Page 478 - Applied Numerical Methods Using MATLAB

ROW ECHELON FORM 467
B.8 PERMUTATION MATRIX

Amatrix P having only one nonzero element of value 1 in each row and column
is called a permutation matrix and has the following properties.

ž Premultiplication/postmultiplication of a matrix A by a permutation matrix
P (i.e., PA or AP ) yields the row/column change of the matrix A,respec-
tively.
T
ž A permutation matrix A is orthogonal, that is, A A ≡ I.

B.9 RANK

The rank of an M × N matrix is the number of linearly independent
rows/columns and if it equals min(M, N), then the matrix is said to be of
maximal or full rank; otherwise, the matrix is said to be rank-deficient or to
have rank-deficiency.

B.10 ROW SPACE AND NULL SPACE

The row space of an M × N matrix A, denoted by R(A), is the space spanned
by the row vectors—that is, the set of all possible linear combinations of row
T
vectors of A that can be expressed by A α with an M-dimensional column vector
α. On the other hand, the null space of the matrix A, denoted by N(A),isthe
space orthogonal (perpendicular) to the row space—that is, the set of all possible
linear combinations of the N-dimensional vectors satisfying Ax = 0.

B.11 ROW ECHELON FORM

Amatrixissaidtobeof row echelon form if
ž Each nonzero row having at least one nonzero element has a 1 as its first
nonzero element.
ž The leading 1 in a row is in a column to the right of the leading 1 in the
upper row.
ž All-zero rows are below the rows that have at least one nonzero element.

Amatrixis saidtobeof reduced row echelon form if it satisfies the above
conditions and, additionally, each column containing a leading 1 has no other
nonzero elements.