Eigenvector

transformation of the In Mathematics , an of a Transformation In this context, only Linear Transformation s from a Vector Space to itself are considered. is a Non-null Vector whose direction is unchanged by that transformation. The factor by which the magnitude is Scaled is called the of that vector. A pictorial example is provided in Fig. 1. Often, a transformation is completely described by its eigenvalues and eigenvectors. An eigenspace is a Set of eigenvectors with a common eigenvalue. These concepts play a major role in several branches of both Pure and Applied Mathematics — appearing prominently in Linear Algebra , Functional Analysis , and even a variety of Nonlinear situations. The German word ''eigen'' was first used in this context by Hilbert in 1904 (there was an earlier related usage by Helmholtz ). "Eigen" can be translated as "own", "peculiar to", "characteristic" or "individual"—emphasizing how important eigenvalues are to defining the unique nature of a specific transformation. In English mathematical jargon, the closest translation would be "characteristic"; and some older references do use expressions like "characteristic value" and "characteristic vector", or even "Eigenwert", German for eigenvalue. In the past, the standard translation used to be "proper". Today the more distinctive term "eigenvalue" is standard. DEFINITIONS Transformation s of space—such as Translation (or shifting the origin), Rotation , Reflection , Stretching , Compression , or any combination of these; other transformations could also be listed—may be visualized by the effect they produce on Vector s. Vectors can be visualised as arrows pointing from one Point to another. Eigenvectors of transformations are vectorsSince all linear transformations leave the zero vector unchanged, it is not considered an eigenvector. which are either left unaffected or simply multiplied by a Scale Factor after the transformation. An eigenvector's eigenvalue is the scale factor that it has been multiplied by. An eigenspace is a Space consisting of all eigenvectors which have the same eigenvalue, along with the zero(null) vector which itself is not an eigenvector. The geometric Multiplicity of an eigenvalue is the Dimension of the associated eigenspace. The spectrum of a transformation on finite dimensional Vector Spaces is the Set of all its eigenvalues. For instance, an ''eigenvector'' of a rotation in three dimensions is a vector located within the Axis about which the rotation is performed. The corresponding ''eigenvalue'' is 1 and the corresponding ''eigenspace'' contains all the vectors Parallel to the axis. As this is a one-dimensional space, its ''geometric multiplicity'' is one. This is the only eigenvalue of the ''spectrum'' (of this rotation) that is a Real Number . See Also: Eigenplane EXAMPLES As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows that lie on the axis of rotation. Consider the transformation of the Earth after one hour of rotation: An arrow from the center of the Earth to the Geographic South Pole would be an eigenvector of this transformation, but an arrow from the center of the Earth to anywhere on the Equator would not be an eigenvector. Since the arrow pointing at the pole is not stretched by the rotation of the Earth, its eigenvalue is 1. Another example is provided by a thin metal sheet expanding uniformly about a fixed point in such a way that the distances from any point of the sheet to the fixed point are doubled. This expansion is a transformation with eigenvalue 2. Every vector from the fixed point to a point on the sheet is an eigenvector, and the eigenspace is the set of all these vectors. in a rope fixed at its boundaries can be seen as an example of an eigenvector, or more precisely, an eigenfunction of the transformation corresponding to the passage of time. As time passes, the Standing Wave is scaled but its shape is not modified. In this case the eigenvalue is time dependent.]] However, three-dimensional geometric space is not the only vector space. For example, consider a stressed rope fixed at both ends, like the Vibrating String s of a String Instrument (Fig. 2). The distances of atoms of the vibrating rope from their positions when the rope is at rest can be seen as the Component s of a vector in a space with as many dimensions as there are Atom s in the rope. Assume the rope is a Continuous Medium . If one considers the transformation of the rope as time passes, its eigenvectors, or ''eigenfunctions'', are its Standing Waves —the things that, mediated by the surrounding air, humans can experience as the twang