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Finding the inverse of a diagonal matrix

Finding the inverse of a diagonal matrix

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Finding the inverse of a diagonal matrix

Postby barnamah » Wed 25 May, 2011 16:39

A diagonal matrix is one which all entries other than diagonal are zero. Here is example:

Code: Select all
D =

     5     0     0
     0     3     0
     0     0     1


As you can see in the diagonal we have 5 3 1 and zero everywhere.

finding the inverse of diagonal matrix
To find the inverse of the matrix, invert each entry of the diagonal . Here inverting 5 which is `5/1` becomes `1/5` and 3 becomes `1/3` and 1 becomes `1/1=1`

Code: Select all
      1/5            0             0     
       0            1/3            0     
       0             0             1 


`D*D^-1=D^(-1)*D=I`
It says if we multiply a diagonal matrix to its inverse the product equals identity matrix.
To prove this, let's look. If we multiply D by `D^(-1)` , we are actually multiplying all diagonal entries. If you look at the diagonal of the inverse matrix, we see we have `1/5`, `1/3` and 1.
and in our D matrix we have 5, 3, 1. Multiplying `5*1/5=1` and `3*1/3=1` and 1*1=1 so we get Identity matrix :

Code: Select all
I=
     1     0     0
     0     1     0
     0     0     1
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