Page 41 - Matrices theory and applications

2. Square Matrices
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In other words,
n
det M =0 ⇐⇒ (∃X ∈ K ,X =0,MX =0).
More generally, since MX = λX (λ ∈ K) can also be written (λI n −M)X =
0, one sees that det(λI n − M) is zero if and only if there exists a nonzero
n
vector in K such that MX = λX. One then says that λ is an eigenvalue
of M in K,and that X is an eigenvector associated to λ. An eigenvector
is thus always a nonzero vector. The set of the eigenvalues of M in K is
called the spectrum of M and is denoted by Sp (M).
K
Amatrix in M n (K) may have no eigenvalues in K, as the following
example demonstrates, with K = IR:

0 1
.
−10
In order to understand in detail in the structure of a square matrix M ∈
M n (K), one is thus led to consider M as a matrix with entries in K.One
then writes Sp(M) instead of Sp (M), and one has Sp (M)= K ∩Sp(M),
K K
since the eigenvalues are characterized by det(λI n − M) = 0, and this
equality has the same meaning in K as in K when λ ∈ K.
2.5 The Characteristic Polynomial
The previous calculations show that the eigenvalues of M ∈ M n (K)are
the roots of the polynomial
P M (X):= det(XI n − M).

Let us observe in passing that if X is an indeterminate, then XI n − M ∈
M n (K(X)). Its determinant P M is thus well-defined, since K(X)is a
commutative integral domain with a unit element. One calls P M the charac-
teristic polynomial of M. Substituting 0 for X, one sees that the constant
n
term in P M is simply (−1) det M. Since the term corresponding to the
permutation σ = id in the computation of the determinant is of degree
n (it is (X − m ii )) and since the products corresponding to the other
i
permutations are of degree less than or equal to n − 2, one sees that P M is
of degree n,with
n

n n−1 n
P M (X)= X − m ii X + ··· +(−1) det M.
i=1
The coefficient
n

m ii
i=1