of a Bow String or the plink of a Guitar . The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as time passes. Each component of the vector associated with the rope is multiplied by this time-dependent factor. The amplitude (eigenvalues) of the standing waves decrease with time if Damping is considered. One can then associate a Lifetime with the eigenvector, and relate the concept of an eigenvector to the concept of Resonance . EIGENVALUE EQUATION Mathematically, v_''λ'' is an eigenvector and ''λ'' the corresponding eigenvalue of a transformation ''T'' if the Equation : : $T(\mathbf{v}_\lambda)=\lambda\,\mathbf{v}_\lambda$ is true, where''T''(v_''λ'') is the vector obtained when applying the transformation ''T'' to v_''λ''. Suppose ''T'' is a Linear Transformation (which means that $T(a\mathbf{v}+b\mathbf{w})=aT(\mathbf{v})+bT(\mathbf{w})$ for all Scalar s ''a'', ''b'', and vectors v, '''w'''). Consider a Basis in that vector space. Then, ''T'' and v_''λ'' can be represented relative to that basis by a Matrix ''A''_''T''—a two-dimensional Array —and respectively a column vector ''v''_''λ''—a one-dimensional vertical array. The eigenvalue equation in its Matrix representation is written : $A_T\,v_\lambda=\lambda\,v_\lambda$ where the juxtaposition is Matrix Multiplication . Since in this circumstance the transformation ''T'' and its matrix representation ''A''_''T'' are equivalent, we can often use just ''T'' for the matrix representation and the transformation. This is equivalent to a set of ''n'' linear equations, where ''n'' is the number of basis vectors in the Basis Set . In this equation both the eigenvalue ''λ'' and the ''n'' components of v_''λ'' are Unknown s. However, it is sometimes unnatural or even impossible to write down the eigenvalue equation in a matrix form. This occurs for instance when the vector space is infinite dimensional, for example, in the case of the rope above. Depending on the nature of the transformation ''T'' and the space to which it applies, it can be advantageous to represent the eigenvalue equation as a set of Differential Equation s. If ''T'' is a Differential Operator , the eigenvectors are commonly called eigenfunctions of the differential operator representing ''T''. For example, Differentiation itself is a linear transformation since (if ''f''(''t'') and ''g''(''t'') are Differentiable functions, and ''a'' and ''b'' are Constant s) : $\displaystylerac{d}{dt}(af+bg) = a rac{df}{dt} + b rac{dg}{dt}$ Consider differentiation with respect to $t$ . Its eigenfunctions ''h''(''t'') obey the eigenvalue equation: : $\displaystylerac{dh}{dt} = \lambda h$ , where ''λ'' is the eigenvalue associated with the function. Such a function of time is constant if $\lambda = 0$ , grows proportionally to itself if $\lambda$ is positive, and decays proportionally to itself if $\lambda$ is negative. For example, an idealized population of rabbits breeds faster the more rabbits there are, and thus satisfies the equation with a positive lambda. The solution to the eigenvalue equation is $g(t)= \exp (\lambda t)$ , the Exponential Function ; thus that function is an eigenfunction of the differential operator ''d/dt'' with the eigenvalue ''λ''. If ''λ'' is Negative , we call the evolution of ''g'' an Exponential Decay ; if it is Positive , an Exponential Growth . The value of ''λ'' can be any Complex Number . The spectrum of ''d/dt'' is therefore the whole Complex Plane . In this example the vector space in which the operator ''d/dt'' acts is the space of the Differentiable functions of one Variable . This space has an Infinite dimension (because it is not possible to express every differentiable function as a Linear Combination of a finite number of Basis Function s). However, the eigenspace associated with any given eigenvalue ''λ'' is one dimensional. It is the set of all functions $g(t)= A \exp (\lambda t)$ , where ''A'' is an arbitrary constant, the initial population at ''t=0''. SPECTRAL THEOREM The ''spectral theorem'' depicts the importance of the eigenvalues and eigenvectors for characterizing a linear transformation in a unique way. In its simplest version, the spectral theorem states that, under precise conditions, a linear transformation of a vector can be expressed as the Linear Combination of the eigenvectors with Coefficients equal to the eigenvalues times the Scalar Product (or Dot Product ) of the eigenvectors with the vector on which the transformation is applied. Mathematically, it can be written as: : $\mathcal{T}(\mathbf{v})= \lambda_1 (\mathbf{v}_1 \cdot \mathbf{v}) \mathbf{v}_1 + \lambda_2 (\mathbf{v}_2 \cdot \mathbf{v}) \mathbf{v}_2 + \dots$ where $\mathbf{v}_1, \mathbf{v}_2, \dots$ and $\lambda_1, \lambda_2, \dots$ stand for the eigenvectors and eigenvalues of $\mathcal{T}$ . The simplest case in which the theorem is valid is the case where the linear transformation is given by a Real Symmetric Matrix or Complex Hermitian Matrix . If one defines the ''n''th power of a transformation as the result of applying it ''n'' times in succession, one can also define Polynomial s of transformations. A more general version of the theorem is that any polynomial ''P'' of $\mathcal{T}$ is equal to: : $P(\mathcal{T})(\mathbf{v})= P(\lambda_1) (\mathbf{v}_1 \cdot \mathbf{v}) \mathbf{v}_1 + P(\lambda_2) (\mathbf{v}_2 \cdot \mathbf{v}) \mathbf{v}_2 + \dots$ The theorem can be extended to other functions of transformations like Analytic Function s, the most general case being Borel Functions . EIGENVALUES AND EIGENVECTORS OF MATRICES Computing eigenvalues and eigenvectors of matrices Suppose that we want to compute the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the Characteristic Polynomial . However, this is often impossible for larger matrices, in which case we must use a Numerical Method . Symbolic computations ;Finding eigenvalues An important tool for describing eigenvalues of square matrices is the (''A'' – ''λI'') ''v'' = 0 (where ''I'' is the Identity Matrix ) has a non-zero solution ''v'' (an eigenvector), and so it is equivalent to the Determinant : : $\det(A - \lambda I) = 0 \!\$ The function ''p''(''λ'') = det(''A'' – ''λI'') is a Polynomial in ''λ'' since determinants are defined as sums of products. This is the characteristic polynomial of ''A'': the eigenvalues of a matrix are the zeros of its Characteristic Polynomial . All the eigenvalues of a matrix ''A'' can be computed by solving the equation $p_A(\lambda) = 0$ . If ''A'' is an ''n''×''n'' matrix, then $p_A$ has degree ''n'' and ''A'' can therefore have at most ''n'' eigenvalues. Conversely, the Fundamental Theorem Of Algebra says that this equation has exactly ''n'' Root s (zeroes), counted with multiplicity. All real polynomials of odd degree have a real number as a root, so for odd n, every real matrix has at least one real eigenvalue. In the case of a real matrix, for even and odd ''n'', the non-real eigenvalues come in Conjugate Pairs . ;Finding eigenvectors Once the eigenvalues λ are known, the eigenvectors can then be found by solving: : $(A - \lambda I) v = 0 \!\$ An example of a matrix with no real eigenvalues is the 90-degree clockwise rotation: : $\begin{bmatrix}0 & 1\ -1 & 0\end{bmatrix}$ whose characteristic polynomial is $\lambda^2+1$ and so its eigenvalues are the pair of complex conjugates ''i'', -''i''. The associated eigenvectors are also not real. Numerical computations In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Computing the polynomial becomes expensive in itself, and exact (symbolic) roots of a high-degree polynomial can be difficult to compute and express: the vector $v$ is chosen and a sequence of Unit Vector s is computed as
	Clearly If ''&lambda'' Is An Eigenvalue Of ''T'', ''&lambda'' Is In The Spectrum Of ''T'' In General, The Converse Is Not True There Are Operators On	"http://wwwinformationdelightinfo/encyclopedia/entry/Hilbert_space_" class="copylinks">Hilbert or Banach Space s which have no eigenvectors at all This can be seen on the following example The Bilateral Shift on the Hilbert space <math>\ell^2(\mathbf{Z})</math> (the space of all sequences of scalars <math>\dots a_{-1}, a_0, a_1,a_2,\dots</math> such that <math>\dots a_{-1}^2 + a_0^2 + a_1^2 + a_2^2 +\dots</math> converge) has no eigenvalue but has spectral values
	The	"http://wwwinformationdelightinfo/encyclopedia/entry/Dirac_notation" class="copylinks">Dirac Notation often used in this context stresses the difference between the vector or state <math>\Psi_E angle</math> and its representation, the function <math>\Psi_E</math> In this context one writes the Schrödinger equation
	:<math>H\Psi E Angle	E\Psi_E angle</math>
	And Call <math>\Psi E Angle</math> An '''eigenstate''' Of ''H'' (sometimes Written <math>\hat{H}</math> In Introductory Textbooks) Which Is Seen As A Transformation (''see	"http://wwwinformationdelightinfo/encyclopedia/entry/Observable" class="copylinks">Observable '') instead of particular representation of it in terms of differential operators In the equation above <math>H\Psi_E angle</math> is understood as the vector obtained by application of the transformation ''H'' to <math>\Psi_E angle</math>

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Eigenvector