Maths Spl

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that does not change its direction, when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and V is a vector in V that is not the zero vector, then v is an eigenvector of T if T (v) is a scalar multiple of v. This condition can be written as the equation
{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,} {\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}
Where is a scalar in the field F, known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector v.
If the vector space V is finite-dimensional, then the linear transformation T can be represented as a square matrix A, and the vector v by a column vector, rendering the above mapping as a matrix multiplication on the left hand side and a scaling of the column vector on the right hand side in the equation
{\displaystyle A\mathbf {v} =\lambda \mathbf {v} .} {\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}
There is a correspondence between n by n square matrices and linear transformations from an n-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations.

In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by Descartes, who distinguished between real and imaginary roots of polynomials.
The real numbers include all the rational numbers, such as the integer 5 and the fraction 4/3, and all the irrational numbers, such as 2 (1.41421356, the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as (3.14159265).
Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one-tenth the size of the previous one. The real line can be thought of as a part of the complex plane, and complex numbers include real numbers.