Difference between revisions of "Matrix"
Line 12: | Line 12: | ||
As already stated before, the columns of <math>A</math> form a subset of <math>F^m</math>. The subspace of <math>F^m</math> generated by these columns is said to be the column space of <math>A</math>, written as <math>C(A)</math>. Similarly, the transposes of the rows form a subset of the vector space <math>F^n</math>. The subspace of <math>F^n</math> generated by these is known as the row space of <math>A</math>, written as <math>R(A)</math>. | As already stated before, the columns of <math>A</math> form a subset of <math>F^m</math>. The subspace of <math>F^m</math> generated by these columns is said to be the column space of <math>A</math>, written as <math>C(A)</math>. Similarly, the transposes of the rows form a subset of the vector space <math>F^n</math>. The subspace of <math>F^n</math> generated by these is known as the row space of <math>A</math>, written as <math>R(A)</math>. | ||
+ | |||
+ | <math>y \in C(A) \Leftrightarrow \exists x <\math><math>\ni y_{m \times 1} = A_{{m \times n} x_{n \times 1}</math> | ||
+ | |||
+ | == Rank and nullity == | ||
+ | |||
+ | The dimension of |
Revision as of 07:53, 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
.
$y \in C(A) \Leftrightarrow \exists x <\math><math>\ni y_{m \times 1} = A_{{m \times n} x_{n \times 1}$ (Error compiling LaTeX. Unknown error_msg)
Rank and nullity
The dimension of