Difference between revisions of "Matrix"
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===Further matrix concepts=== | ===Further matrix concepts=== | ||
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*[[Determinant]] | *[[Determinant]] | ||
| − | + | *[[Eigenspace]] | |
| + | *[[Eigenvalue]] | ||
| + | *[[Eigenvector]] | ||
| + | *[[Identity matrix]] | ||
| + | *[[Jordan canonical form]] | ||
*[[Inverse matrix]] | *[[Inverse matrix]] | ||
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*[[Null, column and row space]] | *[[Null, column and row space]] | ||
| − | + | *[[Resultant]] | |
| − | *[[ | + | *[[Systems of linear equations]] |
| + | *[[Transpose matrix]] | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
[[Category:Computers]] | [[Category:Computers]] | ||
Revision as of 20:21, June 8, 2008
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.
Contents
Mathematics
In mathematics, matrices can be manipulated in a variety of ways, including addition and multiplication.
Addition of matrices
For example, to add two matrices, one would add their respective elements, thus:
would equal
Multiplication of matrices
To multiply two matrices, one uses the rule "go along the rows and down the columns". This is best illustrated by a specific example: a matrix times a vector:
It is important to note that matrix multiplication is not commutative: in general, 
for two matrices 
and 
. This has important consequences in quantum mechanics.
To see why matrix multiplication works the way it does, we will use suffix notation. Consider first forming the product of two matrices, 
, which is itself a matrix. Then form the product 
. Matrix multiplication is associative, so we can consider this as either 
or 
. In suffix notation,
The vector 
is arbitrary, so we can therefore deduce the rule for finding the product of two matrices:
