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11/24/2018 Generalized eigenvector - Wikipedia Generalized eigenvector In linear algebra, a generalized eigenvector of an n × n matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.[1] Let be an n-dimensional vector space; let be a linear map in L(V), the set of all linear maps from into itself; and let be the matrix representation of with respect to some ordered basis. There may not always exist a full set of n linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.[2][3] This happens when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity (the nullity of the matrix , or the dimension of its nullspace). In this case, is called a defective eigenvalue and is called a defective matrix.[4] A generalized eigenvector corresponding to , together with the matrix generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of .[5][6][7] Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for .[8] This basis can be used to determine an "almost diagonal matrix" in Jordan normal form, similar to , which is useful in computing certain matrix functions of .[9] The matrix is also useful in solving the system of linear differential equations where need not be diagonalizable.[10][11] Contents Overview and definition Examples Example 1 Example 2 Jordan chains Canonical basis Computation of generalized eigenvectors Example 3 Generalized modal matrix Jordan normal form Example 4 Example 5 Applications Matrix functions Differential equations Notes References https://en.wikipedia.org/wiki/Generalized_eigenvector 1/15 11/24/2018 Generalized eigenvector - Wikipedia Overview and definition There are several equivalent ways to define an ordinary eigenvector.[12][13][14][15][16][17][18][19] For our purposes, an eigenvector associated with an eigenvalue of an × matrix is a nonzero vector for which , where is the × identity matrix and is the zero vector of length .[20] That is, is in the kernel of the transformation . If has linearly independent eigenvectors, then is similar to a diagonal matrix . That is, there exists an invertible matrix such that is diagonalizable through the similarity transformation .[21][22] The matrix is called a spectral matrix for . The matrix is called a modal matrix for .[23] Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.[24] On the other hand, if does not have linearly independent eigenvectors associated with it, then is not diagonalizable.[25][26] Definition: A vector is a generalized eigenvector of rank m of the matrix and corresponding to the eigenvalue if but [27] Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector.[28] Every × matrix has linearly independent generalized eigenvectors associated with it and can be shown to be similar to an "almost diagonal" matrix in Jordan normal form.[29] That is, there exists an invertible matrix such that .[30] The matrix in this case is called a generalized modal matrix for .[31] If is an eigenvalue of algebraic multiplicity , then will have linearly independent generalized eigenvectors corresponding to .[32] These results, in turn, provide a straightforward method for computing certain matrix functions of .[33] Note: For an matrix over a field to be expressed in Jordan normal form, all eigenvalues of must be in . That is, the characteristic polynomial must factor completely into linear factors. For example, if has real-valued elements, then it may be necessary for the eigenvalues and the components of the eigenvectors to have complex values.[34][35][36] The set spanned by all generalized eigenvectors for a given , forms the generalized eigenspace for .[37] Examples Here are some examples to illustrate the concept of generalized eigenvectors. Some of the details will be described later. Example 1 This example is simple but clearly illustrates the point. This type of matrix is used frequently in textbooks.[38][39][40] Suppose Then there is only one eigenvalue, , and its algebraic multiplicity is m = 2. https://en.wikipedia.org/wiki/Generalized_eigenvector 2/15 11/24/2018 Generalized eigenvector - Wikipedia Notice that this matrix is in Jordan normal form but is not diagonal. Hence, this matrix is not diagonalizable. Since there is one superdiagonal entry, there will be one generalized eigenvector of rank greater than 1 (or one could note that the vector space is of dimension 2, so there can be at most one generalized eigenvector of rank greater than 1). Alternatively, one could compute the dimension of the nullspace of to be p = 1, and thus there are m – p = 1 generalized eigenvectors of rank greater than 1. The ordinary eigenvector is computed as usual (see the eigenvector page for examples). Using this eigenvector, we compute the generalized eigenvector by solving Writing out the values: This simplifies to The element has no restrictions. The generalized eigenvector of rank 2 is then , where a can have any scalar value. The choice of a = 0 is usually the simplest. Note that so that is a generalized eigenvector, so that is an ordinary eigenvector, and that and are linearly independent and hence constitute a basis for the vector space . Example 2 This example is more complex than Example 1. Unfortunately, it is a little difficult to construct an interesting example of low order.