Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array -- of numbers, symbols, or expressions, arranged in rows and columns -- that is treated in certain prescribed ways.

Description

One such way is to state the order of the matrix. For example, the order of the matrix below is a 2x3 matrix because there are two rows and three columns. The individual items in a matrix are called its elements or entries.

\begin{bmatrix}1 & 9 & -13 \\20 & 5 & -6 \end{bmatrix}.

(TO DO: fix math display.)

Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element.

The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.

Linear transformation

A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x.

For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation.

The product of two transformation matrices is a matrix that represents the composition of two linear transformations.

Systems of linear equations

Another application of matrices is in the solution of systems of linear equations.

If the matrix is square, it is possible to deduce some of its properties by computing its determinant.

For example, a square matrix has an inverse if and only if its determinant is not zero.

Geometry of a linear transformation

Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.

Applications

Applications of matrices are found in most scientific fields.

In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies.

Computer graphics

In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen.

Probability theory and statistics

In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.

Matrix calculus

Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.

Numerical analysis

A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research.

Matrix decomposition methods

Matrix decomposition methods simplify computations, both theoretically and practically.

Algorithms tailored matrix structures

Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations.

Infinite matrices

Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function.

See also

• Algebraic multiplicity
• Array data structure
• Expression (mathematics)
• Geometric multiplicity
• Gram-Schmidt process
• Linear algebra
• List of matrices
• Mathematics
• Matrix calculus
• Matrix function
• Normal-form game
• Number
• Numerical analysis
• Periodic matrix set
• Symbol
• Tensor
• Vector (mathematics)

External links

• Matrix (mathematics) @ Wikipedia