Math 550 Notes - Chapter 10 Chapter 10.pdf · Title: Math 550 Notes - Chapter 10 Author: Jesse Crawford Created Date: 12/7/2010 4:23:54 PM

Math 550 NotesChapter 10

Jesse Crawford

Department of MathematicsTarleton State University

Fall 2010

(Tarleton State University) Math 550 Chapter 10 Fall 2010 1 / 28

Notation

F denotes R or C.V is a finite-dimensional, nonzero vector space over F.I denotes all identity operators and all identity matrices.

Chapter 10 covers the trace and determinant and places a greateremphasis on matrices than the rest of the book.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 2 / 28

Outline

1 Change of Basis

2 Trace

3 Determinant of an Operator

4 Determinant of a Matrix

(Tarleton State University) Math 550 Chapter 10 Fall 2010 3 / 28

Reminder: The Matrix of a Linear Map

ReminderSuppose U is a vector space with basis u1, . . . ,up, andV is a vector space with basis v1, . . . , vn.If T ∈ L(U,V ), the matrix of T wrt. these bases is denoted

M(T , (u1, . . . ,up), (v1, . . . , vn)).

If W is a third space with basis (w1, . . . ,wm), andS ∈ L(V ,W ), then ST ∈ L(U,W ), and

M(ST , (u1, . . . ,up), (w1, . . . ,wm))

=M(S, (v1, . . . , vn), (w1, . . . ,wm))M(T , (u1, . . . ,up), (v1, . . . , vn)).

(Tarleton State University) Math 550 Chapter 10 Fall 2010 4 / 28

Example

The matrix of the identity operator on F2

wrt. the bases ((4,2), (5,3)) and ((1,0), (0,1)) is

M(I, ((4,2), (5,3)), ((1,0), (0,1))) =(

4 52 3

)Proposition (10.2)

If (u1, . . . ,un) and (v1, . . . , vn) are bases of V ,thenM(I, (u1, . . . ,un), (v1, . . . , vn)) is invertible, and

M(I, (u1, . . . ,un), (v1, . . . , vn))−1 =M(I, (v1, . . . , vn), (u1, . . . ,un)).

Example

M(I, ((1,0), (0,1)), ((4,2), (5,3))) =(

4 52 3

)−1

=

( 32 −5

2−1 2

).

(Tarleton State University) Math 550 Chapter 10 Fall 2010 5 / 28

Change of Basis

Theorem (10.3)Suppose T ∈ L(V ), andlet (u1, . . . ,un) and (v1, . . . , vn) be bases of V .Let A =M(I, (u1, . . . ,un), (v1, . . . , vn)).Then,

M(T , (u1, . . . ,un)) = A−1M(T , (v1, . . . , vn))A.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 6 / 28

Suppose the matrix of T ∈ L(F2) wrt. the basis ((1,0), (0,1)) is

M(T , ((1,0), (0,1))) =(

1 23 4

).

Then the matrix of T wrt. ((4,2), (5,3)) is

M(T , ((4,2), (5,3))) =(

4 52 3

)−1( 1 23 4

)(4 52 3

)

=

(−38 −5132 43

).

(1 23 4

)(10

)= 1

(10

)+ 3

(01

)(−38 −5132 43

)( 32−1

)= 1

( 32−1

)+ 3

(−5

22

)(Tarleton State University) Math 550 Chapter 10 Fall 2010 7 / 28

Rewriting Theorems in Terms of Matrices

Real Spectral Theorem (7.13)Suppose that V is a real inner-product space, andT ∈ L(V ).Then V has an orthonormal basis consisting of evec.’s of T iffT is self-adjoint.

