Linear Equation Solutions: A Systematic Approach for n1xn2 and n1xm2 Matrices

[phynewb]

Hi guys

I wonder if you know any linear algebra formalism or something to solve the following question systematically?

Give A,B with A=n1xn2 matrix and B=n1xm2 matrix.
How do we get C=n2xm2 matrix such that A*C=B.
A simple example if A=(1,2)^t, B=(2,4)^t, then C=2

The question is how to solve C given A and B.
Thanks

 

[HallsofIvy]

That depends upon how many dimensions A and B have. In the example you give the simplest method is the write out the components and solve the resulting system of equations. Since A is "1 by 2" and B is "2 by 1", C must be a 1 by 1 matrix (a single number) and AC= B becomes
$$\begin{bmatrix}1 \\ 2\end{bmatrix}\begin{bmatrix}c\end{bmatrix}= \begin{bmatrix}c \\ 2c\end{bmatrix}= \begin{bmatrix}2 \\ 4\end{bmatrix}$$
which gives the two equations c= 2 and 2c= 4 so that c= 2. Of course, if B has NOT been a simple multiple of A, there would not have been any solution.

 

[AlephZero]

If you think about how matrix multiplication AC is defined, the i'th column of C only affects the i'th column of B.

So you only need to consider the simpler problem where C and B are vectors.

This is just a set of linear equations, which may have zero, one, or more than one solution depending on the row and column dimensions of A and the rank of A. The details should be covered in any course on linear algebra, or numerical methods for solving linear equations.

 

[phynewb]

Thanks AlephZero and HallsofIvy.

I guess the example is too simple.
I like to consider the general case.
Here is the way I solve it.
Want to solve C with AC=B.
Multiply A^t on both sides A^t.A.C=A^t.B
Now A^t.A^t is a square matrix so I can calculate its inverse (if it is not singular)
Then I get C=(A^t.A)^-1.A^t.B (1)

However if I calculate C in this way, something goes wrong!
For example,
Say A=[1,2]^t, B=[[1,0,2],[2,1,0]]
By(1) C=1/5[5,2,2]
But A.C=1/5[[5,2,2],[10,4,4]]\=B.
So I wonder what is wrong with (1).
Why does (1) get correct C such that A.C=B?

 

[blue_raver22]

for your question! There are several methods for solving linear equations involving matrices. One systematic approach is to use the Gaussian elimination method, which involves performing row operations on the augmented matrix [A|B] until it is in row-echelon form. This will give you a system of equations that can be easily solved.

Another approach is to use matrix inversion, where you find the inverse of matrix A and then multiply it by B to get C. However, this method may not always be feasible if A is not invertible.

There are also other techniques such as using the LU decomposition or the QR decomposition, which can be helpful in solving linear equations involving matrices.

In summary, there are various systematic approaches that can be used to solve linear equations involving matrices. It ultimately depends on the specific problem and the properties of the matrices involved. I hope this helps!