Items tagged with matrix

Hi there,

            I am new to maple. I want to ask a simple question.

            If I have a array, and I want its each component to take natural logarithm. How can I do?

            Eg:[2 3 4]->[ln(2) ln(3) ln(4)]

            Thanks in advance.

Dirichlet

Simple test of GaussianElimination function

Why doesn't this work? 

Also can I just confirm that for GaussianElimination (according to the help this uses LUDecomposition function with the output=['U'] option) the input is the augmented matrix of the system, the coefficient matrix augmented with the RHS)

In a separate test I got an example working using this code, but I've never seen this syntax before for A Matrix (using << it seems?)

 

Thank you for your help.

 

 

ORIGINAL ATTEMPT

with(LinearAlgebra):

A := matrix(3, 3, [-3, 2, 1, 1, -2, 1, 1, 2, -3]);

Matrix(3, 3, {(1, 1) = -3, (1, 2) = 2, (1, 3) = 1, (2, 1) = 1, (2, 2) = -2, (2, 3) = 1, (3, 1) = 1, (3, 2) = 2, (3, 3) = -3})

(1)

GaussianElimination(A)

Error, (in LinearAlgebra:-GaussianElimination) invalid input: LinearAlgebra:-GaussianElimination expects its 1st argument, A, to be of type Matrix() but received A

 

``

 

Download gauss_elimination_2.mw

I want to analyze the runtimes on certain Linear Algebra functions in Maple, so I need a (large) set of matrices to input into these functions.

I have written the below code, which does succesfully generate a file of matrices:

The resulting file looks like:

However, I am unable to read the matrices from this file back into Maple. When using the code below, I get an error.


I think the error is that %a in fscanf scans up to the next whitespace, so the spacing in Matrix(3, 3, [[9,1,-4],[-5,6,-10],[-10,-4,-4]]) is throwing fscanf off. Do you guys know of any way I can fix this?

 

Or, is there a better way for me to generate these matrices so that they can be easily read into Maple? I've considered using ImportMatrix/ExportMatrix, but I believe that they only work for a single matrix, not the numerous ones that I would need. 

I have data matrix in text file.

I opened it with Maple and replaced decimal commas to decimal points + replaced column separator to semicolon.

Then I set 'Convert to plain text'.

After these modifications, I would like to export that Maple worksheet back to .txt  file with content exactly as shown on the display.

But if I export the worksheet as Maple text, extra pound signs are added to the beginning of each row. In this case I cannot import data to matrix datatype=float.

If I export the worksheet as Plain text, all the rows are destroyed and I cannot import data into matrix as well.

How can I export worksheet content, exactly as shown on display, into text file?

Hi everybody

In the attached file, when I run the code, a confusing error message appears. Obviously, the product of a 3*2 matrix by a 2*1 matrix is possible, but Maple gives an error. These 2 matrices are multiplied correctly in another worksheet. What's the source of this error?

Thanks in advance

 

F1.mw

Hi,

I have this program which when I run gives me five unique matrices as I want it. I however, when I run it again the enries of the matrices changes and I get 5 identical matrices, how can I prevent this from happening?

Vic

Hello everybody,

I would like to ask: How many ways to impose the rank deficiency of a matrix J?

1. First is the determinant(J) = 0

2. Multiply with a non-zero vector V: so that we have J*V = 0;

3. ...to be listed......

 

something about the minors of the matrix? 

I hope to have as many methods as possible!

Hi,

I have this 5 by 2 matrix, and I want to form  lists of lengths 5, whereby the ith entry in my list should be any of the elements from the ith row of the matrix. How do i get all the possible lists? (I am expecting a total of 32lists, each of length 5):

Hello! 

In an assigment I have been asked to use the Eigenvectors command to find the eigenvalues and eigenvectors of a particular matrix. 

As highlighted in the following image, my questions are:

1. What is the meaning of the suffix "+0.I"? Does it mean that there are further decimal digits which are not displayed?

2. How do the first and third eigenvalues, which are equal, result in different eigenvectors? As per my understanding, equal eigenvalues should have equal corresponding eigenvectors. Please help.

I have a system of equations e.g.

A^2+B*A+C=0

where A,B,C are Matrices and I want to solve for A.

Sure I can write every equations in brakets [..=0], but isn'T it possible to just use the matrix notation?

I've been trying to numerically solve (and plot) this equation. Maple tells me that some matrix is singular - I have no idea, what I can do.

eq := diff(y(x), `$`(x, 3))-(diff(y(x), x))*(diff(y(x), x))+1 = 0;

cond := (D(y))(0) = 0, (D(y))(1) = 1, ((D@@2)(y))(0) = 0

de := dsolve({cond, eq}, y(x), numeric);

Error, (in dsolve/numeric/bvp) matrix is singular

Hi,

 

I am trying to solve a simple system of the form AX=0, where A is a N*N matrix, X is an N*1 vector (and the right-hand side of the equation is an N*1 vector of zeros, I apologize for the inexact notation). The difficulty comes from the fact that the values of A are parameterized by 2*N parameters (that I will write as the 2*N vector P), and I would like to get a solution in the form X=f(P).

