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Are all of the Fs real? If so, try this:

Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]] 

You'll get an answer, but it'll be ugly...

You can use Position to test for zero elements like this (in this case I'm applying it to your original matrix to show that it works):

Position[Ftemp,x_/;PossibleZeroQ[x]] {{1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 2}, {3, 4}, {4, 1}, {4, 2}, {4, 3}, {5, 1}, {5, 2}} 

So for the matrix you're interested in:

FtempInv = Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]]; Position[FtempInv,x_/;PossibleZeroQ[x]] 

Unfortunately, when I do that MMA spends a great deal of time thinking and I have yet to see an answer. There may be better test to use here than PossibleZeroQ; if so, I'm sure someone else will suggest one.

It turns out that PossibleZeroQ is Listable, so you can you just do

PossibleZeroQ[FtempInv] 

But that doesn't solve the speed problem...

Are all of the Fs real? If so, try this:

Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]] 

You'll get an answer, but it'll be ugly...

You can use Position to test for zero elements like this (in this case I'm applying it to your original matrix to show that it works):

Position[Ftemp,x_/;PossibleZeroQ[x]] {{1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 2}, {3, 4}, {4, 1}, {4, 2}, {4, 3}, {5, 1}, {5, 2}} 

So for the matrix you're interested in:

FtempInv = Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]]; Position[FtempInv,x_/;PossibleZeroQ[x]] 

Unfortunately, when I do that MMA spends a great deal of time thinking and I have yet to see an answer. There may be better test to use here than PossibleZeroQ; if so, I'm sure someone else will suggest one.

It turns out that PossibleZeroQ is Listable, so you can you just do

PossibleZeroQ[FtempInv] 

But that doesn't solve the speed problem...

I let PossibleZeroQ[FtempInv] run for a while. Here's what I got:

{{False, False, False, False, False}, {False, False, False, False, False}, {False, False, False, False, False}, {False, False, False, False, False}, {False, False, False, False, False}}

Are all of the Fs real? If so, try this:

Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]] 

You'll get an answer, but it'll be ugly...

You can use Position to test for zero elements like this (in this case I'm applying it to your original matrix to show that it works):

Position[Ftemp,x_/;PossibleZeroQ[x]] {{1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 2}, {3, 4}, {4, 1}, {4, 2}, {4, 3}, {5, 1}, {5, 2}} 

So for the matrix you're interested in:

FtempInv = Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]]; Position[FtempInv,x_/;PossibleZeroQ[x]] 

Unfortunately, when I do that MMA spends a great deal of time thinking and I have yet to see an answer. There may be better test to use here than PossibleZeroQ; if so, I'm sure someone else will suggest one.

Are all of the Fs real? If so, try this:

Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]] 

You'll get an answer, but it'll be ugly...

You can use Position to test for zero elements like this (in this case I'm applying it to your original matrix to show that it works):

Position[Ftemp,x_/;PossibleZeroQ[x]] {{1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 2}, {3, 4}, {4, 1}, {4, 2}, {4, 3}, {5, 1}, {5, 2}} 

So for the matrix you're interested in:

FtempInv = Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]]; Position[FtempInv,x_/;PossibleZeroQ[x]] 

Unfortunately, when I do that MMA spends a great deal of time thinking and I have yet to see an answer. There may be better test to use here than PossibleZeroQ; if so, I'm sure someone else will suggest one.

It turns out that PossibleZeroQ is Listable, so you can you just do

PossibleZeroQ[FtempInv] 

But that doesn't solve the speed problem...

Are all of the Fs real? If so, try this:

Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]] 

You'll get an answer, but it'll be ugly...

Are all of the Fs real? If so, try this:

Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]] 

You'll get an answer, but it'll be ugly...

You can use Position to test for zero elements like this (in this case I'm applying it to your original matrix to show that it works):

Position[Ftemp,x_/;PossibleZeroQ[x]] {{1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 2}, {3, 4}, {4, 1}, {4, 2}, {4, 3}, {5, 1}, {5, 2}} 

So for the matrix you're interested in:

FtempInv = Assuming[{F11, F12, F13, F21, F22, F24, F33, F35, F44, F45, F53, F54, F55} \[Element] Reals, MatrixPower[Ftemp, 1/2]]; Position[FtempInv,x_/;PossibleZeroQ[x]] 

Unfortunately, when I do that MMA spends a great deal of time thinking and I have yet to see an answer. There may be better test to use here than PossibleZeroQ; if so, I'm sure someone else will suggest one.