Basics
Fields.Let (K,+,ã) be a field It could beIR, the field of real numbers,CC
(complex numbers), or, more rarely,QQ(rational numbers) Other choices are possible, of course The elements ofK are calledscalars.
In the context of a field \( k \), one can construct larger fields such as algebraic extensions \( k(\alpha_1, \ldots, \alpha_n) \), fields of rational fractions \( k(X_1, \ldots, X_n) \), and fields of formal power series \( k[[X_1, \ldots, X_n]] \) However, these concepts are infrequently addressed in this book, so we will not provide definitions here; readers are encouraged to refer to their preferred abstract algebra textbooks for further information.
In a field K, the digits 0 and 1 represent their standard meanings, where 0 + x equals x and 1 + x equals x We can define the subring ZZ1, which consists of all possible sums of the form ±(1 + + 1) This subring ZZ1 is isomorphic to either the integers ZZ or to a field ZZ/pZZ, where p is a prime number known as the characteristic of K If K is isomorphic to ZZ, it is classified as having characteristic 0.
Vector spaces are defined as commutative groups, denoted as (E,+), where E typically does not reside within the field K It is important to note that using the symbol "+" for the additive operations of both E and K can be considered a notational convenience.
2 1 Elementary Theory be a map such that
A vector space over a field K, often referred to as a K-vector space, is defined by the properties that for all elements a, b in K and x in the vector space E, the equations a(bx) = (ab)x and 1x = x hold true The elements within this space are known as vectors, and it is important to note that the zero vector is represented as 0x = 0, or more specifically, 0_K x = 0_E.
WhenP, Q ⊂K and F, G ⊂E, one denotes by P Q (respectively P +
A linear subspace of a vector space \(E\) is defined as a subgroup \((F,+)\) that remains stable under scalar multiplication, meaning \(KF \subset F\) This ensures that \(F\) is a \(K\)-vector space and includes the zero vector \(0_E\), confirming its nonemptiness Additionally, the intersection of any collection of linear subspaces results in another linear subspace, and the sum \(F + G\) of two linear subspaces is also a linear subspace The associative property of addition, expressed as \((F + G) + H = F + (G + H)\), allows for a clear definition of the sum of multiple linear subspaces.
F+G+H and, by induction, the sum of any finite family of subsets ofE.
When linear subspaces are combined, their sum also forms a linear subspace Let \( I \) be a set, and define \( K_I \) as the collection of maps \( a = (a_i)_{i \in I} : I \rightarrow K \), where only a finite number of \( a_i \) are nonzero This collection inherently possesses a \( K \)-vector space structure through defined addition and multiplication operations.
In a vector space \( E \), a map \( i \to f_i \) from an index set \( I \) to \( E \) allows for the formation of linear combinations represented as \( \sum_{i \in I} a_i f_i \), where the coefficients \( a_i \) are scalars and only finitely many are nonzero This results in a vector in \( E \) The family \( (f_i)_{i \in I} \) is termed free if every linear combination, except the trivial one where all coefficients are zero, yields a nonzero vector Conversely, it is considered a generating family if every vector in \( E \) can be expressed as a linear combination of its elements.
An indexed family of vectors \((a_i)_{i \in I}\) is injective (or onto) if the corresponding mapping \(i \in I \to a_i f_i\) meets these criteria Furthermore, the family \((f_i)_{i \in I}\) is considered a basis of the vector space \(E\) if it is both free and generating In this scenario, the mapping becomes bijective, establishing an isomorphism between the vector spaces.
IfG ⊂E, one often identifiesGand the associated family (g) g ∈G The set
Gof linear combinations of elements ofGis a linear subspaceE, called the linear subspacespannedbyG It is the smallest linear subspaceEcontaining
G, equal to the intersection of all linear subspaces containingG The subset
Every K-vector space has at least one basis, a fact that stems from the axiom of choice All bases of a vector space E share the same cardinality, referred to as the dimension of E, denoted as dim E This dimension serves as both an upper and lower bound for the cardinality of free and generating families, respectively.
In this book we shall only use finite-dimensional vector spaces IfF, Gare two linear subspaces ofE, the following formula holds: dimF+ dimG= dimF∩G+ dim(F+G).
IfF ∩G={0}, one writesF ⊕Ginstead ofF +G, and one says that F andGare indirect sum One has then dimF⊕G= dimF+ dimG.