 

One solution is to try to use LinearAlgebra[LinearSolve], but it only returns the trivial solution X=0, which I am not interested in.

Another solution is to compute analytically the Moore-Penrose pseudoinverse Ag of A, as the general solution is of the form

(I - Ag A)f ;

where f is a vector of free parameters. However, even for a small matrix size (N=4), Maple is still computing after 3 hours on my (fairly powerful) machine, and it is taking more and more memory over time. As the results are polynomial/rational equations in the parameters P, I was actually expecting Maple to be more powerful than other softwares, but for this particular problem, Matlab's symbolic toolbox (muPAD) gives quick solutions until N=6. I need, in the end, to solve additional polynomial/rational equations that are derived from the solutions X=f(P), where Matlab fails. This is why I would really like to be able to solve the above-mentioned problem AX=0 with Maple in order to try to solve the subsequent step of the problem (polynomial system) with Maple.

 

Any suggestions on how to do this would be highly appreciated! Thank you very much for your time and help.

 

Laureline

When I print 2 matrix it always try to multiply them. How to force just to show?

I want to create a matrix (B) from entries of other matrices (A) with a helper-function (helper). The helper function is defined such that it returns a certain matrix depending on the index variables. This is necessary because the inner matrices are constructed with another function.

Since the helper-function returns matrices, the big matrix is of datatype=matrix. Unfortunately, creating the big matrix with the correct size and forcing the datatype=float, does not yield the desired result. However, the manual definition using the constructor with a list of matrices does create the desired matrix.

How do I resolve a matrix of matrices?

Note: I know that I could write a convert function that copies the entries to a corresponding matrix, though this seems to be unnecessary effort to me.

This might not be minimal but shows the issue. (Compare B and test)

MWE_matrix_of_matrices.mw

restart;
with(LinearAlgebra):

size_A := 2;
size_B := 3;

2

 

3

(1)

helper2 := proc(i::integer,j::integer);
  if i=j then
    a;
  elif i=j-1 or i=j+1 then
    b;
  else
    c;
  end if;
end proc:

helper3 := proc(i::integer,j::integer);
  if i=j then
    Matrix(size_A,size_A,helper2);
  elif i=j-1 or i=j+1 then
    -IdentityMatrix(size_A);
  else
    Matrix(size_A);
  end if;
end proc:

A := Matrix(size_A, size_A, helper2);
B := Matrix(size_B, size_B, helper3);
B := Matrix(size_B,size_B, helper3, datatype = float);
B := Matrix(size_B*size_A, size_B*size_A,[Matrix(size_B,size_B,helper3)], datatype = float)

A := Matrix(2, 2, {(1, 1) = a, (1, 2) = b, (2, 1) = b, (2, 2) = a})

 

B := Matrix(3, 3, {(1, 1) = Matrix(2, 2, {(1, 1) = a, (1, 2) = b, (2, 1) = b, (2, 2) = a}), (1, 2) = Matrix(2, 2, {(1, 1) = -1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), (1, 3) = Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}), (2, 1) = Matrix(2, 2, {(1, 1) = -1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), (2, 2) = Matrix(2, 2, {(1, 1) = a, (1, 2) = b, (2, 1) = b, (2, 2) = a}), (2, 3) = Matrix(2, 2, {(1, 1) = -1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), (3, 1) = Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 0}), (3, 2) = Matrix(2, 2, {(1, 1) = -1, (1, 2) = 0, (2, 1) = 0, (2, 2) = -1}), (3, 3) = Matrix(2, 2, {(1, 1) = a, (1, 2) = b, (2, 1) = b, (2, 2) = a})})

 

Error, (in Matrix) unable to store 'Matrix(2, 2, {(1, 1) = a, (1, 2) = b, (2, 1) = b, (2, 2) = a})' when datatype=float[8]

 

Error, (in Matrix) unable to store 'Matrix(2, 2, {(1, 1) = a, (1, 2) = b, (2, 1) = b, (2, 2) = a})' when datatype=float[8]

 

test := Matrix(4, 4, [
                [Matrix([[1,2],[0,9]]), Matrix([[3,6],[0,9]])],
                [Matrix([[3,4],[7,8]]), Matrix([[7,6],[5,5]])]
               ]); # is converted to a matrix of floats

test := Matrix(4, 4, {(1, 1) = 1, (1, 2) = 2, (1, 3) = 3, (1, 4) = 6, (2, 1) = 0, (2, 2) = 9, (2, 3) = 0, (2, 4) = 9, (3, 1) = 3, (3, 2) = 4, (3, 3) = 7, (3, 4) = 6, (4, 1) = 7, (4, 2) = 8, (4, 3) = 5, (4, 4) = 5})

(2)


Hi

We know determinant of a square matrix A[ij] (i,j ∈ {1,2,3}) is equal to the following expression

det(A) = 1/6 * e[ijk] * e[pqr] * A[ip] * A[jq] * A[kr] 

in which e[ijk] is a third order Tensor (Permutation notation or Levi-Civita symbol) and has a simple form as follows:

e[mnr] = 1/2 * (m-n) * (n-r) * (r-m).

