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I have the following code:

COV1 = {{a1,0},{0,a2}}; COV2 = {{b1,0},{0,b2}};
T = MatrixPower[MatrixPower[COV1,1/2],-1]*MatrixPower[MatrixPower[COV1,1/2]*COV2*MatrixPower[COV1,1/2],1/2]*MatrixPower[MatrixPower[COV1,1/2],-1];
Thalf = 1/2*{{1,0},{0,1}} + 1/2*T;
COV3 = Thalf*COV1*Transpose[Thalf]+{{0,t},{t,0}}
d1 = Sqrt[Tr[COV1 + COV2 - 2*MatrixPower[MatrixPower[COV1,1/2]*COV2*MatrixPower[COV1,1/2],1/2]]]
d2 =Sqrt[Tr[COV1 + COV3 - 2*MatrixPower[MatrixPower[COV1,1/2]*COV3*MatrixPower[COV1,1/2],1/2]]]*2
Simplify[d2 - d1]

When I plug in numbers such as

a1 = 5, a2 = 7, b1 = 9, b2 = 45

The last expression evaluates to zero. However when I leave everything as variable, the expresssion doesn't reduce to zero. However, it should. What is going on?

My guess is I need to impose conditions like a1, a2, b1, b2 are strictly positive reals -- but I don't know how to do this.

Also I tried this powerExpand trick I found -- it didn't work (perhaps I used it incorrectly?)

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    $\begingroup$ Simplify[d2 - d1, Assumptions -> Element[{a1, a2, b1, b2}, Reals]] $\endgroup$
    – Sumit
    Commented Sep 8, 2020 at 15:00
  • $\begingroup$ ah, this still didnt work $\endgroup$
    – yoshi
    Commented Sep 8, 2020 at 15:09
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    $\begingroup$ Your final expression is quite complicated. Perhaps mathematica can't do further simplification. You can try FullSimplify and Assumptions->{a1>0, a2>0,b1>0,b2>0}. $\endgroup$
    – Sumit
    Commented Sep 8, 2020 at 15:13
  • $\begingroup$ ya, i tried fixing 3 variables, (a2 = 3, b1 = 5, b2 = 7) and working with the resulting expression -- it still doesn't simplify to zero $\endgroup$
    – yoshi
    Commented Sep 8, 2020 at 15:40
  • $\begingroup$ @yoshi for a given assignment you can do RootReduce[ FullSimplify[(d2 - d1) /. {b1 -> 5, b2 -> 3}, (a1 | a2) \[Element] PositiveReals] /. {a1 -> 4, a2 -> 6} ]. $\endgroup$
    – flinty
    Commented Sep 8, 2020 at 17:06

1 Answer 1

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Since a1,a2,b1,b2 are all positive, we can substitute them with other numbers $x^2, y^2, z^2, w^2$.

FullSimplify[d2 - d1 /. {a1 -> x^2, a2 -> y^2, b1 -> z^2, b2 -> w^2},
 Element[{x,y,z,w}, PositiveReals]]

(* result: 0 *)
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  • $\begingroup$ i think there's a bracket missing at the end, but ya, thanks! $\endgroup$
    – yoshi
    Commented Sep 8, 2020 at 17:35
  • $\begingroup$ @yoshi thanks, fixed. $\endgroup$
    – flinty
    Commented Sep 8, 2020 at 17:40

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