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Equivalent Matrices


Date: 7/12/96 at 1:1:29
From: Anonymous
Subject: Equivalent Matrices

Dr. Math,

I was just brushing up on my linear algebra and I ran across 
equivalent matrices again. I had problems with this when I went 
through high school. Now working on my senior thesis in particle 
physics, I need to use some equivalent matrices to get where I need to 
go. Could you assist this member of the U.S. Air Force?


Date: 7/12/96 at 11:2:24
From: Doctor Anthony
Subject: Re: Equivalent Matrices

If you consider a matrix M1 as an operator mapping a set of elements 
into itself, and M2 another matrix also mapping the set of elements, 
then the product M2*M1 of the operators M1 and M2 in that order is 
defined by the relation

       (M2*M1)x = M2{(M1)x}

In other words, (M2*M1)x denotes the element obtained when x is first 
operated on by M1 and (M1x) is then operated on by M2.  This idea can 
be extended to any number of matrices  (M4*M3*M2*M1)x, so that a 
series of operations can be represented by the product of matrices (in 
reverse order), that is M1 is the first operation and M4 the last.

We now consider E-matrices, which correspond to ELEMENTARY operations.

An elementary operation on a matrix is an operation of one of the 
following three types:

  (i) The interchange of two rows (or columns)
 (ii) The multiplication of a row (or column) by a non-zero scalar.
(iii) The addition of a multiple of one row (or column) to another row 
      (or column).

We distinguish between row operations and column operations according 
as E-operations in question apply to rows or to columns.

An elementary matrix is any matrix derived from a unit matrix by a 
single E-operation.

Examples  |0  1  0|      |1  0  0|      |1  0  0|
          |1  0  0|      |0  5  0|      |0  1  0|
          |0  0  1|      |0  0  1|      |3  0  1|

In the first of these we have interchanged rows 1 and 2.
In the second we have multiplied row 2 by 5.
In the third we have added 3*row 1 to row 3.  

If we have a 3*4 matrix |a1  a2  a3  a4|
                        |b1  b2  b3  b4|
                        |c1  c2  c3  c4|

and wish to interchange rows 2 and 3, we premultiply by the E-matrix

|1  0  0| |a1  a2  a3  a4|   |a1  a2  a3  a4|
|0  0  1|*|b1  b2  b3  b4| = |c1  c2  c3  c4| 
|0  1  0| |c1  c2  c3  c4|   |b1  b2  b3  b4|

If we wish to do column operations, we post-multiply by the 
appropriate E-matrix.

We can finally get onto equivalent matrices.

A matrix A is EQUIVALENT to a matrix B if it is possible to pass from 
A to B by a chain of E-operations.

The relation of equivalence defined here meets the usual criteria for 
equivalence relations, namely 

  (i) reflexive   A equiv. A
 (ii) symmetric   A equiv B implies B equiv A.
(iii) transitive  A equiv B,  B equiv C, implies A equiv C.


-Doctor Anthony,  The Math Forum
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