Why doesn't mathematica replace $\frac{I_0(\kappa)}{I_0(\kappa)}$ by 1

Question

I have a matrix of the form:
$$\left( \begin{array}{ccc} 1 & \frac{\delta I_1(\kappa )}{I_0(\kappa )} & 0 \\ \frac{I_1(\kappa ) \delta ^*}{I_0(\kappa )} & \frac{\delta (I_1(\kappa )+\kappa I_2(\kappa )) \delta ^*}{\kappa I_0(\kappa )} & 0 \\ 0 & 0 & \frac{\delta I_1(\kappa ) \delta ^*}{\kappa I_0(\kappa )} \\ \end{array} \right)$$
That I want to convert to the form:
$$\begin{pmatrix}1&g_c\delta&0\\g_c\delta^*&\frac{(1+g)}{2}|\delta|^2&0\\0&0&\frac{(1-g)}{2}|\delta|^2\end{pmatrix}$$
where $g_c=\frac{I_1(\kappa)}{I_0(\kappa)}$ and $g=\frac{I_2(\kappa)}{I_0(\kappa)}$
For the time being I don't speak about converting $\delta\delta^*$ to $|\delta|^2$. I've asked that question here
But about the other parts, I've used the following code:

Tvol[[1, 2]] /. {BesselI[1, κ]/BesselI[0, κ] -> Subscript[g, c]} // Simplify // TraditionalForm

$$\delta g_c$$

Tvol[[2, 1]] /. {BesselI[1, κ]/BesselI[0, κ] -> Subscript[g, c]} // Simplify // TraditionalForm  

$$g_c \delta ^*$$

Tvol[[2, 2]] /. {BesselI[1, κ] -> κ/2 (BesselI[0, κ] - BesselI[2, κ])} // FullSimplify // TraditionalForm  

$$\frac{\delta \delta ^* (I_0(\kappa )+I_2(\kappa ))}{2 I_0(\kappa )}$$

Tvol[[2, 2]] /. {BesselI[2, κ] -> g*BesselI[0, κ]} //FullSimplify // TraditionalForm  

$$\delta \delta ^* \left(g+\frac{I_1(\kappa )}{\kappa I_0(\kappa )}\right)$$
I wonder why at this step mathematica doesn't simply replace $\frac{I_0(\kappa)}{I_0(\kappa)}=1$ and $\frac{I_2(\kappa)}{I_0(\kappa)}=g$ to reach to the simple form:
$$\delta \delta ^*\frac{ 1+g}{2}$$
and returns back to $I_1(\kappa)$ ?
I have changed the code several times but none of them helped!!!

Answer 1

I'm voting to close this question, but I think you could use a little guidance here nonetheless, so I am posting this as CW. It is important that you learn the basic ins-and-outs of MMA. I'm sure you've been directed there before, but make sure you carefully peruse What are the most common pitfalls awaiting new users?

I feel like maybe, in this case, the particular problem is that you are Assuming commands will have side effects which they don't:

The /. command (otherwise known as ReplaceAll) does not have the side effect of writing over the original expression. For instance, evaluate

Clear[a, b]
a = b;
a /. b -> 1
a

and carefully try to understand the results.

In your case, you need to either update Tvol at each stop, or do both replacements at once, and what you choose to do depends on what you need to do later. Either:

Tvol[[2, 2]] = Tvol[[2, 2]] /. {BesselI[1, κ] -> κ/2 (BesselI[0, κ] - BesselI[2, κ])} // FullSimplify
Tvol[[2, 2]] = Tvol[[2, 2]] /. {BesselI[2, κ] -> g*BesselI[0, κ]} //FullSimplify

or

Tvol[[2, 2]] //. {BesselI[2, κ] -> g*BesselI[0, κ], BesselI[1, κ] -> κ/2 (BesselI[0, κ] - BesselI[2, κ])} // FullSimplify

will result in the expression that you want. The first will change the the value of Tvol, while the second will not.