Given a setI, the family (e i ) i∈I , defined by
1, j =i, is a basis ofK I , called thecanonical basis The dimension ofK I is therefore equal to the cardinality ofI.
In a vector space, every generating family contains at least one basis of
E Similarly, given a free family, it is contained in at least one basis ofE.
This is theincomplete basis theorem.
Let L be a field and K a subfield of L If F is an L-vector space, it can also be considered a K-vector space Additionally, L functions as a K-vector space, leading to the relationship dim K F = dim L F / dim K L.
The most common example (the only one that we shall consider) isK=IR,
L=CC, for which we have dim IR F= 2 dim C C F.
Conversely, if Gis anIR-vector space, one builds its complexification G C C as follows:
G C C =G×G, with the induced structure of an additive group An element (x, y) ofG C C is also denotedx+iy One defines multiplication by a complex number by
(λ=a+ib, z=x+iy)→λz:= (ax−by, ay+bx).
One verifies easily thatG C C is aCC-vector space, with dim C C G C C = dim IR G.
Furthermore,Gmay be identified with anIR-linear subspace ofG C C by x→(x,0).
In this context, we define \( G_{C} = G + iG \) More broadly, we can examine two fields \( K \) and \( L \) where \( K \subset L \), rather than just \( \mathbb{R} \) and \( \mathbb{C} \) However, the construction of \( G_{L} \) becomes more complex and requires the concept of tensor products, which will not be addressed in this book.
One says that a polynomial P ∈L[X] splitsover Lif it can be written as a product of the form a r i=1
A field L is considered algebraically closed if every nonconstant polynomial P in L[X] has a root, meaning that all polynomials in L[X] can be factored completely If a field K contains L and every polynomial P in K[X] has a root in K, then the set of roots of polynomials in K[X] forms an algebraically closed field that is the smallest such field containing K, known as the algebraic closure of K Every field K has a unique algebraic closure, denoted by K The fundamental theorem of algebra states that the real numbers IR are equivalent to the complex numbers CC For example, the algebraic closure of the rational numbers QQ consists of all algebraic complex numbers, which are the roots of polynomials P in ZZ[X].
Let K be a field, and consider a matrix M of size n × m with entries in K, where n and m are both greater than or equal to 1 This matrix can be viewed as a mapping from the set {1, , n} × {1, , m} to K, represented as an array consisting of n rows and m columns Each entry in the matrix, denoted as m_ij, corresponds to the element located at the intersection of the ith row and the jth column.
In particular circumstances (extraction of matrices or minors, for example) the rows and the columns can be numbered in a different way, using non-
1.1 Basics 5 consecutive numbers One needs only two finite sets, one for indexing the rows, the other for indexing the columns.
The set of matrices of size n×m with entries in K is denoted by
The set of n×m matrices over a field K, denoted as M n×m (K), forms an additive group where the sum of two matrices M is defined by adding their corresponding entries, represented as m_ij = m_ij + m_ij Scalar multiplication is defined for any scalar a in K, resulting in a new matrix M := aM with entries m_ij = am_ij This structure satisfies key properties, such as sa(bM) = (ab)M, a(M + M) = (aM) + (aM), and (a + b)M = (aM) + (bM), establishing M n×m (K) as a K-vector space The zero matrix is denoted by 0 or 0_nm to prevent ambiguity.
Whenm =n, one writes simply M n (K) instead of M n × n (K), and 0 n instead of 0 nn The matrices of sizesn×nare calledsquarematrices One writesI n for theidentity matrix, defined by m ij =δ i j 0, ifi=j,
The identity matrix is a specific type of permutation matrix, characterized as a square matrix with precisely one nonzero entry (a 1) in each row and column Formally, a permutation matrix M can be expressed as m_ij = δ_iσ(j), where σ represents a permutation from the symmetric group S_n.
Change of Basis
LetE be aK-vector space, in which one chooses a basisβ ={e 1 , , e n }.
Let \( P \in M_n(K) \) be an invertible matrix, defining the set \( \beta = \{ e_1, \ldots, e_n \} \) where \( e_i = \sum_{j=1}^n p_{ji} e_j \) This set serves as a basis for \( E \) The matrix \( P \) is referred to as the change-of-basis matrix from \( \beta \) to \( \beta' \) If a vector \( x \in E \) has coordinates \( (x_1, \ldots, x_n) \) in the basis \( \beta \) and \( (x_1', \ldots, x_n') \) in the basis \( \beta' \), the relationship between these coordinates is expressed by the formula \( x_j' = \sum_{i=1}^n p_{ji} x_i \).
