# Matrix

Matrices of the same size can be added or subtracted element by element. But the rule for matrix multiplication is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. 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. If R is a rotation matrix and 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 matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of a system 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. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations. 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. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. 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.[3] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. 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 simplify computations, both theoretically and practically. 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 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. Contents [hide]

1 Definition

1.1 Size

2 Notation

3 Basic operations

3.1 Addition, scalar multiplication and transposition

3.2 Matrix multiplication

3.3 Row operations

3.4 Submatrix

4 Linear equations

5 Linear transformations

6 Square matrices

6.1 Main types

6.1.1 Diagonal or triangular matrix

6.1.2 Identity matrix

6.1.3 Symmetric or skew-symmetric matrix

6.1.4 Invertible matrix and its inverse

6.1.5 Definite matrix

6.1.6 Orthogonal matrix

6.2 Main operations

6.2.1 Trace

6.2.2 Determinant

6.2.3 Eigenvalues and eigenvectors

7 Computational aspects

8 Decomposition

9 Abstract algebraic aspects and generalizations

9.1 Matrices with more general entries

9.2 Relationship to linear maps

9.3 Matrix groups

9.4 Infinite matrices

9.5 Empty matrices

10 Applications

10.1 Graph theory

10.2 Analysis and geometry

10.3 Probability theory and statistics

10.4 Symmetries and transformations in physics

10.5 Linear combinations of quantum states

10.6 Normal modes

10.7 Geometrical optics

10.8 Electronics

11 History

11.1 Other historical usages of the word “matrix” in mathematics 12 See also

13 Notes

14 References

14.1 Physics references

14.2 Historical references

15 External...

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