Assume A is k x n-matrix and B is n x l-matrix and AB = 0. Prove that rank(A) + rank(B) < n.
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Q: Assume A is k × n-matrix and B is n × l-matrix. Prove that rank(AB) < rank(A) and rank(AB) < rank(B)
A: Assume A is k x n-matrix and B is n x l- matrix. Prove that rank(AB)
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Q: ) Assume A is k × n-matrix and Q is n x n-invertible matrix. Prove that rank(AQ) = rank(A).
A: Solution:
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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?make an adjacency matrix for the following: E= {(1,2),(2,1),(3,2),(4,3),(4,5),(5,6),(6,7)}Assume A is k x n-matrix and B is n x l-matrix. Prove that rank(AB) < rank(A) and rank(AB) < rank(B)
- Let A be the n x n matrix with n = 1000 and entries A(i, i) = i, A(i, i + 1) = A(i + 1, i) = 1/2, A(i, i + 2) = A(i + 2, i) = 1/2 for all i that fit within the matrix.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 030läs 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.
- Convergence of iterative methods like the Conjugate Gradient Method can be accelerated by the use of a technique called preconditioning. The convergence rates of iterative methods often depend, directly or indirectly, on the condition number of the coefficient matrix A. The idea of preconditioning is to reduce the effective condition number of the problem. The preconditioned form of the n X n linear system Ax = b is where M is an invertible n x n matrix called the preconditioner. When A is a symmetric positive-definite n x n matrix, we will choose a symmetric positive-definite matrix M for use as a preconditioner. A particularly simple choice is the Jacobi preconditioner M = D, where D is the diagonal of A. The Preconditioned Conjugate Gradient Method is now easy to describe: Replace Ax = b with the preconditioned equation M-¹Ax = M-¹b, and replace the Euclidean inner product with (v, w) M. To convert Algorithm 2 in Section 3.3 to the preconditioned version, let zk = M-¹b - M-¹ Axk =…If two rows are identical (same) in a square matrix then the determinant value of the matrix is а. —1 b. 0 c. 1 d. null matrix aSolve and program a matrix using the following definition: if i = j aj ={i-j if i>j |2i + j otherwise %3D
- Given a matrix of dimension m*n where each cell in the matrix can have values 0, 1 or 2 which has the following meaning: 0: Empty cell 1: Cells have fresh oranges 2: Cells have rotten oranges So we have to determine what is the minimum time required so that all the oranges become rotten. A rotten orange at index [i,j] can rot other fresh orange at indexes [i-1,j], [i+1,j], [i,j-1], [i,j+1] (up, down, left and right). If it is impossible to rot every orange then simply return -1. Examples: Input: arr[][C] = { {2, 1, 0, 2, 1}, {1, 0, 1, 2, 1}, {1, 0, 0, 2, 1}}; Output: All oranges cannot be rotten. Below is algorithm. 1) Create an empty Q. 2) Find all rotten oranges and enqueue them to Q. Also enqueue a delimiter to indicate beginning of next time frame. 3) While Q is not empty do following 3.a) While delimiter in Q is not reached (i) Dequeue an orange from queue, rot all adjacent oranges. While rotting the adjacents, make sure that time frame is incremented only once. And time frame is…matrix of dimension m*n where each cell in the matrix can have values 0, 1 or 2 which has the following meaning: 0: Empty cell 1: Cells have fresh oranges 2: Cells have rotten oranges So we have to determine what is the minimum time required so that all the oranges become rotten. A rotten orange at index [i,j] can rot other fresh orange at indexes [i-1,j], [i+1,j], [i,j-1], [i,j+1] (up, down, left and right). If it is impossible to rot every orange then simply return -1. Examples: Input: arr[][C] = { {2, 1, 0, 2, 1}, {1, 0, 1, 2, 1}, {1, 0, 0, 2, 1}}; Output: All oranges can become rotten in 2 time frames.Input: arr[][C] = { {2, 1, 0, 2, 1}, {0, 0, 1, 2, 1}, {1, 0, 0, 2, 1}}; Note: solve as soon as possible use c++ languageIf 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