 solution or infinity^(n) solution. So the geometric multiplicity is the exponent ”n” or, in other words, the domension of the space of the solutions. For example consider the matrix
A=[1 0;2 1]
The charaterictic equation is (1-p)^(2)=1 so the only one eigenvalue is p=1 with algebraic multiplicity 2. If we resolve the system A*x=p*x we have
x=x
2x+y=y
So the solution is x=0 and y=k (whit k a generic real number) so we have an infinity^(1) dimension of the space of the solutions and so geometric multiplicity is 1. In other words the solution is a line. If you had have x=h, y=k the dimension is 2 or a ”plane”.
It is fundamental to know that a linear application is digonalizable if and only if the algebraic multiplicity is equal to geometric multiplicity (in my example is not true and so it is not diagonalizable)
square root of negative one
