Information AboutEigenvalue |
| CATEGORIES ABOUT EIGENVALUE, EIGENVECTOR AND EIGENSPACE | |
| fundamental physics concepts | |
| abstract algebra | |
| linear algebra | |
| matrices | |
| singular value decomposition | |
| german loanwords | |
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In Mathematics , a Vector may be thought of as an arrow. It has a length, called its ''magnitude'', and it points in some particular direction. A Linear Transformation may be considered to operate on a vector to change it, usually changing both its magnitude and its direction. An of a given linear transformation is a vector which is simply multiplied by a constant called the during that transformation. The direction of the eigenvector is either unchanged by that transformation (for positive eigenvalues) or reversed (for negative eigenvalues). For example, an eigenvalue of +2 means that the eigenvector is doubled in length and points in the same direction. An eigenvalue of +1 means that the eigenvector stays the same, while an eigenvalue of −1 means that the eigenvector is reversed in direction. An eigenspace of a given transformation is the set of all eigenvectors of that transformation that have the same eigenvalue, together with the zero vector (which has no direction). An eigenspace is an example of a Subspace of a Vector Space . In Linear Algebra , every linear transformation can be given by a Matrix , which is a rectangular array of numbers arranged in rows and columns. Standard methods for finding eigenvalues, '''eigenvectors''', and '''eigenspaces''' of a given matrix are discussed below. These concepts play a major role in several branches of both Pure and Applied Mathematics — appearing prominently in Linear Algebra , Functional Analysis , and to a lesser extent in Nonlinear mathematics. Many kinds of mathematical objects can be treated as vectors: Functions , Harmonic Modes , Quantum States , and Frequencies , for example. In these cases, the concept of ''direction'' loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract ''direction'' is unchanged by a given linear transformation, the prefix "eigen" is used, as in ''eigenfunction'', ''eigenmode'', ''eigenstate'', and ''eigenfrequency''. HISTORY Eigenvalues are often introduced in the context of Matrix Theory . Historically, however, they arose in the study of Quadratic Form s and Differential Equation s. In the first half of the 18th century, Johann and Daniel Bernoulli , D'Alembert and Euler encountered eigenvalue problems when studying the motion of a rope, which they considered to be a weightless string loaded with a number of masses. Laplace and Lagrange continued their work in the second half of the century. They realized that the eigenvalues are related to the stability of the motion. They also used eigenvalue methods in their study of the Solar System .See Hawkins (1975), §2; Kline (1972), pp. 807+808. Euler had also studied the rotational motion of a Rigid Body and discovered the importance of the Principal Axes . As Lagrange realized, the principal axes are the eigenvectors of the inertia matrix.See Hawkins (1975), §2. In the early 19th century, Cauchy saw how their work could be used to classify the Quadric Surface s, and generalized it to arbitrary dimensions.See Hawkins (1975), §3. Cauchy also coined the term ''racine caractéristique'' (characteristic root) for what is now called ''eigenvalue''; his term survives in '' Characteristic Equation ''.See Kline (1972), pp. 807+808. Fourier used the work of Laplace and Lagrange to solve the Heat Equation by Separation Of Variables in his famous 1822 book ''Théorie analytique de la chaleur''.See Kline (1972), p. 673. Sturm developed Fourier's ideas further and he brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that symmetric matrices have real eigenvalues. This was extended by Hermite in 1855 to what are now called Hermitian Matrices . Around the same time, Brioschi proved that the eigenvalues of Orthogonal Matrices lie on the unit circle, and Clebsch found the corresponding result for Skew-symmetric Matrices . Finally, Weierstrass clarified an important aspect in the Stability Theory started by Laplace by realizing that Defective Matrices can cause instability. In the meantime, Liouville had studied similar eigenvalue problems as Sturm; the discipline that grew out of their work is now called '' Sturm-Liouville Theory ''.See Kline (1972), pp. 715+716. Schwarz studied the first eigenvalue of Laplace's Equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's Equation a few years later.See Kline (1972), pp. 706+707. At the start of the 20th century, Hilbert studied the eigenvalues of Integral Operator s by considering them to be infinite matrices.See Kline (1972), p. 1063. He was the first to use the German word ''eigen'' to denote eigenvalues and eigenvectors in 1904, though he may have been following a 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. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.See Aldrich (2006). The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the Power Method . One of the most popular methods today, the QR Algorithm , was proposed independently by Francis and Kublanovskaya in 1961. See Golub and Van Loan (1996), §7.3; Meyer (2000), §7.3. DEFINITIONS See Also: Eigenplane Linear Transformation s of space—such as Rotation , Reflection , Stretching , Compression , Shear or any combination of these—may be visualized by the effect they produce on Vector s. Vectors can be visualized as arrows pointing from one Point to another.
