Let M be an n x n matrix with each entry equal to either 0 or 1. Let mij denote the entry in row i and column j. A diago...
Let M be an n x n matrix with each entry equal to either 0 or 1. Let mij denote the entry in row i and column j. A diagonal entry is one of the form mii for some i.
Swapping rows i and j of the matrix M denotes the following
action: we swap the values mik and mjk for k
= 1,2, ... , n. Swapping two columns is defined analogously.
We say that M is rearrangeable if it is possible to swap some of
the pairs of rows and some of the pairs of columns (in any
sequence) so that, after all the swapping, all the diagonal entries
of M are equal to 1.
(a) Give an example of a matrix M that is not rearrangeable, but
for wbich at least one entry in each row and each column is equal
to 1.
(b) Give a polynomial-time algorithm that determines whether a
matrix M with 0-1 entries is rearrangeable.
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Let M be an n x n matrix with each entry equal to either 0 or 1. Let mij denote the entry in row i and column j. A diago...
Linear Algebra: Show that for each i = 1, ..., n there is a natural number p. j- 1v1, . . . , Vnf is a canonical Let be...
Linear Algebra: Show that for each i = 1, ..., n there is a
natural number p.
j- 1v1, . . . , Vnf is a canonical Let be a linear operator on V and Jordan basis, ie. ΤΊβ is a canonical Jordan form. Show that for each i-1, . . . ,n there is some p є N such that (T-ÀI)" (vi-0, where is the diagonal entry of the matrix [T]β on the ith column.
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Let A be an n × n real symmetric matrix with its row and column sums both equal to 0. Let λ1, . . . , λn be the eigenvalues of A, with λn = 0, and with corresponding eigenvectors v1,...,vn (these exist because A is real symmetric). Note that vn = (1, . . . , 1). Let A[i] be the result of deleting the ith row and column. Prove that detA[i] = (λ1···λn-1)/n. Thus, the number of spanning...
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Let A be an m x n matrix with entries 0 and 1. We say that A is even if the number of 1's in each row is even and the number of 1's in each column is even. Let A and B be distinct even mx n matrices. Show that A and B must differ in at least four entries. Note: the integer 0 is even.
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About linear algebra,matrix;
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-entry of the matrix A equals 0 if i + j is divisible by 5 and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u 9,64,-71,...
please answer 2a(i) only 2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 × 9 matrices. The (i, j)-entry of the matrix B is given by i *j - 1. The (i,j)-en...
please answer 2a(i) only
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 × 9 matrices. The (i, j)-entry of the matrix B is given by i *j - 1. The (i,j)-entry of the matrix A equals 0 if i +j is divisible by and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u- [9,...
An m×n array A of real numbers is a Monge array if for all i,j,k, and l such that 1≤i<k≤m and ...
An m×n
array A
of real numbers is a Monge array if for all i,j,k,
and l
such that 1≤i<k≤m
and 1≤j<l≤n
, we have
>A[i,j]+a[k,l]≤A[i,l]+A[k,j]>
In other words, whenever we pick two rows and two columns of a
Monge array and consider the four elements at the intersections of
the rows and columns, the sum of the upper-left and lower-right
elements is less than or equal to the sum of the lower-left and
upper-right elements. For example, the following...
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j- 1v1, . . . , Vnf is a canonical Let be a linear operator on V and Jordan basis, ie. ΤΊβ is a canonical Jordan form. Show that for each i-1, . . . ,n there is some p є N such that (T-ÀI)" (vi-0, where is the diagonal entry of the matrix [T]β on the ith column.
j- 1v1, . ....
Enter Tor F depending on whether the statement is true or false (You must enter T or F -- True and False will not work.) 1. If the corresponding entries of two matrices A and B are equal, then A = B 2. A matrix with dimensions m byn, where m > n has fewer rows than columns. 3. The ith row jth column entry of a square matrix, where i>j. is called a diagonal entry
Let A be an m x n matrix with entries 0 and 1. We say that A is even if the number of 1's in each row is even and the number of 1's in each column is even. Let A and B be distinct even mx n matrices. Show that A and B must differ in at least four entries. Note: the integer 0 is even.
About linear algebra,matrix;
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 x 9 matrices. The (i.j)-entry of the matrix B is given by i *j -1. The (i. j)-entry of the matrix A equals 0 if i + j is divisible by 5 and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u 9,64,-71,...
please answer 2a(i) only
2. (a) Use Octave as a Calculator to answer this question. Suppose that A and B are two 8 × 9 matrices. The (i, j)-entry of the matrix B is given by i *j - 1. The (i,j)-entry of the matrix A equals 0 if i +j is divisible by and equals the (i,j)-entry of the matrix B otherwise. i. What are the rank and nullity of matrices A and B? ii. Is vector u- [9,...
An m×n
array A
of real numbers is a Monge array if for all i,j,k,
and l
such that 1≤i<k≤m
and 1≤j<l≤n
, we have
>A[i,j]+a[k,l]≤A[i,l]+A[k,j]>
In other words, whenever we pick two rows and two columns of a
Monge array and consider the four elements at the intersections of
the rows and columns, the sum of the upper-left and lower-right
elements is less than or equal to the sum of the lower-left and
upper-right elements. For example, the following...
LU be an LU factorization of matrix A e Fn×n computed by the Gaussian elimination Let PA with partial pivoting (GEPP). Let us denote Prove that (a) leyl 1, for all i >j S 2-1 maxij laijl You may assume P-1, i.e., in each step of the Gaussian elimination process the absolute value of the diagonal entry is already the largest among those of the entries below the diagonal entry on the same column You may prove the results with...
(a) Let A be a real n x m matrix. (i) State what conditions on n and m, if any, are needed such that the matrix AAT exists. Justify your statement. (ii) Assuming that the matrix AA exists, find its size. (iii) Assuming that the matrix AAT exists, prove using index notation that all diagonal elements of AAT are positive or equal to zero. (iv) Let 12 5 -3 A= 3-4 2 Calculate (AAT) -- (show all your working). 2)
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