number of nonzero singular values
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Let A be an m × n matrix with m > n.
(a) What is the maximum number of nonzero singular values that A can have?
(b) If rank(A) = k, how many nonzero singular values does A have?
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- Use a software program or a graphing utility with matrix capabilities to find the eigenvalues of the matrix. (Enter your answers as a comma-separated list.) 1 0 -1 1 1 1 -3 0 3 -3 0 3 3 λ = 0 030make an adjacency matrix for the following: E= {(1,2),(2,1),(3,2),(4,3),(4,5),(5,6),(6,7)}If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. #UsePython
- . The determinant of an n X n matrix can be used in solving systems of linear equations, as well as for other purposes. The determinant of A can be defined in terms of minors and cofactors. The minor of element aj is the determinant of the (n – 1) X (n – 1) matrix obtained from A by crossing out the elements in row i and column j; denote this minor by Mj. The cofactor of element aj, denoted by Cj. is defined by Cy = (-1y**Mg The determinant of A is computed by multiplying all the elements in some fixed row of A by their respective cofactors and summing the results. For example, if the first row is used, then the determi- nant of A is given by Σ (α(CI) k=1 Write a program that, when given n and the entries in an n Xn array A as input, computes the deter- minant of A. Use a recursive algorithm.Assume A is k x n-matrix and B is n x l-matrix and AB = 0. Prove that rank(A) + rank(B) < n.If matrix A is a 2 x 3 matrix, it can be multiplie by matrix B to obtain AB only if matrix B has:A. 2 rowsB. 2 columnsC. 3 rowsD. 3 columns
- Perform matrix manipulation: Given that (AB)T = BTAT show that (ABC)T = CTBTATsuppose a computer solves a 100x100 matrix using Gauss elimination with partial pivoting in 1 second, how long will it take to solve a 300x300 matrix using Gauss elimination with partial pivoting on the same computer? and if you have a limit of 100 seconds to solve a matrix of size (N x N) using Gauss elimination with partial pivoting, what is the largest N can you do? show all the steps of the solutionläs 5 Find the SUM for mean of the matrix * ? F F= 1 1 1 1 2 2 2 2 3 3 3 4 4 4 4 sum(mean (F)) O sum (sum F) O mean(mean(F)) O Jalbo läl li 3.
- subject : analysis of algorithm Q.No.1: Consider the following chain of matrices having matrices A, B, C and D. You have to consider the digits of your Registration Number in the order of the matrix as given. Add 2 to the digit if its zero. For example, your Reg_No. 19-Arid-797 has last digit 7, 2nd last digit 9 and 3rd last digit 7. A B C 2 X last digit last digit X 2nd last digit 2nd last digit X 3nd last digit D 3rd Last digit X 4 What will be the minimum number of multiplication to multiply these matrices? Show the order of multiplication as well.The join of two zero-one matrices A and B is described as AvB = [aij v bij] AAB = [aij a bij] AAB = [aij v bij] AvB = [aij a bij]Pls Use Python If there is a non-singular matrix P such as P-1AP=D , matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A. A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) : 1 -1 -1 1 3 1 -3 1 -1 a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy. b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy. Ps: Please also explain step by step with " # "