Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Matrix"

(→‎Vector spaces associated with a matrix)
(→‎Rank and nullity)
Line 20: Line 20:
  
 
== Rank and nullity ==
 
== Rank and nullity ==
 +
 +
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>.
 +
 +
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 15:38, 5 November 2006

A matrix is a rectangular array of scalars from any field, such that each column belongs to the vector space $F^m$, where $m$ is the number of rows. If a matrix $A$ has $m$ rows and $n$ columns, its order is said to be $m \times n$, and it is written as $A_{m \times n}$.

The element in the $i^{th}$ row and $j^{th}$ column of $A$ is written as $(A)_{ij}$. It is more often written as $a_{ij}$, in which case $A$ can be written as $[a_{ij}]$.

Transposes

Let $A$ be $[a_{ij}]$. Then $[a_{ji}]$ is said to be the transpose of $A$, written as $A^T$ or simply $A'$. If A is over the complex field, replacing each element of $A^T$ by its complex conjugate gives us the conjugate transpose $A^*$ of $A$. In other words, $A^*=[\bar {a_{ji}}]$

$A$ is said to be symmetric if and only if $A=A^T$. $A$ is said to be hermitian if and only if $A=A^*$. $A$ is said to be skew symmetric if and only if $A=-A^T$. $A$ is said to be skew hermitian if and only if $A=-A^*$.

Vector spaces associated with a matrix

As already stated before, the columns of $A$ form a subset of $F^m$. The subspace of $F^m$ generated by these columns is said to be the column space of $A$, written as $C(A)$. Similarly, the transposes of the rows form a subset of the vector space $F^n$. The subspace of $F^n$ generated by these is known as the row space of $A$, written as $R(A)$.

$y \in C(A)$implies $\exists x$ such that $y_{m \times 1} = A_{m \times n} x_{n \times 1}$

Similarly, $y \in C(A)$implies $\exists x$ such that $y_{n \times 1} = A^T_{n \times m} x_{m \times 1}$

The set $\{x:A_{m \times n}x_{n \times 1} = \phi\}$ forms a subspace of $F^n$, known as the null space $N(A)$ of $A$.

Rank and nullity

The dimension of $C(A)$ is known as the column rank of $A$. The dimension of $R(A)$ is known as the row rank of $A$. These two ranks are found to be equal, and the common value is known as the rank $r(A)$ of $A$.

The dimension of $N(A)$ is known as the nullity $\eta (A)$ of A.

If $A$ is a square matrix of order $n \times n$, then $r(A) + \eta (A) = n$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Matrix&oldid=11050"