Page 55 - Matrices theory and applications

2. Square Matrices
38
(d) Generalize this result to M n(k): for every N 1 ,... ,N n+1 ∈
M n (k), one has

(π)T π (N 1 ,... ,N n+1 )= 0.
π∈S n+1
Note: Polynomial identities satisfied by every n×n matrix have
been studied for decades. See [15] for a thorough account. One
should at least mention the theorem of Amitsur and Levitzki:
Theorem 2.9.1 Consider the free algebra ZZ[x 1 ,... ,x r ] (where
x 1 ,... ,x r are noncommuting indeterminates) define the stan-
dard polynomial S r by

S r (x 1 ,... ,x r )= (π)x π(1) ··· x π(r) .
π∈S r
Then, given a commutative ring A, one has the polynomial
identity
S 2n (Q 1 ,... ,Q 2n )= 0 n , ∀Q 1 ,... ,Q 2n ∈ M n (A).
26. Let k be a field and let A ∈ M n (k) be given. For every set J ⊂
{1,... ,n},denote by A J the matrix extracted from A by keeping
only the indices i, j ∈ J. Hence, A J ∈ M p (k)for p =cardJ.Let
λ ∈ k.
(a) Assume that for every J whose cardinality is greater than or
equal to n − p, λ is an eigenvalue of A J . Show that λ is an
eigenvalue of A, of algebraic multiplicity greater than or equal
to p+1 (express the derivatives of the characteristic polynomial).
(b) Conversely, let q be the geometric multiplicity of λ as an eigen-
value of A. Show that if cardJ> n − q,then λ is an eigenvalue
of A J .

27. Let A ∈ M n (k)and l ∈ IN be given. Show that there exists a poly-
l
nomial q l ∈ k[X], of degree at most n − 1, such that A = q l (A). If
A is invertible, show that there exists r l ∈ k[X], of degree at most
n − 1, such that A −l = r l (A).

28. Let k be a field and A, B ∈ M n (k). Assume that λ = µ for every
λ ∈ Sp A, µ ∈ Sp B. Show, using the Cayley–Hamilton theorem,
that the linear map M → AM − MB is an automorphism of M n (k).
29. Let k be a field and (M jk ) 1≤j,k≤n a set of matrices of M n(k), at
k
least one of which is nonzero, such that M ij M kl = δ M il for all
j
1 ≤ i, j, k, l ≤ n.
(a) Show that none of the matrices M jk vanishes.
(b) Verify that each M ii is a projector. Denote its range by E i .