« previous | Tuesday, September 11, 2012 | next »
System of equations now in form , where is , and and are
For a non-singular, invertible matrix , , and this system will be equivalent to the original.
could even be product of several matrices: , where is an elementary matrix.
Types of elementary matrices (each corresponds to rules of matrix):
- obtained by interchanging two rows of identity matrix
- permutes rows
- permutes columns
- obtained by multiplying row of identity matrix by a nonzero constant
- scales corresponding row by
- scales corresponding column by .
- obtained from by adding a multiple of one row to another row :
- adds multiple of row to row
- adds multiple of column to column
Note: (and thus ) is always a square matrix, but can be of any size
is a matrix: we can think of it as being obtained from by either a row operation or column operation. If is a matrix, premultiplying by has the effect of performing the same row operation on . If is a matrix, postmultiplying by is equivalent to performing that same column operation on .
If is an elementary matrix, then is nonsingular and is an elementary matrix of the same type as
- , since interchanging two rows twice undoes the effect, so
- if has an at position . will have at
- has a at position , where . will have at . In multiplying by , position will be product of th row and th column:
is row equivalent to (written iff there exists a finite sequence of elementary matrices such that . In other words, can be obtained from by a finite number of row operations.
By extension, two augmented matrices and are row equivalent iff and are equivalent systems.
Equivlent Conditions for Nonsingularity
Let be a matrix. Then the following are equivalent:
- is nonsingular
- has only trivial solution
- is row equivalent to identity matrix of size ()
(1) → (2): Let be solution of . So can become . , so
(2) → (3): Rewrite in row echelon form as . If one diagonal entry of is 0, then there is at least one free variable and infinitely many solutions, one of which is nonzero. This leads to a contradiction, so all diagonal entries of must be 1, so rref will be identity matrix.
(3) → (1): If is row equivalent to , then there exists a sequence of elementary matrices such that . Therefore, (note reverse order), so is nonsingular.
The system of equations has a unique solution iff is nonsingular.
- is nonsingular, can be rewritten as
- Assume unique solution exists. If were singular then homogeneous system would have a nonzero solution (by Theorem 1.4.2). If this were the case, ... there would be another solution, which is a contradiction.