Matrix VersionSuppose M is an n × n real matrix.Then there exists an n × n orthogonal matrix Rand an n × n diagonal matrix D, such that

D = RtMR,

if and only if M is symmetric.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 8 / 28

Outline

1 Change of Basis

2 Trace

3 Determinant of an Operator

4 Determinant of a Matrix

(Tarleton State University) Math 550 Chapter 10 Fall 2010 9 / 28

Revisiting the Characteristic Polynomial

Let V be a complex vector space.Recall that the characteristic polynomial of T ∈ L(V ) is

(z − λ1) · · · (z − λn),

where the λj ’s are the eval.’s of T , each repeated a number oftimes equal to its multiplicity.Expanding the characteristic polynomial yields

zn − (λ1 + · · ·+ λn)zn−1 + · · ·+ (−1)n(λ1 · · ·λn).

The negative of the coefficient of zn−1 in the characteristicpolynomial is called the trace of T , denoted trace T .trace T = λ1 + · · ·+ λn.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 10 / 28

DefinitionSuppose V is a complex vector space, andlet T ∈ L(V ).Then trace T is the sum of the eigenvalues of T includingmultiplicities.

Observation: If a basis is chosen for V such thatM(T ) isupper-triangular, then trace T is the sum of the diagonal entries ofM(T ).Conjecture: this is always true regardless of the basis for V .

DefinitionSuppose A is a square matrix.Then trace A is the sum of the diagonal entries of A.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 11 / 28

Proving the Conjecture

Proposition (10.9)If A and B are square matrices of the same size, then

trace (AB) = trace (BA).

Corollary (10.10)Suppose T ∈ L(V ).If (u1, . . . ,un) and (v1, . . . , vn) are bases of V , then

traceM(T , (u1, . . . ,un)) = traceM(T , (v1, . . . , vn)).

Theorem (10.11)If T ∈ L(V ), then trace T = traceM(T ).

(Tarleton State University) Math 550 Chapter 10 Fall 2010 12 / 28

Additional Facts About the Trace

Corollary (10.12)If S,T ∈ L(V ), then

trace (ST ) = trace (TS), and

trace (S + T ) = trace S + trace T .

Corollary (10.13)There do not exist operators S,T ∈ L(V ) such that

ST − TS = I.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 13 / 28

Outline

1 Change of Basis

2 Trace

3 Determinant of an Operator

4 Determinant of a Matrix

(Tarleton State University) Math 550 Chapter 10 Fall 2010 14 / 28

Determinant of an Operator

Let V be a complex vector space, T ∈ L(V ),and let λ1, . . . , λn be the eval.’s of T ,each repeated a number of times equal to its multiplicity.Expanding the characteristic polynomial yields

zn − (λ1 + · · ·+ λn)zn−1 + · · ·+ (−1)n(λ1 · · ·λn),

where the λj ’s are the eval.’s of T , each repeated a number oftimes equal to its multiplicity.

DefinitionIf T ∈ L(V ), the determinant of T , denoted det Tis (−1)dim V times the constant term in the characteristicpolynomial of T .If V is a complex vector space, det T = λ1 · · ·λn,the product of the eval.’s of T , including multiplicities.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 15 / 28

Example

Let T ∈ L(C3) be the operator whose matrix is 3 −1 −23 2 −31 2 0

.

The eval.’s of this operator are 1, 2 + 3i , and 2− 3i , sodet T = 1(2 + 3i)(2− 3i) = 13.

PropositionSuppose V is a complex vector space, and T ∈ L(V ).If a basis is chosen for V wrt. which T has an upper triangularmatrix,then det T is the product of the diagonal entries of this matrix.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 16 / 28

Proposition (10.14)An operator is invertible if and only if its determinant is nonzero.

PropositionIf T ∈ L(V ), and λ, z ∈ F, thenλ is an eigenvalue of T if and only ifz − λ is an eigenvalue of zI − T .The multiplicities are the same.

Theorem (10.17)Suppose T ∈ L(V ).The characteristic polynomial of T is det(zI − T ).

