10.3
Week 12
Ali Mousavidehshikh
Department of Mathematics University of Toronto
Ali Mousavidehshikh
Week 12
Outline
1 10.3
10.3
Week 12
Ali Mousavidehshikh
10.3
Theorem: Let T : V → V be a linear operator on a finite dimensional vector space V . Then the following are equivalent.
Week 12
Ali Mousavidehshikh
10.3
Theorem: Let T : V → V be a linear operator on a finite dimensional vector space V . Then the following are equivalent.
(1) V has a basis consisting of eigenvectors of T.
Week 12
Ali Mousavidehshikh
10.3
Theorem: Let T : V → V be a linear operator on a finite dimensional vector space V . Then the following are equivalent.
(1) V has a basis consisting of eigenvectors of T.
(2) There exists a basis B of V such that MB (T ) is diagonal.
Week 12
Ali Mousavidehshikh
10.3
Theorem: Let T : V → V be a linear operator on a finite dimensional vector space V . Then the following are equivalent.
(1) V has a basis consisting of eigenvectors of T.
(2) There exists a basis B of V such that MB (T ) is diagonal. Proof. WehaveMB(T)=[CB(T(b1) ··· CB(T(bn))]where B = {b1,…,bn} is any basis of V. By comparing columns: MB(T) = diag(λ1,…,λn) if and only if T(bi) = λibi for each i.
Week 12
Ali Mousavidehshikh
10.3
Theorem: Let T : V → V be a linear operator on a finite dimensional vector space V . Then the following are equivalent.
(1) V has a basis consisting of eigenvectors of T.
(2) There exists a basis B of V such that MB (T ) is diagonal. Proof. WehaveMB(T)=[CB(T(b1) ··· CB(T(bn))]where B = {b1,…,bn} is any basis of V. By comparing columns: MB(T) = diag(λ1,…,λn) if and only if T(bi) = λibi for each i.
Definition: A linear operator T on a finite dimensional vector space V is called diagonalizable if and only if V has a basis consisting of eigenvectors of T.
Week 12
Ali Mousavidehshikh
10.3
Theorem: Let T : V → V be a linear operator on a finite dimensional vector space V . Then the following are equivalent.
(1) V has a basis consisting of eigenvectors of T.
(2) There exists a basis B of V such that MB (T ) is diagonal. Proof. WehaveMB(T)=[CB(T(b1) ··· CB(T(bn))]where B = {b1,…,bn} is any basis of V. By comparing columns: MB(T) = diag(λ1,…,λn) if and only if T(bi) = λibi for each i.
Definition: A linear operator T on a finite dimensional vector space V is called diagonalizable if and only if V has a basis consisting of eigenvectors of T.
Example: Let T : P2 → P2 be given by T(a+bx+cx2)=(a+4c)−2bx+(3a+2c)x2. Findthe eigenspaces of T and hence a basis of eigenvectors.
Ali Mousavidehshikh
Week 12
10.3
1 0 4 Solution. If B0 = {1,x,x2}, then MB0(T) = 0 −2 0.
302 So CT (x) = (x + 2)2(x − 5), and the eigenvalues of T are
λ = −2,5.
Week 12
Ali Mousavidehshikh
10.3
1 0 4 Solution. If B0 = {1,x,x2}, then MB0(T) = 0 −2 0.
302
So CT (x) = (x + 2)2(x − 5), and the eigenvalues of T are
λ = −2, 5. Finding the eigenvectors corresponding to these
041
eigenvalues gives 1 , 0 , 0 is a basis of
0 −3 1 eigenvectors of MB0 (T ),
Week 12
Ali Mousavidehshikh
10.3
1 0 4 Solution. If B0 = {1,x,x2}, then MB0(T) = 0 −2 0.
302
So CT (x) = (x + 2)2(x − 5), and the eigenvalues of T are
λ = −2, 5. Finding the eigenvectors corresponding to these
041
eigenvalues gives 1 , 0 , 0 is a basis of
0 −3 1
eigenvectors of MB0(T), so B = {x,4−3×2,1+x2} is a basis
of P2 consisting of eigenvectors of T.
Week 12
Ali Mousavidehshikh
10.3
1 0 4 Solution. If B0 = {1,x,x2}, then MB0(T) = 0 −2 0.
302
So CT (x) = (x + 2)2(x − 5), and the eigenvalues of T are
λ = −2, 5. Finding the eigenvectors corresponding to these
041
eigenvalues gives 1 , 0 , 0 is a basis of
0 −3 1
eigenvectors of MB0(T), so B = {x,4−3×2,1+x2} is a basis
of P2 consisting of eigenvectors of T.
Let T : V → V be a linear operator on an inner product space V. If B = {b1,…,bn} is an orthogonal basis of V,
⟨bi,T(bj)⟩ then(MB(T))ij = ∥bi ∥2 .
