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### Equivalent Matrices

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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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Definitions
College Linear Algebra

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