derivative of matrix in mathematica

[yashar]

hi
in mathematica
how i can produce a general n by n matrix so that each of its elements be a variable of t???
how i can take derivative of this matrix with respect to variable t?
how i can with equating this n by n matrix with another n by n matrix which its elements are known obtain unknown elements of this general n by n matrix?

[Bill Rowe]

On 7/9/11 at 7:32 AM, physi...@yahoo.com (yashar) wrote:

>in mathematica
>how i can produce a general n by n matrix so that each of its elements be a variable of t???

Any number of ways. It depends on what it is you want. An couple
of examples would be:

In[1]:= m = t RandomInteger[5, {2, 2}]

Out[1]= {{0, 5*t}, {2*t, 0}}

In[2]:= m1 = t^RandomInteger[5, {2, 2}]

Out[2]= {{t, 1}, {t^3, t^2}}

>how i can take derivative of this matrix with respect to variable t?

The same way as taking the derivative of any function with
respect to t. Using the examples above;

In[3]:= D[m, t]

Out[3]= {{0, 5}, {2, 0}}

In[4]:= D[m1, t]

Out[4]= {{1, 0}, {3*t^2, 2*t}}

>how i can with equating this n by n matrix with another n by n matrix which its elements are known obtain unknown elements of this general n by n matrix?

Here, I am not clear on what it is you want to do. I can find a
solution by setting corresponding elements equal as follows

In[5]:= y = RandomInteger[5, {2, 2}];
MapThread[Solve[#1 == #2, t] &, {Flatten@y, Flatten@m}]

Out[6]= {{}, {{t -> 1}}, {{t -> 2}}, {{}}}

But it is trivial to find examples where this won't work.

[Heike Gramberg]

You could do something like

n = 3;
matrix := Table[p[i, j][#], {i, 1, n}, {j, 1, n}] &

then matrix'[t] gives

{{Derivative[1][p[1, 1]][t], Derivative[1][p[1, 2]][t],
Derivative[1][p[1, 3]][t]}, {Derivative[1][p[2, 1]][t],
Derivative[1][p[2, 2]][t],
Derivative[1][p[2, 3]][t]}, {Derivative[1][p[3, 1]][t],
Derivative[1][p[3, 2]][t], Derivative[1][p[3, 3]][t]}}

To set each of the elements of this matrix equal to the elements of another matrix
you can do something like

MapThread[
SetDelayed, {matrix[t_], Table[Sin[i t] j, {i, 1, n}, {j, 1, n}]}, 2];

Heike

On 9 Jul 2011, at 12:32, yashar wrote:

> hi

> in mathematica
> how i can produce a general n by n matrix so that each of its elements be a variable of t???

> how i can take derivative of this matrix with respect to variable t?

[W. Craig Carter]

Hello Yashar,
Here is an example:
5x5 where each element is a function of t

fiveByfive = Array[matrixFuncs[#1, #2][t] &, {5, 5}]

The derivative of the matrix with respect to t
fiveByfive = Array[matrixFuncs[#1, #2][t] &, {5, 5}]

I am not sure what you are asking for with your third question, but perhaps this gives you a direction:

anotherfiveByfive = Array[otherFuncs[#1, #2][t] &, {5, 5}]
Table[fiveByfive[[i, j]] == anotherfiveByfive[[i, j]], {i, 5}, {j, 5}]