Difference between revisions of "Matrix"
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<math>A</math> is said to be symmetric if and only if <math>A=A^T</math>. <math>A</math> is said to be hermitian if and only if <math>A=A^*</math>. <math>A</math> is said to be skew symmetric if and only if <math>A=-A^T</math>. <math>A</math> is said to be skew hermitian if and only if <math>A=-A^*</math>. | <math>A</math> is said to be symmetric if and only if <math>A=A^T</math>. <math>A</math> is said to be hermitian if and only if <math>A=A^*</math>. <math>A</math> is said to be skew symmetric if and only if <math>A=-A^T</math>. <math>A</math> is said to be skew hermitian if and only if <math>A=-A^*</math>. | ||
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+ | == Matrix Product == | ||
+ | |||
+ | If <math>A</math> is of order <math>m_1 \times n</math> and <math>B</math> is of order <math>n \times m_2</math>, <math>C_{m_1 \times m_2}</math> is said to be <math>AB</math> if and only if <math>(C)_{ij}=\displaystyle \sum ^n _{k=1} (A)_{ik} (B)_{kj}</math> | ||
== Vector spaces associated with a matrix == | == Vector spaces associated with a matrix == |
Revision as of 14:48, 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
.
Contents
[hide]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
.
Matrix Product
If is of order
and
is of order
,
is said to be
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
.