[41] The matrix https://en.wikipedia.org/wiki/Generalized_eigenvector 3/15 11/24/2018 Generalized eigenvector - Wikipedia has eigenvalues and with algebraic multiplicities and , but geometric multiplicities and . The generalized eigenspaces of are calculated below. is the ordinary eigenvector associated with . is a generalized eigenvector associated with . is the ordinary eigenvector associated with . and are generalized eigenvectors associated with . This results in a basis for each of the generalized eigenspaces of . Together the two chains of generalized eigenvectors span the space of all 5-dimensional column vectors. https://en.wikipedia.org/wiki/Generalized_eigenvector 4/15 11/24/2018 Generalized eigenvector - Wikipedia An "almost diagonal" matrix in Jordan normal form, similar to is obtained as follows: where is a generalized modal matrix for , the columns of are a canonical basis for , and .[42] Jordan chains Definition: Let be a generalized eigenvector of rank m corresponding to the matrix and the eigenvalue . The chain generated by is a set of vectors given by (1) Thus, in general, (2) https://en.wikipedia.org/wiki/Generalized_eigenvector 5/15 11/24/2018 Generalized eigenvector - Wikipedia The vector , given by (2), is a generalized eigenvector of rank j corresponding to the eigenvalue . A chain is a linearly independent set of vectors.[43] Canonical basis Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains. Thus, once we have determined that a generalized eigenvector of rank m is in a canonical basis, it follows that the m − 1 vectors that are in the Jordan chain generated by are also in the canonical basis.[44] Let be an eigenvalue of of algebraic multiplicity . First, find the ranks (matrix ranks) of the matrices . The integer is determined to be the first integer for which has rank (n being the number of rows or columns of , that is, is n × n). Now define The variable designates the number of linearly independent generalized eigenvectors of rank k corresponding to the eigenvalue that will appear in a canonical basis for . Note that .[45] Computation of generalized eigenvectors In the preceding sections we have seen techniques for obtaining the n linearly independent generalized eigenvectors of a canonical basis for the vector space associated with an n × n matrix . These techniques can be combined into a procedure: Solve the characteristic equation of for eigenvalues and their algebraic multiplicities ; For each Determine ; Determine ; Determine for ; Determine each Jordan chain for ; Example 3 The matrix https://en.wikipedia.org/wiki/Generalized_eigenvector 6/15 11/24/2018 Generalized eigenvector - Wikipedia has an eigenvalue of algebraic multiplicity and an eigenvalue of algebraic multiplicity . We also have n = 4. For we have . The first integer for which has rank is . We now define Consequently, there will be three linearly independent generalized eigenvectors; one each of ranks 3, 2 and 1. Since corresponds to a single chain of three linearly independent generalized eigenvectors, we know that there is a generalized eigenvector of rank 3 corresponding to such that (3) but (4) https://en.wikipedia.org/wiki/Generalized_eigenvector 7/15 11/24/2018 Generalized eigenvector - Wikipedia Equations (3) and (4) represent linear systems that can be solved for . Let Then and Thus, in order to satisfy the conditions (3) and (4), we must have and . No restrictions are placed on and . By choosing , we obtain as a generalized eigenvector of rank 3 corresponding to . Note that it is possible to obtain infinitely many other generalized eigenvectors of rank 3 by choosing different values of , and , with . Our first choice, however, is the simplest.[46] Now using equations (1), we obtain and as generalized eigenvectors of rank 2 and 1 respectively, where and https://en.wikipedia.org/wiki/Generalized_eigenvector 8/15 11/24/2018 Generalized eigenvector - Wikipedia The simple eigenvalue can be dealt with using standard techniques and has an ordinary eigenvector A canonical basis for is and are generalized eigenvectors associated with . is the ordinary eigenvector associated with . It should be noted that this is a fairly simple example. In general, the numbers of linearly independent generalized eigenvectors of rank k will not always be equal. That is, there may be several chains of different lengths corresponding to a particular eigenvalue.[47] Generalized modal matrix Let be an n × n matrix.
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