Ali Mousavidehshikh
Week 12
10.3
Example. Let T : R3 → R3 be given by T(a,b,c)=(a+2b−c,2a+3c,−a+3b+2c). Ifthedot product in R3 is used as our inner product, find the matrix of T with respect to the standard basis.
Week 12
Ali Mousavidehshikh
10.3
Example. Let T : R3 → R3 be given by T(a,b,c)=(a+2b−c,2a+3c,−a+3b+2c). Ifthedot product in R3 is used as our inner product, find the matrix of T with respect to the standard basis.
1 2 −1 Solution. MB(T) = 2 0 3 .
−1 3 2
Week 12
Ali Mousavidehshikh
10.3
Example. Let T : R3 → R3 be given by T(a,b,c)=(a+2b−c,2a+3c,−a+3b+2c). Ifthedot product in R3 is used as our inner product, find the matrix of T with respect to the standard basis.
1 2 −1 Solution. MB(T) = 2 0 3 .
−1 3 2
Theorem: Let V be a finite dimensional inner product vector space. The following are equivalent for a linear operator T:V→V.
Ali Mousavidehshikh
Week 12
10.3
Example. Let T : R3 → R3 be given by T(a,b,c)=(a+2b−c,2a+3c,−a+3b+2c). Ifthedot product in R3 is used as our inner product, find the matrix of T with respect to the standard basis.
1 2 −1 Solution. MB(T) = 2 0 3 .
−1 3 2
Theorem: Let V be a finite dimensional inner product vector space. The following are equivalent for a linear operator T:V→V.
(1) ⟨v,T(w)⟩ = ⟨T(v),w⟩ for all v,w ∈ V.
Ali Mousavidehshikh
Week 12
10.3
Example. Let T : R3 → R3 be given by T(a,b,c)=(a+2b−c,2a+3c,−a+3b+2c). Ifthedot product in R3 is used as our inner product, find the matrix of T with respect to the standard basis.
1 2 −1 Solution. MB(T) = 2 0 3 .
−1 3 2
Theorem: Let V be a finite dimensional inner product vector space. The following are equivalent for a linear operator T:V→V.
(1) ⟨v,T(w)⟩ = ⟨T(v),w⟩ for all v,w ∈ V.
(2) The matrix of T is symmetric with respect to every orthonormal basis of V .
Ali Mousavidehshikh
Week 12
10.3
Example. Let T : R3 → R3 be given by T(a,b,c)=(a+2b−c,2a+3c,−a+3b+2c). Ifthedot product in R3 is used as our inner product, find the matrix of T with respect to the standard basis.
1 2 −1 Solution. MB(T) = 2 0 3 .
−1 3 2
Theorem: Let V be a finite dimensional inner product vector space. The following are equivalent for a linear operator T:V→V.
(1) ⟨v,T(w)⟩ = ⟨T(v),w⟩ for all v,w ∈ V.
(2) The matrix of T is symmetric with respect to every orthonormal basis of V .
(3) The matrix of T is symmetric with respect to some orthonormal basis of V .
Ali Mousavidehshikh
Week 12
10.3
Example. Let T : R3 → R3 be given by T(a,b,c)=(a+2b−c,2a+3c,−a+3b+2c). Ifthedot product in R3 is used as our inner product, find the matrix of T with respect to the standard basis.
1 2 −1 Solution. MB(T) = 2 0 3 .
−1 3 2
Theorem: Let V be a finite dimensional inner product vector space. The following are equivalent for a linear operator T:V→V.
(1) ⟨v,T(w)⟩ = ⟨T(v),w⟩ for all v,w ∈ V.
(2) The matrix of T is symmetric with respect to every orthonormal basis of V .
(3) The matrix of T is symmetric with respect to some orthonormal basis of V .
(4) There is an orthonormal basis B = {f1,…,fn} of V such that ⟨fi,T(fj)⟩ = ⟨T(fi),fj⟩ for all i,j.
Ali Mousavidehshikh
Week 12
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Week 12
Ali Mousavidehshikh
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Theorem: A symmetric linear operator on a finite dimensional inner product space has real eigenvalues. Proof is similar to the one for symmetric matrices.
Week 12
Ali Mousavidehshikh
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Theorem: A symmetric linear operator on a finite dimensional inner product space has real eigenvalues. Proof is similar to the one for symmetric matrices.
Theorem: Let T : V → V be a symmetric linear operator on an inner product vector space V, and let U be a T-invariant subspace of V . Then:
Week 12
Ali Mousavidehshikh
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Theorem: A symmetric linear operator on a finite dimensional inner product space has real eigenvalues. Proof is similar to the one for symmetric matrices.
Theorem: Let T : V → V be a symmetric linear operator on an inner product vector space V, and let U be a T-invariant subspace of V . Then:
(1) The restriction of T to U: T : U → U is a symmetric operator on U.
Week 12
Ali Mousavidehshikh
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Theorem: A symmetric linear operator on a finite dimensional inner product space has real eigenvalues. Proof is similar to the one for symmetric matrices.
