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where A is an nxn matrix and kappa is the condition number
P2.6.1 Show that if...
a) suppose that the nxn matrix A has its n eigenvalues arranged in decreasing order of absolute size, so that >>....
a) suppose that the nxn matrix A has its n eigenvalues arranged
in decreasing order of absolute size, so that >>....
each eigenvalue has its corresponding eigenvector, x1,x2,...,xn.
suppose we make some initial guess y(0) for an eigenvector.
suppose, too, that y(0) can be written in terms of the actual
eigenvectors in the form y(0)=alpha1.x1 +alpha2.x2
+...+alpha(n).x(n), where alpha1, alpha2, alpha(n) are constants.
by considering the "power method" type iteration y(k+1)=Ay(k) argue
that (see attached image)
b) from an nxn...
υΣνΤ. Answer the following questions: Suppose a matrix A E Rmxn has an SVD A (i) Show that the rank of the miatrix A E Rmxn is equal to the number of its nonzero singular values. (ii) Show that miult...
υΣνΤ. Answer the following questions: Suppose a matrix A E Rmxn has an SVD A (i) Show that the rank of the miatrix A E Rmxn is equal to the number of its nonzero singular values. (ii) Show that miultiplication by an orthogonal matrix on the left and multiplication by an orthogonal matrix on the right, i.e., UA and BU, where A E Rmxn and B ERnm are general matrices, and U Rxm is an orthogonal matrix, preserve the Frobenius...
If A is a symmetric nxn matrix with n distinct characteristic numbers 2, show that any...
If A is a symmetric nxn matrix with n distinct characteristic numbers 2, show that any polynomial P(A) can be expressed in the form P(A) = CAP-+ +cAn-2 +...+C1_A+CH where the c’s are determined by the n simultaneous linear equations P(2) = 2** (i = 1, 2,...,n) 1)=20:45
Do the following using Matlab: Let A be a matrix where each row corresponds to an...
Do the following using Matlab: Let A be a matrix where each row
corresponds to an article, and each column to how often a specific
word appears in the article. Patterns among the articles and words
can often be identified by analyzing the singular values and
vectors of the singular value decomposition of A.
(a) Use the Matlab svd function to find the singular values of
the following
. Note this problem does not ask you to print out the...
Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for...
Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for each n x n matrix A, we define the n x n matrix p(A) by P(A) = ao In + a A+ ... + amA”. Also, for each n, let Onxn E Rnxn be the n x n zero matrix. (a) Show that for all polynomials p and q and square matrices A, we have p(A)q(A) = 9(A)p(A). (b) Show that for every 2...
10.10 If A is an 'n x n matrix, and x is an n x 1 vector, then the linear transformation y = Ar maps* n to·m, so th...
10.10 If A is an 'n x n matrix, and x is an n x 1 vector, then the linear transformation y = Ar maps* n to·m, so the linear transformation should have a condition number, condAx (x). Assume that l a subordinate norm a. Show that we can define condar (x)-[All Irl/IArll for every x 0. b. Find the condition number of the linear transformation atx [ - 2 using the oo-norm ng the oo-norm. T-3 2 1 .12...
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the...
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the...
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
(a) Suppose we want to solve the linear vector-matrix equation Ax b for the vector x. Show that the Gauss elimination algorithm may be written bAbm,B where m 1, This process produces a matrix equa...
(a) Suppose we want to solve the linear vector-matrix equation Ax b for the vector x. Show that the Gauss elimination algorithm may be written bAbm,B where m 1, This process produces a matrix equation of the form Ux = g , in which matrix U is an upper-triangular matrix. Show that the solution vector x may be obtained by a back-substitution algorithm, in the form Jekel (b) Iterative methods for solving Ax-b work by splitting matrix A into two...
I need 7.3. This is a matlab question and I don't know what simple number is....
I need 7.3. This is a matlab question and I don't know what
simple number is.
7.3. Examine the eigenvalues of the family of matrices 「-2 1 0 0 1 0 0 一2 1 Dr =-N- 10 where Dv is N x N, for growing values of N, for example, N 32,64,128. The smallest eigenvalues converge to integer multiples of a simple number. 7.4. Verify, for a random 10 x 5 matrix, the identity i=1 where K-minrn, n), {σ1, ....
a) suppose that the nxn matrix A has its n eigenvalues arranged
in decreasing order of absolute size, so that >>....
each eigenvalue has its corresponding eigenvector, x1,x2,...,xn.
suppose we make some initial guess y(0) for an eigenvector.
suppose, too, that y(0) can be written in terms of the actual
eigenvectors in the form y(0)=alpha1.x1 +alpha2.x2
+...+alpha(n).x(n), where alpha1, alpha2, alpha(n) are constants.
by considering the "power method" type iteration y(k+1)=Ay(k) argue
that (see attached image)
b) from an nxn...
υΣνΤ. Answer the following questions: Suppose a matrix A E Rmxn has an SVD A (i) Show that the rank of the miatrix A E Rmxn is equal to the number of its nonzero singular values. (ii) Show that miultiplication by an orthogonal matrix on the left and multiplication by an orthogonal matrix on the right, i.e., UA and BU, where A E Rmxn and B ERnm are general matrices, and U Rxm is an orthogonal matrix, preserve the Frobenius...
If A is a symmetric nxn matrix with n distinct characteristic numbers 2, show that any polynomial P(A) can be expressed in the form P(A) = CAP-+ +cAn-2 +...+C1_A+CH where the c’s are determined by the n simultaneous linear equations P(2) = 2** (i = 1, 2,...,n) 1)=20:45
Do the following using Matlab: Let A be a matrix where each row
corresponds to an article, and each column to how often a specific
word appears in the article. Patterns among the articles and words
can often be identified by analyzing the singular values and
vectors of the singular value decomposition of A.
(a) Use the Matlab svd function to find the singular values of
the following
. Note this problem does not ask you to print out the...
Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for each n x n matrix A, we define the n x n matrix p(A) by P(A) = ao In + a A+ ... + amA”. Also, for each n, let Onxn E Rnxn be the n x n zero matrix. (a) Show that for all polynomials p and q and square matrices A, we have p(A)q(A) = 9(A)p(A). (b) Show that for every 2...
10.10 If A is an 'n x n matrix, and x is an n x 1 vector, then the linear transformation y = Ar maps* n to·m, so the linear transformation should have a condition number, condAx (x). Assume that l a subordinate norm a. Show that we can define condar (x)-[All Irl/IArll for every x 0. b. Find the condition number of the linear transformation atx [ - 2 using the oo-norm ng the oo-norm. T-3 2 1 .12...
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
(a) Suppose we want to solve the linear vector-matrix equation Ax b for the vector x. Show that the Gauss elimination algorithm may be written bAbm,B where m 1, This process produces a matrix equation of the form Ux = g , in which matrix U is an upper-triangular matrix. Show that the solution vector x may be obtained by a back-substitution algorithm, in the form Jekel (b) Iterative methods for solving Ax-b work by splitting matrix A into two...
I need 7.3. This is a matlab question and I don't know what
simple number is.
7.3. Examine the eigenvalues of the family of matrices 「-2 1 0 0 1 0 0 一2 1 Dr =-N- 10 where Dv is N x N, for growing values of N, for example, N 32,64,128. The smallest eigenvalues converge to integer multiples of a simple number. 7.4. Verify, for a random 10 x 5 matrix, the identity i=1 where K-minrn, n), {σ1, ....
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