Homework Answers
1: 1 131 2 Given matrix A 2 2 2. matrix P and I S set 2. a) Show that matrix P diaqonalizes A and find D(diagonal m...
1: 1 131 2 Given matrix A 2 2 2. matrix P and I S set 2. a) Show that matrix P diaqonalizes A and find D(diagonal matnx) that matches. 6) Find the eigen values of A Observe that the columns of P form set S c) orthogonal Set using the inner product standard show that set S is not an Use the Gram- Schmidt process to get an orthonormal set from S using inner product standard
1: 1 131...
Note that is the -Jordan block of size m with 's in the diagonal and 1 to the right of the diagonal. 3. A Mercato...
Note that is
the -Jordan block of
size m with 's in the diagonal
and 1 to the right of the diagonal.
3. A Mercator matrix is a matrix A E Matn,n(R) such that pA(x) = (x-aj (x-4) for some α' E R, which are not necessarily distinct, such that 0 < α' < 2. Let Mn(R) be the set of Mercator matrices. As in Tutorial 1, if A EM(R) define the logarithm of A to be the matrix given...
(a) Reduce the following matrices to diagonal form and find a g-inverse of each 120-11 4 5 6 2 2 3 -1 A=158 O 11 and...
(a) Reduce the following matrices to diagonal form and find a g-inverse of each 120-11 4 5 6 2 2 3 -1 A=158 O 11 and B-1084 7 1o-2 3 21 6 (5+5 (b) () For any n x I vector a 0, show that a (ii) Find the g-inverse of the vector a, where a' = [1 a'a 5 2] 3 1
(a) Reduce the following matrices to diagonal form and find a g-inverse of each 120-11 4 5...
6. (Strang 2.7.22) Find P, L and U, such that PA = LU (ie the LU...
6. (Strang 2.7.22) Find P, L and U, such that PA = LU (ie the LU decomposition with row exchanges), where: [1 2 07 A= 2 4 11 1 1 1 Note: P must be a permutation matrix, L must be lower triangular with l's on the diagonal, and U must be upper triangular (with any values allowed on the diagonal).
1. 2. The matrix A is given by 7 The diagonal terms are 8, 7 and...
1. 2. The matrix A is given by 7 The diagonal terms are 8, 7 and 5. The non-diagonal terms are unknown Two of the eigenvalues of A are 6 and 4 What is the of the third eigenvalue. 2. Using Newtons method find the value of x that minimizes the function X -sin x is in radians. Start with x 0.5 4. Use the golden section rule to find the value of that minimizes the function f(x) x-14x+607-70x in...
Question B 7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i)...
Question B
7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
1 2 1 Show that A=0 0 a diagonal D. 2 is diagonalizable by finding P...
1 2 1 Show that A=0 0 a diagonal D. 2 is diagonalizable by finding P and D such that P-1AP = D for 3 0
PlOBlem 4 (10 points) 2 m 3 m The frame is made up of members ACE, BCD and DE. The frame is pin- 6 m supported at A...
PlOBlem 4 (10 points) 2 m 3 m The frame is made up of members ACE, BCD and DE. The frame is pin- 6 m supported at A and roller-supported at B. On member ACE, determine the 2 m magnitudes and directions of forces acting at pins C and E Hint: Begin with overall free body 3 m diagram 600 N 5 m
PlOBlem 4 (10 points) 2 m 3 m The frame is made up of members ACE, BCD...
please answer the five questions clearly. I have provided the data. 7 Diagonal Matrices Diagonal Matrices...
please answer the five questions clearly. I have provided the
data.
7 Diagonal Matrices Diagonal Matrices If A = (a) is a square matrix, then the entries and are called diagonal entries. A square matrix is called diagonal if all non-diagonal entries are zeros. Explore what happens if we add, subtract or multiply diagonal matrices. A and B are the same matrices in previous sections ( section 5.) Type D-diag(diag(A)) to create a diagonal matrix from A. Type E-diag(diag(B)) to...
Help with question 6 please!! a diagonal matrix D such that A = PDP-1 6. If...
Help with question 6 please!!
a diagonal matrix D such that A = PDP-1 6. If u is an eigenvector of an invertible matrix A corresponding to ), show that z is also an eigenvector of A-1. What is the corresponding eigenvalue? lu Drou that if 42 - IT is
1: 1 131 2 Given matrix A 2 2 2. matrix P and I S set 2. a) Show that matrix P diaqonalizes A and find D(diagonal matnx) that matches. 6) Find the eigen values of A Observe that the columns of P form set S c) orthogonal Set using the inner product standard show that set S is not an Use the Gram- Schmidt process to get an orthonormal set from S using inner product standard
1: 1 131...
Note that is
the -Jordan block of
size m with 's in the diagonal
and 1 to the right of the diagonal.
3. A Mercator matrix is a matrix A E Matn,n(R) such that pA(x) = (x-aj (x-4) for some α' E R, which are not necessarily distinct, such that 0 < α' < 2. Let Mn(R) be the set of Mercator matrices. As in Tutorial 1, if A EM(R) define the logarithm of A to be the matrix given...
(a) Reduce the following matrices to diagonal form and find a g-inverse of each 120-11 4 5 6 2 2 3 -1 A=158 O 11 and B-1084 7 1o-2 3 21 6 (5+5 (b) () For any n x I vector a 0, show that a (ii) Find the g-inverse of the vector a, where a' = [1 a'a 5 2] 3 1
(a) Reduce the following matrices to diagonal form and find a g-inverse of each 120-11 4 5...
6. (Strang 2.7.22) Find P, L and U, such that PA = LU (ie the LU decomposition with row exchanges), where: [1 2 07 A= 2 4 11 1 1 1 Note: P must be a permutation matrix, L must be lower triangular with l's on the diagonal, and U must be upper triangular (with any values allowed on the diagonal).
1. 2. The matrix A is given by 7 The diagonal terms are 8, 7 and 5. The non-diagonal terms are unknown Two of the eigenvalues of A are 6 and 4 What is the of the third eigenvalue. 2. Using Newtons method find the value of x that minimizes the function X -sin x is in radians. Start with x 0.5 4. Use the golden section rule to find the value of that minimizes the function f(x) x-14x+607-70x in...
Question B
7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
1 2 1 Show that A=0 0 a diagonal D. 2 is diagonalizable by finding P and D such that P-1AP = D for 3 0
PlOBlem 4 (10 points) 2 m 3 m The frame is made up of members ACE, BCD and DE. The frame is pin- 6 m supported at A and roller-supported at B. On member ACE, determine the 2 m magnitudes and directions of forces acting at pins C and E Hint: Begin with overall free body 3 m diagram 600 N 5 m
PlOBlem 4 (10 points) 2 m 3 m The frame is made up of members ACE, BCD...
please answer the five questions clearly. I have provided the
data.
7 Diagonal Matrices Diagonal Matrices If A = (a) is a square matrix, then the entries and are called diagonal entries. A square matrix is called diagonal if all non-diagonal entries are zeros. Explore what happens if we add, subtract or multiply diagonal matrices. A and B are the same matrices in previous sections ( section 5.) Type D-diag(diag(A)) to create a diagonal matrix from A. Type E-diag(diag(B)) to...
Help with question 6 please!!
a diagonal matrix D such that A = PDP-1 6. If u is an eigenvector of an invertible matrix A corresponding to ), show that z is also an eigenvector of A-1. What is the corresponding eigenvalue? lu Drou that if 42 - IT is
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