Equivalent MatricesDate: 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/ |
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