Suppose we have a matrix A of dimension n x n of rank n - 1. Explain two numeri cally stable ways to find a nonzero vector z so that Az = 0.
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- 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. #UsePythonFind the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.) Sufficient Condition for Diagonalization If an n xn matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable. Find the eigenvalues. (Enter your answers as a comma-separated list.) Is there a sufficient number to guarantee that the matrix is diagonalizable? O Yes O No Need Help? Read it. 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.
- 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?Consider the following. 0 1 -3 A = 0 4 0 4 1 2 2 2 -1 2 (a) Verify that A is diagonalizable by computing P-AP. p-1AP = (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues.Assume A is k x n-matrix and P is k × k-invertible matrix. Prove that rank(PA) = rank(A).
- "onsdensiks Consider the m x n-matrix A and the vector be Rm that are given by A = [aij], 6 = [bi] where aij = (–1)++i(i – j), b; = (-1)' for i = 1,... m and j = 1,..· , n. %3D Note that aij is the entry of A at the i-th row and the j-th column. Consider the following condition on vectors iE R": (Condition1) Aa õ. Write and run a python Jupyter notebook using the PULP package, to check whether there is a vector iE R" that satisfies (Condition1), • when m =n = 10, and if it exists, find one. Submit your screenshots or the pdf file of your Jupyter Notebook; it should show both the codes and the results. [For credits, you must write a python Jupyter notebook to solve this problem.] Python Hint: Start with giving М-10 N=10 You may want to use Python lists combined with 'forloop'. For example, column=[j+1 for j in range(N)] row=[i+1 for i in range(M)] Then, one can define the matrix A and the vector 6, using Python dictionaries combined with 'forloop' as a= {i:{j:(-1)**(i+j)*(i-j) for j in…Let f ∈ C+ 2π with a zero of order 2p at z. Let r>p and m = n/r. Then there exists a constant c > 0 independent of n such that for all nsufficiently large, all eigenvalues of the preconditioned matrix C−1 n (Km,2r ∗ f)Tn(f) are larger than c.The matrix [A] has a norm of ||A||=3.2 x 10° and the norm of the inverse of that matrix is ||A-4||=2.5 x 1012 If a solution is obtained for a system of equations [A]{x}={b} using this same matrix [A] and double precision computations, then O the solution is very sensitive to input error O the solution is insensitive to input error O small residuals indicate accurate answers O small residuals do not indicate accurate answers O the system is ill-conditioned
- Consider the following. -4 2 0 1 -3 A = 0 4 0 4 2 2 -1 1 2 2 (a) Verify that A is diagonalizable by computing P-1AP. p-1AP = (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n x n matrices, then they have the same eigenvalues.Current Attempt in Progress Suppose that a 4x 4 matrix A has eigenvalues A = 1,A2=-7. As-10, and A4=-10. Use the following method to find tr(A). If A is a square matrix and p(A) = det (Al - A) is the characteristic polynomial of A, then the coefficient of in p) is the negative of the trace of A. tr(A) =For the matrix A, find (if possible) a nonsingular matrix P such that p-lAP is diagonal. (If not possible, enter IMPOSSIBLE.) 6 -3 A = -2 P = Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP =