If we know that a square matrix A has a particular eigenvalue l, then we know how to find the corresponding eigenvectors. But how do we determine which choices of l (if any) actually are the eigenvalues for a given square matrix A?
Notice that, for l to be an eigenvalue, there must be nonzero solutions to the homogeneous linear system (A - lIn )x = 0. But that means that the system does not have a unique solution, and since this is a system of n equations in n unknowns, that is the same as saying the matrix (A - lIn) is not invertible. But we already know that is the same as saying the determinant of (A - lIn) is zero.
Eigenvalue Theorem
Let A be an n x n matrix.
The eigenvalues of A are the
solutions l
to the equation det(A -
tIn
)= 0. This equation is
called the characteristic equation of the matrix
A. It is a
polynomial in t, called the characteristic
polynomial.
Notice that the characteristic polynomial is a polynomial in t of degree n, so it has at most n roots. Since we have been considering only real matrices and vector spaces, we will treat only the real foots of the characteristic polynomial. (If we move to the complex numbers as our scalars, then in fact the characteristic polynomial will have exactly n roots, although they may not all be distinct.)
Examples of finding characteristic polynomials, eigenvalues, and eigenspaces
Example
If A and B are n x n matrices such that
there is an invertible n x n matrix P
with B = P-1
AP, then A and B are called similar.
(We will give a geometric interpretation to similar matrices
later.)
Notice that in this case,
det(B
- tIn
)= det(P-1
AP
-
tIn
)= det(P-1
AP
- P-1
tIn
P
)
=
det(P-1
(A
- tIn
)P)=
det(P-1
)det(A
- tIn
)det(P)
= det(A -
tIn
)
by what we have previously
done. In other words,any two similar matrices have the same
characteristic polynomial.
Now, since the eigenvalues of a matrix are the (real)roots of a polynomial, we can consider their multiplicity when considered as a root of the polynomial. (Remember, a root l of a polynomial in t is said to have multiplicity k if (t - l)k is the highest power of (t - l) which is a factor of the polynomial.)
Def. The multiplicity of an eigenvalue l of a square matrix A is the multiplicity of l as a root of the characteristic polynomial.
Examples
There is a connection between the multiplicity of an eigenvalueand the size of the corresponding eigenspace.
Multiplicity
Theorem
Let l
be an eigenvalue of the n x n matrix A.
The dimension of the eigenspace of A
corresponding to l
is less than or equal to the multiplicity of
l.
Examples
In order to prove the multiplicity theorem, we consider the eigenspace of A corresponding to l. If it has dimension k, then we know that it has a basis of vectors v1 , . . ., vk . Moreover, we know by the Extension Theorem that this basi s can be extendedto a basis v1, . . . , vk, . . . , vn of Rn . Now, we have Avj = lvj for 1 < j < k. If V = [v1 . . . vk . . . vn ] is the matrix with the basis vectors as columns, then
and since V is invertible (its columns are linearly independent), V-1 AV = B is the corresponding block matrix. Now, A and B are similar, so they have the same characteristic polynomial. But it is easily seen that (t-l)k is a factor of the characteristic polynomial of B, and hence the multiplicity of the eigenvalue l is at least k.
Examples of Complex Eigenvalues
