Help proving equality of matrix expressions

[Chuck37]

I've been beating my head against this problem for hours. I see numerically that two expressions are equal, but I can't prove it:

(J$^{T}$R$^{-1}$J + P$^{-1}$)$^{-1}$J$^{T}$R$^{-1}$

=

PJ$^{T}$(JPJ$^{T}$ + R)$^{-1}$

J is arbitrary size, P and R are square though not necessarily equal. Can anyone help? I thought binomial inverse theorem would save me but I haven't been able to get rid of the extraneous terms.

 

[Chuck37]

Well, right after I posted I tried a new tack of setting them equal and doing operations to both sides until it came out the same. Not quite as satisfying as transforming one into the other, but I guess it works. If someone can see how to make one into the other, I'd love to see it.

 

[AlephZero]

This is equivalent to

J$^{T}$R$^{-1}$ JPJ$^{T}$ + J$^{T}$
=
J$^{T}$R$^{-1}$ JPJ$^{T}$ + J$^{T}$

Factorize it two different ways:

J$^{T}$R$^{-1}$ (JPJ$^{T}$ + R)
=
(J$^{T}$R$^{-1}$J + P$^{-1}$) PJ$^{T}$

And the final step to get to your equation should be obvious.

(Of course this was invented by working backwards.)

 

[Fredrik]

Well, right after I posted I tried a new tack of setting them equal and doing operations to both sides until it came out the same. Not quite as satisfying as transforming one into the other, but I guess it works. If someone can see how to make one into the other, I'd love to see it.

As you just said, you don't actually have to "transform one into the other" to prove this result. However, it's not hard to just rewrite your left-hand side until you end up with the right-hand side. You just have to use the identity $(AB)^{-1}=B^{-1}A^{-1}$ repeatedly (and the fact that matrix multiplication is distributive over matrix addition). I recommend that you prove this identity first. It's not hard.

I can't tell you the complete solution because of the forum rules about textbook-style problems, so for now I will only suggest that you start by applying this identity to the factor $J^TR^{-1}$. To get more help, you will have to show your work up to the point where you get stuck.

By the way, you shouldn't put itex tags around each symbol. Instead, put an opening tag (itex or tex) before each formula and a closing tag after it. See the LaTeX guide for more information.Edit: The method I suggested only works when J is invertible (and therefore square). When J isn't square, I think you have to do something like what AlephZero suggested.

 

[blue_raver22]

It can be challenging to prove equality between two matrix expressions, especially when dealing with arbitrary sizes and different matrices. I understand that you have been working on this problem for hours and have not been able to find a solution using the binomial inverse theorem. It is important to remember that proving equality between matrix expressions requires a rigorous mathematical approach and may not always be straightforward.

One possible way to approach this problem is to start by simplifying both expressions as much as possible. For example, you can try expanding the left-hand side expression using the binomial inverse theorem and see if that leads you to a simpler form. Similarly, you can try manipulating the right-hand side expression to make it closer to the left-hand side expression.

Another helpful strategy is to use properties of matrices, such as the associative, distributive, or commutative properties, to rearrange the expressions and potentially simplify them. Additionally, you can try using known identities or theorems related to matrices to see if they can be applied to your problem.

If you are still having trouble proving equality, it may be helpful to seek assistance from a colleague or a math tutor who has experience with matrix operations and proofs. They may be able to offer a fresh perspective and provide guidance on how to approach the problem.

In conclusion, proving equality between matrix expressions can be a challenging task, but with patience, perseverance, and the use of various mathematical techniques, it is possible to find a solution. I hope these suggestions will help you in your pursuit of proving the equality of the two expressions.