The (i,j) minor of A, denoted Mij, is the determinant of the (n − 1)×(n − 1) matrix that results from deleting row i and column j of A. The cofactor matrix of A is the n×n matrix C whose (i, j) entry is the (i, j) cofactor of A,

C[ij]= -1 i+j * M[ij]

A-1=CT/det(A)

The general form of Levi-Civita symbol is as bellow:

 

I want to write a program for finding inverse of (NxN) matrix:

N=2 →

restart;
N := 2:
with(LinearAlgebra):
f := (1/2)*(sum(sum(sum(sum((m-n)*(p-q)*A[m, p]*A[n, q], q = 1 .. 2), p = 1 .. 2), n = 1 .. 2), m = 1 .. 2)):
A := Matrix(N, N, proc (i, j) options operator, arrow; evalf((37*i^2+j^3)/(2*i+4*j)) end proc):
f/Determinant(A);

N=3 →

restart;
N := 3:
with(LinearAlgebra):
f := (1/24)*(sum(sum(sum(sum(sum(sum((m-n)*(n-r)*(r-m)*(p-q)*(q-z)*(z-p)*A[m, p]*A[n, q]*A[r, z], m = 1 .. N), n = 1 .. N), r = 1 .. N), p = 1 .. N), q = 1 .. N), z = 1 .. N)):
A := Matrix(N, N, proc (i, j) options operator, arrow; 10*i^2/(20*i+j) end proc):
f/Determinant(A);

The results of above programs are equal to 1 and validation of method is observed.

If we can write the general form of determinant then we can find the inverse of any square non-singular matrices.

Now I try to write the mentioned program.

restart;
with(linalg):
N := 7:
Digits := 40:
e := product(product(signum(a[j]-a[i]), j = i+1 .. N), i = 1 .. N):
ML := product(A[a[k], b[k]], k = 1 .. N):
s[0] := e*subs(`$`(a[q] = b[q], q = 1 .. N), e)*ML:
for i to N do
s[i] := sum(sum(s[i-1], a[i] = 1 .. N), b[i] = 1 .. N)
end do:
A := Matrix(N, N, proc (i, j) options operator, arrow; evalf((3*i+j)/(i+2*j)) end proc): # arbitrary matrix
CN:=simplify(s[N]/det(A));

Therefore det(A)= CN-1 * e[a1,a2,..an] * e [b1,b2,.., bn] * A[a1,b1] * A[a2,b2] * ... * A[an,bn].

The correction coefficient is CN(for N)/CN(for N-1) = N!/(N-1)! =N.

restart:
with(linalg): N := 4: Digits := 20:
e := product(product(signum(a[j]-a[i]), j = i+1 .. N), i = 1 .. N):
ML := product(A[a[k], b[k]], k = 1 .. N):
s[0] := e*subs(`$`(a[q] = b[q], q = 1 .. N), e)*ML:
for r to N do s[r] := sum(sum(s[r-1], a[r] = 1 .. N), b[r] = 1 .. N) end do:
A := Matrix(N, N, proc (i, j) options operator, arrow; evalf((3*i+2*j)/(i+2*j)) end proc):
DET:=S[N]:
for x to N do for y to N do
e := product(product(signum(a[j]-a[i]), j = i+1 .. N-1), i = 1 .. N-1):
ML := product(AA[a[k], b[k]], k = 1 .. N-1):
S[0, x, y] := e*subs(`$`(a[q] = b[q], q = 1 .. N-1), e)*ML:
for r to N-1 do S[r, x, y] := sum(sum(S[r-1, x, y], a[r] = 1 .. N-1), b[r] = 1 .. N-1) end do:
f[y, x] := (-1)^(x+y)*subs(seq(seq(AA[t, u] = delrows(delcols(A, y .. y), x .. x)[t, u], t = 1 .. N-1), u = 1 .. N-1), S[N-1, x, y])
end do: end do:
Matrix(N, N, f)/(DET)*(24/6);
A^(-1);

CN for N=4 and N=3 is 24 and 6 respectively i.e. CN(4)/CN(3)=24/6.

When I use bellow procedure the error "(in S) bad index into Matrix" is occurred.

Please help me to write this algorithm by using procedure.

Thank you 

restart; with(linalg): Digits := 40: n := 7:
S := proc (N) local e, ML, s, i:
e := product(product(signum(a[j]-a[i]), j = i+1 .. N), i = 1 .. N):
ML := product(A[a[k], b[k]], k = 1 .. N):
s[0] := e*subs(`$`(a[q] = b[q], q = 1 .. N), e)*ML:
for i to N do s[i] := sum(sum(s[i-1], a[i] = 1 .. N), b[i] = 1 .. N) end do
end proc:
A := Matrix(n, n, proc (i, j) options operator, arrow; evalf((3*i+j)/(i+2*j)) end proc): # arbitrary matrix
CN := simplify(S(n)/det(A))

3 4 5 6 7 8 9 Last Page 5 of 46