程序代写代做代考 Sections 8.3-8.4

Sections 8.3-8.4
Week 6
Ali Mousavidehshikh
Department of Mathematics University of Toronto
Ali Mousavidehshikh
Week 6

Outline
1 Sections 8.3-8.4
Sections 8.3-8.4
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Proof . Since the det A is the product of all the eigenvalues, the result follows.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Proof . Since the det A is the product of all the eigenvalues, the result follows.
Theorem: A symmetric matrix A is positive definite if and only if xT Ax > 0 for every column x ̸= 0 in Rn.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Proof . Since the det A is the product of all the eigenvalues, the result follows.
Theorem: A symmetric matrix A is positive definite if and only if xT Ax > 0 for every column x ̸= 0 in Rn.
Example: If U is any invertible n × n matrix, show that A = UT U is positive definite.
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Proof . Since the det A is the product of all the eigenvalues, the result follows.
Theorem: A symmetric matrix A is positive definite if and only if xT Ax > 0 for every column x ̸= 0 in Rn.
Example: If U is any invertible n × n matrix, show that A = UT U is positive definite.
Proof . If x ∈ Rn and x ̸= 0, then
xT Ax = xT UT Ux
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Proof . Since the det A is the product of all the eigenvalues, the result follows.
Theorem: A symmetric matrix A is positive definite if and only if xT Ax > 0 for every column x ̸= 0 in Rn.
Example: If U is any invertible n × n matrix, show that A = UT U is positive definite.
Proof . If x ∈ Rn and x ̸= 0, then
xTAx = xTUTUx = (Ux)T(Ux)
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Proof . Since the det A is the product of all the eigenvalues, the result follows.
Theorem: A symmetric matrix A is positive definite if and only if xT Ax > 0 for every column x ̸= 0 in Rn.
Example: If U is any invertible n × n matrix, show that A = UT U is positive definite.
Proof . If x ∈ Rn and x ̸= 0, then
xTAx = xTUTUx = (Ux)T(Ux) =∥ Ux ∥2
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Definition: A square is called positive definite if it is symmetric and all its eigenvalues are positive.
Theorem: If A is positive definite, then it is invertible and detA > 0.
Proof . Since the det A is the product of all the eigenvalues, the result follows.
Theorem: A symmetric matrix A is positive definite if and only if xT Ax > 0 for every column x ̸= 0 in Rn.
Example: If U is any invertible n × n matrix, show that A = UT U is positive definite.
Proof . If x ∈ Rn and x ̸= 0, then
xTAx = xTUTUx = (Ux)T(Ux) =∥ Ux ∥2 > 0, because x ̸= 0 and U is invertible.
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Definition: Let A be an m × n matrix with independent columns. A QR-factorization of A expresses it as A = QR, where Q is m × n with orthonormal columns and R is an invertible upper triangular matrix (so R is n × n) with positive diagonal entries.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Definition: Let A be an m × n matrix with independent columns. A QR-factorization of A expresses it as A = QR, where Q is m × n with orthonormal columns and R is an invertible upper triangular matrix (so R is n × n) with positive diagonal entries.
Theorem: Every m × n matrix A with linearly independent columns has a QR-factorization. The matrices Q and R are uniquely determined by A.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Definition: Let A be an m × n matrix with independent columns. A QR-factorization of A expresses it as A = QR, where Q is m × n with orthonormal columns and R is an invertible upper triangular matrix (so R is n × n) with positive diagonal entries.
Theorem: Every m × n matrix A with linearly independent columns has a QR-factorization. The matrices Q and R are uniquely determined by A.
1 1 0 Example: Find the QR factorization of A = −1 0 1.
 0 1 1  001
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Definition: Let A be an m × n matrix with independent columns. A QR-factorization of A expresses it as A = QR, where Q is m × n with orthonormal columns and R is an invertible upper triangular matrix (so R is n × n) with positive diagonal entries.
Theorem: Every m × n matrix A with linearly independent columns has a QR-factorization. The matrices Q and R are uniquely determined by A.
1 1 0 Example: Find the QR factorization of A = −1 0 1.
001 Solution. Let c1,c2 and c3 be the columns of A (which are
linearly independent).
 0 1 1 
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Definition: Let A be an m × n matrix with independent columns. A QR-factorization of A expresses it as A = QR, where Q is m × n with orthonormal columns and R is an invertible upper triangular matrix (so R is n × n) with positive diagonal entries.
Theorem: Every m × n matrix A with linearly independent columns has a QR-factorization. The matrices Q and R are uniquely determined by A.
1 1 0 Example: Find the QR factorization of A = −1 0 1.
001 Solution. Let c1,c2 and c3 be the columns of A (which are
linearly independent). Let f1 = c1, f2 = c2 − 1f1, and
2
 0 1 1 
f3 = c3 + 1 f1 − f2 (Gram-Schmidt method). 2
Ali Mousavidehshikh
Week 6

Write qi = 1 fi for i = 1, 2, 3. ∥fi ∥
Sections 8.3-8.4
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Write qi = 1 fi for i = 1,2,3. So {q1,q2,q3} is an ∥fi ∥
orthonormal set.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Write qi = 1 fi for i = 1,2,3. So {q1,q2,q3} is an ∥fi ∥
orthonormal set. Set Q = (q1 q2 q3) and let
2 1 −1 √√
1
R=√20 3 √3.
002
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Write qi = 1 fi for i = 1,2,3. So {q1,q2,q3} is an ∥fi ∥
orthonormal set. Set Q = (q1 q2 q3) and let
2 1 −1 √√
1
R=√20 3 √3.
002 You can verify that A = QR.
Week 6
Ali Mousavidehshikh

Sections 8.3-8.4
Write qi = 1 fi for i = 1,2,3. So {q1,q2,q3} is an ∥fi ∥
orthonormal set. Set Q = (q1 q2 q3) and let
2 1 −1 √√
1
R=√20 3 √3.
002 You can verify that A = QR.
Corollary: If A has independent rows, then A factors uniquely as A = LP, where P has an orthonormal rows and L is an invertible lower triangular matrix with positive main diagonal entries.
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Write qi = 1 fi for i = 1,2,3. So {q1,q2,q3} is an ∥fi ∥
orthonormal set. Set Q = (q1 q2 q3) and let
2 1 −1 √√
1
R=√20 3 √3.
002 You can verify that A = QR.
Corollary: If A has independent rows, then A factors uniquely as A = LP, where P has an orthonormal rows and L is an invertible lower triangular matrix with positive main diagonal entries.
Proof . If A has independent rows, then AT has independent columns, so AT has a QR factorization, say AT = QR. Then A = RTQT, letting L = RT and P = QT gives the desired result.
Ali Mousavidehshikh
Week 6

Sections 8.3-8.4
Theorem: Every square, invertible matrix A has factorization A=QR andA=LP whereQ andP areorthogonal,R is upper triangular with positive diagonal entries, and L is lower triangular with positive diagonal entries.
Ali Mousavidehshikh
Week 6