Difference between revisions of "Matrix"

From Conservapedia
Jump to navigation Jump to search
(New page: A '''matrix''' (pl.: "matrices," Latin origin) is a complex ordering, in deliberate fashion, of numerals. In mathematics, a "matrix" is a regular grid of numbers, whic...)
 
m (→‎Advanced concepts: Capatalise)
 
(30 intermediate revisions by 16 users not shown)
Line 1: Line 1:
−
A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral|numerals]].  In [[mathematics]], a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level [[algebra]].  Matrix algebra is usually taught in [[sophomore]] [[high school]] level mathematics.
+
''For the 1999 film, see [[The Matrix]].''
  
−
Alternately, a matrix may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the [[synergy]] of the networked objects.
+
A '''matrix''' (pl.: "matrices," [[Latin]] origin) is a complex ordering, in deliberate fashion, of [[numeral]]s.  In [[mathematics]], a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level [[algebra]].  [[Matrix algebra]] is usually taught in [[sophomore]] [[high school]] level mathematics.
  
−
[[Category:Mathematics]]
+
More formally, a matrix is an example of a rank-2 [[tensor]].
 +
 
 +
Alternately, a '''matrix''' may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the [[synergy]] of the networked objects.
 +
 
 +
==Mathematics==
 +
In mathematics, matrices can be manipulated in a variety of ways, including [[addition]] and [[multiplication]].
 +
 
 +
===Addition of matrices===
 +
To add two matrices, one would add their respective elements.  For example:
 +
 
 +
<math>\begin{bmatrix}
 +
  x      & y & z      \\
 +
  1 & 3 & 5 \\
 +
  0      & 2 & 0
 +
\end{bmatrix} + \begin{bmatrix}
 +
  0      & 3 & 1      \\
 +
  4 & 3 & {x+2} \\
 +
  0      & 4 & v
 +
\end{bmatrix} </math>
 +
 
 +
would equal
 +
 
 +
<math>\begin{bmatrix}
 +
{x+0} & {y+3} & {z+1} \\
 +
{1+4} & {3+3} & {5+(x+2)} \\
 +
{0+0} & {2+4} & {0+v}
 +
\end{bmatrix} = \begin{bmatrix}
 +
x & {y+3} & {z+1} \\
 +
5 & 6 & {x+7} \\
 +
0 & 6 & v
 +
\end{bmatrix} </math>
 +
 
 +
===Multiplication of matrices===
 +
To multiply two matrices, use the rule for finding the product of two matrices:
 +
 
 +
<math>
 +
(AB)_{ij}=\sum_k A_{ik}B_{kj}
 +
</math>
 +
 
 +
However, not every pair of matrices can be multiplied.  In order for matrices <math>A</math> and <math>B</math> to be compatible for multiplication, the number of ''columns'' in <math>A</math> must equal the number of ''rows'' in <math>B</math>.  If <math>A</math> is an <math>m \times n</math> matrix and <math>B</math> is an <math>n \times p</math> matrix, the product matrix <math>AB</math> will have <math>m</math> rows and <math>p</math> columns.
 +
 
 +
Matrix multiplication is [[associativity|associative]].  However, matrix multiplication is ''not'' [[commutativity|commutative]].  That is, it is possible for <math>AB \neq BA</math>.
 +
 
 +
The <math>n \times n</math> [[identity matrix]] <math>I_n</math> satisfies the property:
 +
<center><math>AI_n = I_n A = A</math></center>
 +
for all <math>n \times n</math> matrices <math>A</math>.
 +
 
 +
Moreover, every square matrix <math>A</math> with nonzero [[determinant]] has an inverse matrix <math>B</math> such that
 +
<center><math>AB = BA = I_n</math></center>
 +
 
 +
This means that for every positive integer <math>n</math>, the set of all <math>n \times n</math> matrices with nonzero determinant form a [[group]] under matrix multiplication.  This group is known as the [[general linear group]] <math>GL_{n}(\mathbb{R})</math>.
 +
 
