Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter D.2, Problem 5E
Program Plan Intro
To prove that every entry of Ainverse is real if and only if every entity of the matrix, A is real.
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Students have asked these similar questions
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?
. 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.
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.
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- List any two properties of eigenvalues of a square matrix. Explain it in your own way and provide it with examples.arrow_forwardFind the matrix product of integer random elements and the size of each (4 x 2) of them.arrow_forwardA What is the product of matrices M=(1,0)(0,-3) and N=(0,1)(0,-3)? B What is the difference of matrices M=(1,0,2)(0,-3,1) and N=(0,1,2)(0,0,-3)? C What is the a(3) term of the sequence {a(n)} where a(n) = 2n - 6? 1.D What is the range of a function that assigns to a 16-bit bit string the number of one bits in the bit string? E What is the sum of matrices M=(1,0,2)(0,-3,1) and N=(0,1,2)(0,0,-3)? F If f(x) = 2x + 3, find f(S) if S={0,2,5}. G What is the transpose of matrix M=(1,0,2)(0,-3,1)?arrow_forward
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