(Tarleton State University) Math 550 Chapter 10 Fall 2010 17 / 28

Outline

1 Change of Basis

2 Trace

3 Determinant of an Operator

4 Determinant of a Matrix

(Tarleton State University) Math 550 Chapter 10 Fall 2010 18 / 28

Goal: define determinants for matrices so thatdet T = detM(T ), regardless of the basis chosen.Would be nice if the determinant were just the product of thediagonal entries, but this is not true.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 19 / 28

Permutations

DefinitionA permutation of the list (1, . . . ,n) is a list(m1, . . . ,mn) that contains each of the numbers 1, . . . ,n exactlyonce.perm n = the set of all such permutations.

DefinitionSuppose (m1, . . . ,mn) ∈ perm n.Count the number of pairs (j , k), such that

I 1 ≤ j < k ≤ n, andI j appears after k in the list (m1, . . . ,mn).

The sign of (m1, . . . ,mn) is defined to be 1 if this number is evenand −1 if it is odd.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 20 / 28

Example(2,6,1,3,4,5,7) ∈ perm 7.The pairs (j , k), such that

I 1 ≤ j < k ≤ n, andI j appears after k in the list (m1, . . . ,mn).

are(2,1), (6,1), (6,3), (6,4), (6,5).

The sign of this permutation is (−1)5 = −1.Alternatively, we can count the number of 2-elementtranspositions that transform (1, . . . ,7) into (2,6,1,3,4,5,7).

(1,2,3,4,5,6,7) → (2,1,3,4,5,6,7) → (2,1,3,4,6,5,7)→ (2,1,3,6,4,5,7) → (2,1,6,3,4,5,7)

(2,6,1,3,4,5,7) is an odd permutation.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 21 / 28

Lemma (10.23)Interchanging two entries in a permutation multiplies the sign by −1.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 22 / 28

The Determinant of a Matrix

DefinitionSuppose A is an n × n matrix whose entries are denoted by ai,j .Then the determinant of A, denoted det A, is

det A =∑

(m1,...,mn)∈perm n

sign(m1, . . . ,mn)am1,1 · · · amn,n.

ExampleThe determinant of a 2× 2 matrix A is

a11a22 − a21a12.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 23 / 28

PropositionIf A is an upper-triangular square matrix, the determinant of A is theproduct of the diagonal entries of A.

Lemma (10.28)Suppose A is a square matrix, andB is obtained from A by interchanging two columns.Then det A = −det B.

Lemma (10.29)If A is a square matrix that has two equal columns, then det A = 0.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 24 / 28

det is an Alternating Form

Notation:If A is an n × n square matrix,A = [a1 · · · an] means that the columns of A are a1, . . . ,an.

Lemma (10.30)Suppose A = [a1 · · · an] is an n × n matrix, and(m1, . . . ,mn) ∈ perm n.Then

det[am1 · · · amn ] = (sign(m1, . . . ,mn))det A.

(Tarleton State University) Math 550 Chapter 10 Fall 2010 25 / 28

det is a Multilinear Form

PropositionThe mapping

(Fn)n → F(a1, . . . ,an) 7→ det[a1 · · · an]

is multilinear, that is, linear in each component.

ExampleNote that (

24

)= 2

(10

)+ 4

(01

).

Therefore,

det(

7 23 4

)= 2 det

(7 13 0

)+ 4 det

(7 03 1

)(Tarleton State University) Math 550 Chapter 10 Fall 2010 26 / 28

The Key Result

Theorem (10.31)If A and B are square matrices of the same size, then

det(AB) = det(BA) = (det A)(det B).

(Tarleton State University) Math 550 Chapter 10 Fall 2010 27 / 28

Smooth Sailing

Corollary (10.32)Suppose T ∈ L(V ), and(u1, . . . ,un) and (v1, . . . , vn) are bases of V .Then

detM(T , (u1, . . . ,un)) = detM(T , (v1, . . . , vn)).

Theorem (10.33)If T ∈ L(V ), then

det T = detM(T ).

Corollary (10.34)If S,T ∈ L(V ), then det(ST ) = det(TS) = (det S)(det T ).

(Tarleton State University) Math 550 Chapter 10 Fall 2010 28 / 28