 +
===Matrix concepts===
 +
 
 +
====Basic concepts====
 +
 
 +
*[[Adjoint]]
 +
*[[Determinant]]
 +
*[[Diagonal matrix]]
 +
*[[Identity matrix]]
 +
*[[Inverse matrix]]
 +
*[[Null, column and row space]]
 +
*[[Scalar]]
 +
*[[Trace]]
 +
*[[Transpose matrix]]
 +
*[[Vector]]
 +
*[[Zero matrix]]
 +
 
 +
====Advanced concepts====
 +
 
 +
*[[Basis]]
 +
*[[Diagonalizable]]
 +
*[[Eigenspace]]
 +
*[[Eigenvalue]]
 +
*[[Eigenvector]]
 +
*[[Gram-Schmidt process]]
 +
*[[Hermitian matrix]]
 +
*[[Jordan canonical form]]
 +
*[[Laplacian]]
 +
*[[Linear independence]]
 +
*[[Matrix diagonalization]]
 +
*[[Matrix reformation]]
 +
*[[Matrix transformation]]
 +
*[[Orthogonal matrix]]
 +
*[[Orthonormal matrix]]
 +
*[[Resultant]]
 +
*[[Span]]
 +
*[[Linear equations]]
 +
*[[Transcriptor]]
 +
*[[Wronskian]]
 +
 
 +
[[Category:Linear Algebra]]
 
[[Category:Computers]]
 
[[Category:Computers]]

Latest revision as of 12:23, September 17, 2017

For the 1999 film, see The Matrix.

A matrix (pl.: "matrices," Latin origin) is a complex ordering, in deliberate fashion, of numerals. In mathematics, a "matrix" is a regular grid of numbers, which may be manipulated and solved through intermediate-level algebra. Matrix algebra is usually taught in sophomore high school level mathematics.

More formally, a matrix is an example of a rank-2 tensor.

Alternately, a matrix may also be a complex ordering of a group of equivalent objects, especially where the order is imposed to gain incidental benefit from the synergy of the networked objects.

Mathematics

In mathematics, matrices can be manipulated in a variety of ways, including addition and multiplication.

Addition of matrices

To add two matrices, one would add their respective elements. For example:

<math>\begin{bmatrix} 

 x      & y & z      \\
 1 & 3 & 5 \\ 
 0      & 2 & 0

\end{bmatrix} + \begin{bmatrix} 

 0      & 3 & 1      \\
 4 & 3 & {x+2} \\ 
 0      & 4 & v

\end{bmatrix} </math>

would equal

<math>\begin{bmatrix} {x+0} & {y+3} & {z+1} \\ {1+4} & {3+3} & {5+(x+2)} \\ {0+0} & {2+4} & {0+v} \end{bmatrix} = \begin{bmatrix} x & {y+3} & {z+1} \\ 5 & 6 & {x+7} \\ 0 & 6 & v \end{bmatrix} </math>

Multiplication of matrices

To multiply two matrices, use the rule for finding the product of two matrices:

<math> (AB)_{ij}=\sum_k A_{ik}B_{kj} </math>

However, not every pair of matrices can be multiplied. In order for matrices <math>A</math> and <math>B</math> to be compatible for multiplication, the number of columns in <math>A</math> must equal the number of rows in <math>B</math>. If <math>A</math> is an <math>m \times n</math> matrix and <math>B</math> is an <math>n \times p</math> matrix, the product matrix <math>AB</math> will have <math>m</math> rows and <math>p</math> columns.

Matrix multiplication is associative. However, matrix multiplication is not commutative. That is, it is possible for <math>AB \neq BA</math>.

The <math>n \times n</math> identity matrix <math>I_n</math> satisfies the property:

<math>AI_n = I_n A = A</math>

for all <math>n \times n</math> matrices <math>A</math>.

Moreover, every square matrix <math>A</math> with nonzero determinant has an inverse matrix <math>B</math> such that

<math>AB = BA = I_n</math>

This means that for every positive integer <math>n</math>, the set of all <math>n \times n</math> matrices with nonzero determinant form a group under matrix multiplication. This group is known as the general linear group <math>GL_{n}(\mathbb{R})</math>.

Matrix concepts

Basic concepts

Advanced concepts

Retrieved from "https://www.conservapedia.com/index.php?title=Matrix&oldid=1374660"