418 VECTORS AND MATRICES [APP. A
A.10Elementary Row Operations, Gaussian Elimination (Optional)
This section discusses the Gaussian elimination algorithm in the context of elementary row operations.Elementary Row Operations
Consider a matrixA=[aij]whose rows will be denoted, respectively, byR 1 ,R 2 ,...,Rm. The first nonzero
element in a rowRiis called theleadingnonzero element. A row with all zeros is called azero row. Thus a zero
row has no leading nonzero element.
The following three operations onAare called theelementary row operations:
[E 1 ] Interchange rowRiand rowRj. This operation will be indicated by writing: “InterchangeRiandRj.”
[E 2 ] Multiply each element in a rowRiby a nonzero constantk. This operation will be indicated by writing:
“MultiplyRibyk.”
[E 3 ] Add a multiple of one rowRito another rowRjor, in other words, replaceRjby the sumkRi+Rj.
This operation will be indicated by writing: “AddkRitoRj.”
To avoid fractions, we may perform [E 2 ] and [E 3 ] in one step; that is, we may apply the following operation:
[E] Add a multiple of one rowRito a nonzero multiple of another rowRjor, in other words, replaceRjby
the sumkRi+k′Rjwherek′=0. We indicate this operation by writing: “AddkRitok′Rj.”We emphasize that, in the row operations [E 3 ] and [E], only rowRjis actually changed.
Notation:MatricesAandBare said to berow equivalent, writtenA∼B, if matrixBcan be obtained from matrixA
by using elementary row operations.
Echelon Matrices
A matrixAis called anechelon matrix, or is said to be inechelon form, if the following two conditions hold:
(i) All zero rows, if any, are on the bottom of the matrix.
(ii) Each leading nonzero entry is to the right of the leading nonzero entry in the preceding row.The matrix is said to be inrow canonical formif it has the following two additional properties:
(iii) Each leading nonzero entry is 1.
(vi) Each leading nonzero entry is the only nonzero entry in its column.
The zero matrix 0, for any number of rows or columns, is a special example of a matrix in row canonical
form. Then-square identity matrixInis another example of a matrix in row canonical form.
A square matrixAis said to be intriangular formif its diagonal entriesa 11 ,a 22 ,...,annare the leading
nonzero entries. Thus a square matrix in triangular form is a special case of an echelon matrix. The identity
matrixIis the only example of a square matrix which is in triangular form and in row canonical form.
EXAMPLE A.9 Consider the echelon matrices in Fig. A-4 whose leading nonzero entries have been circled.
(The zeros preceding and below the leading nonzero entries in an echelon matrix form a “staircase” pattern
as indicated above by the shading.) The third matrix is in row canonical form. The second matrix is not in
row canonical form since the third column contains a leading nonzero entry and another nonzero entry. The
first matrix is not in row canonical form since some leading nonzero entries are not 1. The last matrix is in
triangular form.
Fig. A-4