In the context of linear maps, if \( u: E \rightarrow F \) is a linear transformation, one can analyze the matrices associated with \( u \) for various choices of bases in vector spaces \( E \) and \( F \) Let \( \beta \) and \( \beta' \) represent bases of \( E \), while \( \gamma \) and \( \gamma' \) denote bases of \( F \) The change-of-basis matrices from \( \beta \) to \( \beta' \) and from \( \gamma \) to \( \gamma' \) are denoted as \( P \) and \( Q \), respectively The matrices of the linear map \( u \) in the bases \( \beta, \gamma \) and \( \beta', \gamma' \) are represented as \( M \) and \( M' \).
M P =QM , or M =Q −1 M P, whereQ −1 denotes the inverse of Q One says thatM andM areequivalent Two equivalent matrices have same rank.
1 See Section 2.2 for the meaning of this notion.
If E =F and u∈End(E), one may compare the matrices M, M of u in two different bases β, β (here γ=β and γ =β ) The above formula becomes
One says thatM andM aresimilar, or that they areconjugate(the latter term comes from group theory) One also says thatM is the conjugate of
The equivalence and the similarity of matrices are two equivalence relations They will be studied in Chapter 6.
When working with matrices that have entries in a ring A, challenges arise primarily when A is non-commutative It is crucial to carefully select the appropriate formula for operations, particularly for addition and multiplication.
When calculating the product (M M) ik in the context of matrices as A-linear maps from A m to A n, it corresponds to the composition law For the case where m equals n, the product forms a composition law in M n (K), establishing this space as a K-algebra Consequently, it qualifies as a ring, allowing for the consideration of matrices with entries in B = M n (K) Let M ∈ M p × q (B) denote a matrix with entries M ij.
(one chooses uppercase letters in order to keep in mind that the entries are themselves matrices) One naturally identifiesM with the matrixM ∈
M pn×qn (K), whose entry of indices ((i−1)n+k,(j−1)n+l), fori≤p, j≤q, andk, l≤n, is nothing but
One verifies easily that this identification is an isomorphism between
M p×q (B) andM pn×qn (K) asK-vector spaces.
More generally, choosing decompositionsn=n 1 +ã ã ã+n r ,m=m 1 +ã ã ã+ m s with n k , m l ≥1, one may associate to every matrix M ∈ M n×m (K) an array ˜M with r rows ands columns whose element of index (k, l) is a matrix ˜M kl ∈ M n k × m l (K) Defining ν k t n.
11 Compute the number of elements in the groupGL 2(ZZ/2ZZ) Show that it is not commutative Show that it is isomorphic to the symmetric group S m , for a suitable integerm.
12 If (a 0 , , a n − 1) ∈ CC n is given, one defines the circulant matrix circ(a 0 , , a n − 1)∈ M n (CC) by circ(a 0 , , a n−1 ) :
We denote by C n the set of circulant matrices Obviously, the ma- trix circ(1,0,0, ,0) is the identity The matrix circ(0,1,0, ,0) is denoted by π.
The set \( C_n \) is demonstrated to be a subalgebra of \( M_n(\mathbb{C}) \), which is equivalent to \( \mathbb{C}[\pi] \) Consequently, it can be concluded that \( C_n \) is isomorphic to the quotient ring \( \mathbb{C}[X]/(X^n - 1) \) Additionally, for any circulant matrix \( C \), both the conjugate \( C^* \) and the polynomial \( P(C) \) are shown to be circulant If \( C \) is nonsingular, further properties can be derived from this structure.
(c) Show that the elements of C n are diagonalizable in a common eigenbasis.
(d) ReplaceCCby any fieldK IfK contains aprimitive nth rootω of unity (that is, ω n = 1, andω m = 1 implies m ∈nZZ), show that the elements ofC n are diagonalizable.
Note: A thorough presentation of circulant matrices and applications is given in Davis’s book [12].
(e) One assumes that the characteristic of K divides n Show that
C n contains matrices that are not diagonalizable.
13 Show that the Pfaffian is linear with respect to any row or column of an alternate matrix Deduce that the Pfaffian is an irreducible polynomial inZZ[x ij ].
Let \( k \) be an algebraically closed field and \( S \) a subset of \( M_n(k) \) If the only linear subspaces of \( k^n \) invariant under all elements of \( S \) are the trivial subspace \( \{0\} \) and \( k^n \) itself, then for any matrix \( A \in M_n(k) \) that commutes with every element of \( S \), there exists a scalar \( c \in k \) such that \( A = cI_n \).
15 (a) Show thatA∈ M n (K) is irreducible if and only if for every pair
(j, k) with 1≤j, k≤n, there exists a finite sequence of indices j=l 1 , , l r =ksuch thata l p ,l p+1 = 0.
(b) Show that a tridiagonal matrixA∈ M n (K), for which none of thea j,j+1’s anda j+1,j ’s vanish, is irreducible.
16 LetA∈ M n (k) (k=IR orCC) be given, with minimal polynomialq.
I x :={p∈k[X]|p(A)x= 0} is an ideal ofk[X], which is therefore principal.
(a) Show thatI x = (0) and that its monic generator, denoted by p x , divides q.
(b) One writes r j instead of p x when x = e j Show that q is the least common multiple ofr 1 , , r n
(c) Ifp∈k[X], show that the set
(the vectorsxsuch thatpdividesp x ) is open.
Let \( x \) be an element in \( k^n \) for which the polynomial \( p(x) \) has the highest degree It can be demonstrated that \( p(x) = q \) Notably, there exists an element \( x \) such that \( p(x) \) corresponds to the minimal polynomial for every field \( k \).
17 Letkbe a field and A∈ M n×m (k), B∈ M m×n (k) be given.
Show thatX m detM =X n det(X 2 I m −BA) (search for a lower triangular matrixM such thatM M is upper triangular). (b) Find an analogous relation between det(X 2 I n −AB) and detM. Deduce that X n P BA (X) =X m P AB (X).
(c) What do you deduce about the eigenvalues ofAand ofB?
18 Letkbe a field andθ:M n (k)→ka linear form satisfyingθ(AB) θ(BA) for everyA, B∈ M n (k).
(a) Show that there existsα∈ k such that for all X, Y ∈k n , one hasθ(XY T ) =α j x j y j (b) Deduce thatθ=αTr.
19 Let A n be the ring K[X 1 , , X n ] of polynomials in n variables. Consider the matrixM ∈ M n (A n ) defined by
Let us denote by ∆(X 1 , , X n ) the determinant ofM.
(a) Show that for everyi=j, the polynomialX j −X i divides ∆. (b) Deduce that
(c) Determine the value ofaby considering the monomial n j=1
(d) Redo this analysis for the matrix
20 Deduce from the previous exercise that the determinant of the
, a 1 , , a n ∈K, is zero if and only if at least two of thea j ’s coincide.
21 A matrix A ∈ M n (IR) is called a totally positive matrix when all minors
(a) Prove that the product of totally positive matrices is totally positive.
(b) Prove that a totally positive matrix admits an LU factorization (see Chapter 8), and that every “nontrivial” minor of LandU is positive Here, “nontrivial” means
L i 1 i 2 ã ã ã i p j 1 j 2 ã ã ã j p with 1≤p≤n, 1 ≤i 1 0 such that ifA∈ M n (CC) andA< , the sum of algebraic multiplicities of the eigenvalues ofM +Ain D equalsà.
Let us remark that this statement becomes false if one considers the geometric multiplicities.
The theorem regarding eigenvalues states that they are continuous functions of a matrix's entries To interpret this, we adapt the Hausdorff distance for compact sets to account for the multiplicity of eigenvalues For matrices M and N in M n (CC), we denote their eigenvalues as (λ 1 , , λ n ) and (θ 1 , , θ n ), respectively, considering their multiplicities The distance between the spectra of M and N is defined as d(SpM,SpN) := inf σ∈ S n max j |λ j −θ σ(j) |, where S n represents the group of permutations of the indices {1, , n} This formulation allows us to restate Theorem 3.1.2 in a more precise manner.
Proposition 3.1.3 If M ∈ M n (CC) and α > 0, there exists > 0 such thatN−M< impliesd(SpM,SpN)< α.
A useful consequence of Theorem 3.1.2 is the following.