Difference between revisions of "Matrix"
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== Matrix Product == | == Matrix Product == | ||
− | + | Let <math>A</math> be a matrix of order <math>m_1 \times m_2</math> and <math>B</math> a matrix of order <math>n_1 \times n_2</math>. Then the product <math>AB</math> exists if and only if <math>m_2=n_1</math> and in that case we define the product <math>C=AB</math> as the matrix of order <math>m_1 \times n_2</math> for which | |
+ | <cmath>(C)_{ij}=\sum ^{n_1} _{k=1} (A)_{ik} (B)_{kj}</cmath> | ||
+ | for all <math>i</math> and <math>j</math> such that <math>1\le i\le m_1</math> and <math>1\le j\le n_2</math>. | ||
== Vector spaces associated with a matrix == | == Vector spaces associated with a matrix == |
Latest revision as of 14:29, 30 March 2013
A matrix over a field is a function from to , where and are the sets and . A matrix is usually represented as a rectangular array of scalars from the 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]Determinant
If is a matrix over with , a Determinant assigns to a member of and is denoted by or
It is defined recursively.
where is the matrix with the row and column removed.
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
Let be a matrix of order and a matrix of order . Then the product exists if and only if and in that case we define the product as the matrix of order for which for all and such that and .
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 .