Explanation of Solution
a.
Symmetric matrix:
An
Now, consider a matrix
Now the product
If the elements of this product matrix are
To show this, consider a
Then transpose is as follows:
Therefore, multiplication is,
Explanation of Solution
b.
Symmetric matrix:
Now taking the same matrix as in part (a.), we have
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Chapter 2 Solutions
Introduction to mathematical programming
- 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 " # "arrow_forwardFind 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 itarrow_forwardIf a matrix A has size 5x6 and a matrix B has a size 6x4, then what will be the size of a matrix A*B?arrow_forward
- . 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.arrow_forwardConsider 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.arrow_forwardConsider 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.arrow_forward
- USING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.arrow_forwardQ3: Find the eigenvalues of the Matrix: C = 3 21 1 ادة / صباحي : انور عدنان یحییarrow_forward"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…arrow_forward
- 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 columnsarrow_forward+ A Transpose of a symmetric matrix does not have to be a symmetric matrix true Falsearrow_forwardUsing Python, to help solve the following problem. Provide an explanation of your solutions to the problem. 4. A symmetric matrix D is positive definite if x¹TDx > 0 for any nonzero vector x. It can be proved that any symmetric, positive definite matrix D can be factored in the form D = LLT for some lower triangular matrix L with nonzero diagonal elements. This is called the Cholesky factorization of D. Consider the matrix [2.25 -3 4.5 -10 -3 5 4.5 -10 34 a. Is A positive definite? Explain. A = b. Find a lower triangular matrix L such that LLT = A.arrow_forward
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole