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Section 6.1 Definitions

Listed in alphabetical order.

Definition 6.1.1.

Given an n×n matrix A with eigenvalue λ, the algebraic multiplicity of the eigenvalue is the degree d to which the term (x−λ)d occurs in the factorization of the characteristic polynomial for A.

Example 6.1.3. Some standard bases for familiar vector spaces.

  • The standard basis for Fn is B={e1,…,en} where ei is the column vector in Fn with a 1 in the ith coordinate and zeroes in the remaining coordinates.
  • A standard basis for Mm×n(F) is
    B={Eij∣1≤i≤m,1≤j≤n},
    where Eij is the m×n matrix with a 1 in row i and column j, and zeroes in all other entries.
  • A standard basis for Pn(F) is B={1,x,x2,…,xn}, and a standard basis for P(F)=F[x] is B={1,x,x2,x3,…}.

Definition 6.1.4.

The characteristic polynomial of a square matrix A∈Mn(F) is χA=det(xIn−A). One can show that χA is a monic polynomial of degree n with coefficients in the field F.
Note that some authors define the characteristic polynomial as det(A−xIn) in which case the leading coefficient is (−1)n, but since the interest is only in the factorization of χA (in particular any roots it may have), it does not really matter which definition one uses.

Definition 6.1.5.

The column space of an m×n matrix A is the span of the columns of A. As such, it is a subspace of Fm.

Definition 6.1.6.

Given an m×n matrix A with complex entries, the conjugate transpose of A is the n×m matrix A∗ whose ij-entry is given by
(A∗)ij=Aji―=(AT)ij―.

Definition 6.1.7.

The dimension of a vector space is the cardinality (size) of any basis for the vector space.
Implicit in the definition of dimension are theorems which prove that every vector space has a basis, and that any two bases for a given vector space have the same cardinality. In other words, the dimension is a well-defined term not depending upon which basis is chosen to consider. When a vector space has a basis with a finite number of elements, it is called finite-dimensional.

Definition 6.1.8.

An elementary matrix is a matrix obtained by performing a single elementary row (or column) operation to an identity matrix.

Definition 6.1.9.

Elementary row (respectively column) operations on a matrix are one of the following:
  • Interchange two rows (resp. columns) of A.
  • Multiply a row (resp. column) of A by a nonzero scalar.
  • Replace a given row (resp. column) of A by the sum of the given row (resp. column) and a multiple of a different row (resp. column).

Definition 6.1.10.

Given an n×n matrix A with eigenvalue λ, the geometric multiplicity of the eigenvalue is the dimension of the eigenspace associated to λ.

Definition 6.1.11.

A complex matrix A is called Hermitian if A=A∗. Necessarily the matrix needs to be square.

Definition 6.1.12.

The image of a linear map T:V→W is
Im(T):={w∈W∣w=T(v) for some v∈V}.
The image of T is a subspace of W; T is surjective if and only if W=Im(T).

Definition 6.1.13.

A function f:X→Y between sets X and Y is injective if for every x,x′∈X, f(x)=f(x′) implies x=x′.

Definition 6.1.14.

Let F denote the field of real or complex numbers. For z=a+bi∈C (a,b∈R and i2=−1), we have the notion of the complex conjugate of z, denoted z―=a−bi. Note that when z∈R, that is z=a=a+0i∈C, we have z=z―. The magnitude (norm, absolutevalue) of z=a+bi is |z|=a2+b2.
Let V be a vector space over the field F. An inner product is a function:
⟨⋅,⋅⟩:V×V→F
so that for all u,v,w∈V and λ∈F:
  1. ⟨u+v,w⟩=⟨u,w⟩+⟨v,w⟩
  2. ⟨λv,w⟩=λ⟨v,w⟩
  3. ⟨v,w⟩―=⟨w,v⟩, where the bar denotes complex conjugate.
  4. ⟨v,v⟩ is a positive real number for all v≠0.

Definition 6.1.15.

An inner product space is a vector space V defined over a field F=R or C to which is associated an inner product. If F=R, V is called a real inner product space, and if F=C, then V is called a complex inner product space.

Definition 6.1.18.

A linear combination of vectors v1,…,vr∈V is any vector of the form a1v1+⋯+arvr for scalars ai∈F.

Definition 6.1.19.

Let S⊆V be a subset of vectors in a vector space V (finite or infinite). The set S is a linearly dependent subset of V if it is not linearly independent, that is there exists a finite subset {v1,…,vr}⊆S, and nonzero scalars a1,…,ar so that
a1v1+⋯+arvr=0.

Definition 6.1.20.

Let S⊆V be a subset of vectors in a vector space V (finite or infinite). The set S is a linearly independent subset of V if for every finite subset {v1,…,vr}⊆S, a linear dependence relation of the form
a1v1+⋯+arvr=0
forces all the scalars ai=0.

Definition 6.1.21.

Given two vector spaces V and W (defined over the same field F), a linear map (or linear transformation) from V to W is a function T:V→W which is
  • additive: T(v+v′)=T(v)+T(v′) for all v,v′∈V, and
  • preserves scalar multiplication: T(λv)=λT(v) for all vectors v∈V and scalars λ.