Theorem: Let T : V → V be a symmetric linear operator on an inner product vector space V, and let U be a T-invariant subspace of V . Then:
(1) The restriction of T to U: T : U → U is a symmetric operator on U.
(2) U⊥ is also T-invariant.
Week 12
Ali Mousavidehshikh
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Theorem: A symmetric linear operator on a finite dimensional inner product space has real eigenvalues. Proof is similar to the one for symmetric matrices.
Theorem: Let T : V → V be a symmetric linear operator on an inner product vector space V, and let U be a T-invariant subspace of V . Then:
(1) The restriction of T to U: T : U → U is a symmetric operator on U.
(2) U⊥ is also T-invariant.
Theorem: The following are equivalent for a linear operator T on a finite dimensional inner product vector space V .
Ali Mousavidehshikh
Week 12
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Theorem: A symmetric linear operator on a finite dimensional inner product space has real eigenvalues. Proof is similar to the one for symmetric matrices.
Theorem: Let T : V → V be a symmetric linear operator on an inner product vector space V, and let U be a T-invariant subspace of V . Then:
(1) The restriction of T to U: T : U → U is a symmetric operator on U.
(2) U⊥ is also T-invariant.
Theorem: The following are equivalent for a linear operator T on a finite dimensional inner product vector space V .
(1) T is symmetric.
Ali Mousavidehshikh
Week 12
10.3
A linear operator satisfying any (and hence all) of the conditions of the previous theorem is called a symmetric linear operator.
Theorem: A symmetric linear operator on a finite dimensional inner product space has real eigenvalues. Proof is similar to the one for symmetric matrices.
Theorem: Let T : V → V be a symmetric linear operator on an inner product vector space V, and let U be a T-invariant subspace of V . Then:
(1) The restriction of T to U: T : U → U is a symmetric operator on U.
(2) U⊥ is also T-invariant.
Theorem: The following are equivalent for a linear operator T on a finite dimensional inner product vector space V .
(1) T is symmetric.
(2) V has an orthogonal basis consisting of eigenvectors of T.
Ali Mousavidehshikh
Week 12
10.3
Example: LetT :P2 →P2 begivenbyT(a+bx+cx2)= (8a−2b+2c)+(−2a+5b+4c)x +(2a+4b+5c)x2. Using the inner product
⟨a+bx +cx2,a′ +b′x +c′x2⟩ = aa′ +bb′ +cc′, show that T is symmetric and find an orthonormal basis of P2 consisting of eigenvectors.
Ali Mousavidehshikh
Week 12
10.3
Example: LetT :P2 →P2 begivenbyT(a+bx+cx2)= (8a−2b+2c)+(−2a+5b+4c)x +(2a+4b+5c)x2. Using the inner product
⟨a+bx +cx2,a′ +b′x +c′x2⟩ = aa′ +bb′ +cc′, show that T is symmetric and find an orthonormal basis of P2 consisting of eigenvectors.
Solution. If B0 is the standard basis for P2, then 8 −2 2
MB0 (T ) = −2 5 4 is symmetric, so T is symmetric 245
(notice that B0 is an orthonormal basis of P2 with respect to this inner product).
Ali Mousavidehshikh
Week 12
10.3
Example: LetT :P2 →P2 begivenbyT(a+bx+cx2)= (8a−2b+2c)+(−2a+5b+4c)x +(2a+4b+5c)x2. Using the inner product
⟨a+bx +cx2,a′ +b′x +c′x2⟩ = aa′ +bb′ +cc′, show that T is symmetric and find an orthonormal basis of P2 consisting of eigenvectors.
Solution. If B0 is the standard basis for P2, then 8 −2 2
MB0 (T ) = −2 5 4 is symmetric, so T is symmetric 245
(notice that B0 is an orthonormal basis of P2 with respect to this inner product). We can find the following eigenvectors
1 1 1 2 1 −2
2, 1, 2 . 3 −2 3 2 3 1
Ali Mousavidehshikh
Week 12
10.3
Example: LetT :P2 →P2 begivenbyT(a+bx+cx2)= (8a−2b+2c)+(−2a+5b+4c)x +(2a+4b+5c)x2. Using the inner product
⟨a+bx +cx2,a′ +b′x +c′x2⟩ = aa′ +bb′ +cc′, show that T is symmetric and find an orthonormal basis of P2 consisting of eigenvectors.
Solution. If B0 is the standard basis for P2, then 8 −2 2
MB0 (T ) = −2 5 4 is symmetric, so T is symmetric 245
(notice that B0 is an orthonormal basis of P2 with respect to this inner product). We can find the following eigenvectors
1 1 1 2 1 −2
2 , 1, 2 . Because B0 is orthonormal, 3 −2 3 2 3 1
the corresponding orthonormal basis of P2 is
B = {1(1 + 2x − 2×2), 1(2 + x + 2×2), 1(−2 + 2x + x2)}. 333
Ali Mousavidehshikh
Week 12