March 28th, 2018, 09:36 PM  #1 
Member Joined: Nov 2016 From: Ireland Posts: 84 Thanks: 3  Reversing a matrix
Ok I think this should be easy, if I can explain it properly. Let's say I had a 2*2 matrix = matrix A And I was multiplying it 2*1 matrix = matrix B And it would produce a 2*1 matrix = matrix C Right? But let's say  I want to go back ways, reverse it... I have Matrix C and I have Matrix A but I need to find Matrix B. I'd better explain this as well, as I'm not sure the rules are exactly the same. I'm using a nominal Pi model to analyse a transmission line The layout is like this Vs  A B  Vr Is  C D  Ir Where Vs = A*Vr + B*Ir And Is = C*Vr + D*Ir But now I need to find Vr and Ir I have ABCD and I have Vs and Is At first I thought well I'll just rearrange the equation to find Vr. But 2 of the variables are missing, when I do that... Any help would be great guys. Thx 
March 29th, 2018, 06:05 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
So basically, you want to solve a matrix equation of the form AB= C for B? If A is invertible then $\displaystyle B= A^{1}C$. If A is not invertible then there is no solution. Write $\displaystyle A= \begin{bmatrix}a & b \\ c & d \end{bmatrix}$, $\displaystyle B= \begin{bmatrix}x \\ y \end{bmatrix}$, and $\displaystyle C= \begin{bmatrix}p \\ q \end{bmatrix}$. A is invertible if and only if its determinant, ad bc is not 0 and, in that case, its inverse is $\displaystyle A^{1}= \frac{1}{ad bc}\begin{bmatrix}d & b \\ c & a\end{bmatrix}$ and $\displaystyle B= A^{1}C= \frac{1}{ad bc}\begin{bmatrix}d & b \\ c & a\end{bmatrix}\begin{bmatrix}p \\ q \end{bmatrix}= \frac{1}{ad bc}\begin{bmatrix}dp bq \\ aq cp\end{bmatrix}$. Or you could simply treat the matrix equation as a system of two equations in two unkowns. We have $\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}= \begin{bmatrix}ax+ by \\ cx+ dy\end{bmatrix}= \begin{bmatrix}p \\ q\end{bmatrix}$. That is equivalent to the two equations ax+ by= p and cx+ dy= q. Multiply the first equation by d to get adx+ bdy= pd and multiply the second equation by b to get bcx+ bdy= bq. The coefficents of y are now the same, bd, so subtracting one equation from the other eliminates y: (ad bc)x= pd bq. If ad bc is not 0, divide both sides by ad bc to get $\displaystyle x= \frac{pd bq}{ad bc}$ as before. If ad bc= 0 and pd bq is not 0, there is no x that makes that equation true. If both ad bc= 0 and pd bq= 0 then any x works. Much the same thing is true of $\displaystyle V_s = AV_r + BI_r$ and $\displaystyle I_s = CV_r + DI_r$. Multiply the first equation by D to get $\displaystyle DV_s= ADV_r+ BDI_r$ and multiply the second equation by B to get $\displaystyle BI_s= BCV_r+ BDI_r$. Now $\displaystyle I_r$ has the same coefficient in both equations so subtracting one equation from the other eliminates $\displaystyle I_r$: $\displaystyle DV_s BI_s= (AD BC)V_r$ and $\displaystyle V_r= \frac{DV_s BI_s}{AD BC}$, assuming, of course, that AD BC is nonzero. Then $\displaystyle V_s= AV_r+ BI_r= \frac{ADV_s ABI_s}{AD BC}+ BI_r$ so $\displaystyle BI_r= V_s \frac{ADV_S ABI_s}{AD BC}= \frac{ADV_s BCV_s ADV_s+ ABI_s}{AD BC}= \frac{ABI_s BCV_s}{AD BC}$. 
March 29th, 2018, 07:59 AM  #3 
Senior Member Joined: Oct 2009 Posts: 906 Thanks: 354  Why wouldn't there be a solution if A is not invertible? Sure, there isn't always a solution for all A and C. But some choices certainly will have (multiple) solutions!

March 29th, 2018, 11:14 AM  #4  
Member Joined: Nov 2016 From: Ireland Posts: 84 Thanks: 3 
Beautiful Country Boy. I've written it all out, substituting in numbers on one side, with the equations on the other. I think I understand most of it now. This bit at the end however, seems to be off by a factor of 2. Quote:
$\displaystyle V_s = AV_r + BI_r$ = $\displaystyle 92 = 12*3 + 14*4$ and $\displaystyle I_s = CV_r + DI_r$ = $\displaystyle 85 = 11*3 + 13*4$ Therefore: $\displaystyle BI_r = 28 $ But: $\displaystyle (ABI_s  BCV_s) / (ADBC) $ $\displaystyle 112/2 = 56$ Last edited by Kevineamon; March 29th, 2018 at 11:21 AM.  
March 29th, 2018, 12:12 PM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  Yes, I should have said right at the start that there is not a unique solution. If A is not invertible then either there is no solution or there are an infinite number of solutions. I did later say that "If ad bc= 0 and pd bq is not 0, there is no x that makes that equation true. If both ad bc= 0 and pd bq= 0 then any x works."

March 29th, 2018, 08:06 PM  #6 
Member Joined: Nov 2016 From: Ireland Posts: 84 Thanks: 3 
How about my example Country Boy... Those numbers don't meet any of you conditions. However: $\displaystyle BI_r ≠ (ABI_s−BCV_s)/(AD−BC)$ Although it is off by a factor of 2, which is the answer I'm looking for... Am I supposed to guess something here? My math is Ok, when I work on it. But it takes a lot of effort. I'm not naturally gifted with it, like your good self. My mathematically intuition = 0 Can you help me take this final step? 
March 29th, 2018, 10:07 PM  #7  
Senior Member Joined: Sep 2016 From: USA Posts: 684 Thanks: 459 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
The fact is that if $A$ has full rank, then $AB = C$ may have a unique solution even if $A$ is not invertible. This is easily seen by noticing that $A^TA$ is always invertible where $A^T$ is the conjugate transpose. Now suppose $Ax = b$ where $x,b$ are vectors, then you can easily see that a formula for $x$ is given by \[x = (A^TA)^{1} A^T b. \] In this case, $(A^TA)^{1}A^T$ is the weak inverse for $A$ which is also a left inverse (sometimes called a pseudoinverse). Notice that if $A$ is not square, then it doesn't even have a determinant, but that has nothing to do with the question of invertibility.  
March 30th, 2018, 07:31 AM  #8 
Member Joined: Nov 2016 From: Ireland Posts: 84 Thanks: 3 
While this high brow mathematically discussion is, I can guess... absolutely fascinating. There's a n00b engineer over here with a transmission line to analyse. Can one of you geniuses, climb down from the lofty tower and explain to lil old me, how to find $\displaystyle I_r$? Please...

March 30th, 2018, 07:45 AM  #9 
Member Joined: Nov 2016 From: Ireland Posts: 84 Thanks: 3 
Wait... Inspiration has come... I think...

March 30th, 2018, 08:10 AM  #10 
Member Joined: Nov 2016 From: Ireland Posts: 84 Thanks: 3 
$\displaystyle I_r= (CV_sAI_s)/(CBAD)$ Well I think your math is great, Country Boy. Never mind the haters. 

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