Definition 6.1.22.

The minimal polynomial of a square matrix A∈Mn(F) is the monic polynomial, μA, of least degree with coefficients in the field F so that μA(A)=0. The Cayley-Hamilton theorem implies that the minimal polynomial divides the characteristic polynomial.

Definition 6.1.23.

A matrix A∈Mn(C) is normal if it commutes with its conjuate transpose: AA∗=A∗A.

Definition 6.1.24.

The nullity of a linear transformation T:V→W is the dimension of ker(T), that is, the dimension of its nullspace.
If T:Fn→Fm is given by T(x)=Ax for an m×n matrix A, then the nullity of T is the dimension of the set of solutions of Ax=0.

Definition 6.1.25.

The nullspace of a linear transformation T:V→W is the kernel of T that is,
ker(T)={v∈V∣T(v)=0W}.
If T:Fn→Fm is given by T(x)=Ax for an m×n matrix A, then the nullspace of T is often called the nullspace of A, the set of solutions of Ax=0.

Definition 6.1.26.

A matrix A∈Mn(R) is an orthogonal matrix if
ATA=AAT=In.
Note that the condition ATA=In is equivalent to saying that the columns of A form an orthonormal basis for Rn, while the condition AAT makes the analogous statement about the rows of A.

Definition 6.1.27.

The pivot positions of a matrix are the positions (row,column) which correspond to a leading one in the reduced row-echelon form of the matrix. The pivots are the actual entry of the original matrix at the pivot position.
The pivot columns are the columns of the original matrix corresponding to the columns of the RREF containing a leading one.

Definition 6.1.28.

The rank of a linear transformation T:V→W is the dimension of its image, Im(T).
If T:Fn→Fm is given by T(x)=Ax for an m×n matrix A, then the rank of T is the dimension of the column space of A.
By theorem, it is also equal to the dimension of the row space which is the number of nonzero rows in the RREF form of the matrix A.

Definition 6.1.29.

The row space of an m×n matrix A is the span of the rows of A. As such, it is a subspace of Fn.

Definition 6.1.30.

Let A,B∈Mn(F). The matrix B is said to be similar (or conjugate) to A if there exists an invertible matrix P∈Mn(F) so that B=P−1AP. Note that if we put Q=P−1, then B=QAQ−1, so it does not matter which side carries the inverse. Also note that this is a symmetric relationship, so that B is similar to A if and only if A is similar to B. Indeed similarity (conjugacy) is an equivalence relation.

Definition 6.1.31.

Let S⊆V be a subset of vectors in a vector space V (finite or infinite). The span of the set S, denoted Span(S), is the set of all finite linear combinations of the elements of S. That is to say
Span(S)={a1v1+⋯+arvr∣r≥1,ai∈F,vi∈S}

Definition 6.1.32.

Let V be a vector space over a field F, and let W⊆V. W is called a subspace of V if W is itself a vector space with the operations of vector addition and scalar multiplication inherited from V.
Of course checking all the vector space axioms can be quite tedious, but as a theorem you prove much easier criteria to check. Recall that you already know that V is a vector space, so many of the axioms (associativity, distributive laws etc) are inherited from V. Indeed, you prove that to show that W is a subspace of V, it is enough to show that the additive identity of V is in W, and that W is closed under the inherited operations of vector addition and scalar multiplication, i.e, whenever w,w′∈W and λ∈F, we must have w+w′∈W, and λw∈W.

Definition 6.1.33.

A function f:X→Y between sets X and Y is surjective if for every y∈Y, there exists an x∈X such that f(x)=y.

Definition 6.1.34.

A matrix A is called symmetric if A=AT. Necessarily the matrix needs to be square.

Definition 6.1.36.

A matrix A∈Mn(C) is an unitary matrix if
A∗A=AA∗=In.
Note that the condition A∗A=In is equivalent to saying that the columns of A form an orthonormal basis for Cn, while the condition AA∗ makes the analogous statement about the rows of A.

Definition 6.1.37.

A vector space is a non-empty set V and an associated field of scalars F, having operations of vector addition, denoted +, and scalar multiplication, denoted by juxtaposition, satisfying the following properties: For all vectors u,v,w∈V, and scalars λ,μ∈F
  • closure under vector addition
  • u+v∈V
  • addition is commutative
  • u+v=v+u
  • addition is associative
  • (u+v)+w=u+(v+w)
  • additive identity
  • There is a vector 0∈V so that 0+u=u.
  • additive inverses
  • For each u∈V, there is a vector denoted −u∈V so that u+−u=0.
  • closure under scalar multiplication
  • λu∈V.
  • scalar multiplication distributes across vector addition
  • λ(u+v)=λu+λv
  • distributes over scalar addition
  • (λ+μ)v=λv+μv
  • scalar associativity
  • (λμ)v=λ(μv)
  • Vis unital
  • The field element 1∈F satisfies 1v=v.