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 along 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 . EXAMPLES Mona Lisa For the example shown on the right, the matrix that would produce a shear transformation similar to this would be. : The set of eigenvectors for is defined as those vectors which, when multiplied by , result in a simple scaling of . Thus, : If we restrict ourselves to real eigenvalues, the only effect of the matrix on the eigenvectors will be to change their length, and possibly reverse their direction. So multiplying the right hand side by the Identity Matrix ''I'', we have : and therefore : In order for this equation to have non-trivial solutions, we require the Determinant which is called the Characteristic Polynomial of the matrix A to be zero. In our example we can calculate the determinant as : and now we have obtained the Characteristic Polynomial of the matrix A. There is in this case only one distinct solution of the equation , . This is the Eigenvalue of the matrix A. As in the study of roots of polynomials, it is convenient to say that this eigenvalue has multiplicity 2. Having found an eigenvalue , we can solve for the space of eigenvectors by finding the Nullspace of . In other words by solving for vectors which are solutions of : Substituting our obtained eigenvalue , : Solving this new matrix equation, we find that vectors in the nullspace have the form : where ''c'' is an arbitrary constant. All vectors of this form, i.e. pointing straight up or down, are Eigenvectors of the matrix A. The effect of applying the matrix A to these vectors is equivalent to multiplying them by their corresponding eigenvalue, in this case 1. In general, 2-by-2 matrices will have two distinct eigenvalues, and thus two distinct eigenvectors. Whereas most vectors will have both their lengths and directions changed by the matrix, eigenvectors will only have their lengths changed, and will not change their direction, except perhaps to flip through the origin in the case when the eigenvalue is a negative number. Also, it is usually the case that the eigenvalue will be something other than 1, and so eigenvectors will be stretched, squashed and/or flipped through the origin by the matrix. Other examples As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows which are parallel to 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 rubber sheet expanding omnidirectionally 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. is scaled by a Sinusoid al oscillation whose frequency is determined by the eigenvalue, but its overall shape is not modified.]] 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 equation for the Acceleration at every point of the rope, its eigenvectors, or '' Eigenfunction s'', are the Standing Wave s. The standing waves correspond to particular oscillations of the rope such that the acceleration of the rope is simply its shape scaled by a factor—this factor, the eigenvalue, turns out to be where is the Angular Frequency of the oscillation. Each component of the vector associated with the rope is multiplied by a time-dependent factor . If Damping is considered, the Amplitude of this oscillation decreases until the rope stops oscillating, corresponding to a Complex ω. One can then associate a lifetime with the imaginary part of ω, and relate the concept of an eigenvector to the concept of Resonance . Without damping, the fact that the acceleration operator (assuming a uniform density) is Hermitian leads to several important properties, such as that the standing wave patterns are Orthogonal Functions . EIGENVALUE EQUATION Suppose ''T '' is a Linear Transformation of a finite-dimensional space, that is for all Scalars ''a'', ''b'', and vectors v, '''w'''. Then is an eigenvector and ''λ'' the corresponding eigenvalue of ''T'' if the Equation : : is true, where ''T''(v''λ'') is the vector obtained when applying the transformation ''T'' to v''λ''. Consider a Basis of the vector space that ''T'' acts on. 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 : where the juxtaposition is Matrix Multiplication . Since, once a basis is fixed, ''T'' and its matrix representation ''A''''T'' are equivalent, we can often use the same symbol ''T'' for both 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 : (''f''(''t'') and ''g''(''t'') are Differentiable functions, and ''a'' and ''b'' are Constant s). Consider differentiation with respect to . Its eigenfunctions ''h''(''t'') obey the eigenvalue equation: :, where ''λ'' is the eigenvalue associated with the function. Such a function of time is constant if , grows proportionally to itself if is positive, and decays proportionally to itself if 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 , 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 , where ''A'' is an arbitrary constant, the initial population at ''t=0''. SPECTRAL THEOREM In its simplest version, the spectral theorem states that, under certain conditions, a linear transformation of a vector can be expressed as a Linear Combination of the eigenvectors, in which the Coefficient of each eigenvector is equal to the corresponding eigenvalue times the Scalar Product (or Dot Product ) of the eigenvector with the vector . Mathematically, it can be written as: : where and stand for the eigenvectors and eigenvalues of . 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 ; more generally the theorem holds for all Normal Matrices . 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 is given by : 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 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 : : 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 . If ''A'' is an ''n''×''n'' matrix, then has degree ''n'' and ''A'' can therefore have at most ''n'' eigenvalues. If the matrix is over an algebraically closed field, such as the complex numbers, then the Fundamental Theorem Of Algebra says that the characteristic equation has exactly ''n'' Root s (zeroes), counted with multiplicity. Therefore, any matrix over the complex numbers has an eigenvalue. 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. However, if ''n'' is even, a matrix with real entries may not have any real eigenvalues. For any ''n'', the non-real eigenvalues of a real matrix will come in Conjugate Pairs , just as the roots of a polynomial with real coefficients do. ;Finding eigenvectors Once the eigenvalues λ are known, the eigenvectors can then be found by solving: : where v is in the Null Space of . An example of a matrix with no real eigenvalues is the 90-degree clockwise rotation: : whose characteristic polynomial is 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 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/information/entry/Hilbert_space" class="copylinks">Hilbert or Banach Space s which have no eigenvectors at all This can be seen in 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>\cdots + a_{-1}^2 + a_0^2 + a_1^2 + a_2^2 + \cdots</math> converges) has no eigenvalue but has spectral values |
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| The | "http://wwwinformationdelightinfo/information/entry/Dirac_notation" class="copylinks">Dirac Notation is often used in this context A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by <math>\Psi_E
angle</math> In this notation, the Schrödinger equation is: |
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| :<math>H\Psi E Angle | E\Psi_E
angle</math> |
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| Where <math>\Psi E Angle</math> Is An '''eigenstate''' Of ''H'' It Is A | "http://wwwinformationdelightinfo/information/entry/self_adjoint_operator" class="copylinks">Self Adjoint Operator , the infinite dimensional analog of Hermitian matrices (''see Observable '') As in the matrix case, in the equation above <math>H\Psi_E
angle</math> is understood to be the vector obtained by application of the transformation ''H'' to <math>\Psi_E
angle</math> |
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