Matrix Equations

Matrix Equations

Different perspectives of a simple matrix equation of the form 

  which might expand to something like 

Geometric picture

The solution for  , ie the values of  that will satisfy the equation, is the point where the 2 lines, 3 planes etc.. meet.  This is the picture that is normally presented.

Column picture
This is a more interesting and less common perspective.  Think of each column of matrix  as a column vector.

Each of the column vectors of   are scaled by and then added to get .  So is the scaling factor for the first column,  is the scaling factor for second column... etc. 

For example if ,   will be equal to  which is the first column of .  

=

Isn't that nice?

Row picture
Better example to see what is going on here is to look at  

Here  is a scaling factor of the first row,  for the second row etc.   

For example if ,  will be  which is the first row of .

Eigenvalues and Eigenvectors