Difference between revisions of "Matrix"
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== Rank and nullity == | == Rank and nullity == | ||
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+ | The dimension of <math>C(A)</math> is known as the column rank of <math>A</math>. The dimension of <math>R(A)</math> is known as the row rank of <math>A</math>. These two ranks are found to be equal, and the common value is known as the rank <math>r(A)</math> of <math>A</math>. | ||
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+ | The dimension of <math>N(A)</math> is known as the nullity <math>\eta (A)</math> of A. | ||
+ | |||
+ | If <math>A</math> is a square matrix of order <math>n \times n</math>, then <math>r(A) + \eta (A) = n</math>. |
Revision as of 14:38, 5 November 2006
A matrix is a rectangular array of scalars from any field, such that each column belongs to the vector space , where
is the number of rows. If a matrix
has
rows and
columns, its order is said to be
, and it is written as
.
The element in the row and
column of
is written as
. It is more often written as
, in which case
can be written as
.
Transposes
Let be
. Then
is said to be the transpose of
, written as
or simply
. If A is over the complex field, replacing each element of
by its complex conjugate gives us the conjugate transpose
of
. In other words,
is said to be symmetric if and only if
.
is said to be hermitian if and only if
.
is said to be skew symmetric if and only if
.
is said to be skew hermitian if and only if
.
Vector spaces associated with a matrix
As already stated before, the columns of form a subset of
. The subspace of
generated by these columns is said to be the column space of
, written as
. Similarly, the transposes of the rows form a subset of the vector space
. The subspace of
generated by these is known as the row space of
, written as
.
implies
such that
Similarly, implies
such that
The set forms a subspace of
, known as the null space
of
.
Rank and nullity
The dimension of is known as the column rank of
. The dimension of
is known as the row rank of
. These two ranks are found to be equal, and the common value is known as the rank
of
.
The dimension of is known as the nullity
of A.
If is a square matrix of order
, then
.