A diagonal matrix S has all non-diagonal elements equal zero. x�32�3�T0P� bCs �b�U���@tr.��'�~�������U()*M��w Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 3 & 9 & 9 & 9 \\ 9 &3 & 9 & 9 \\ 9 & 9 & 3 & 9 \\ 9 & 9 & 9 & 3 \end{bmatrix}.\] (Harvard University, Linear Algebra Final Exam Problem) Add to solve later . endobj th largest or The diagonal elements of a triangular matrix are equal to its eigenvalues. v << /Filter /FlateDecode Diagonal matrices make calculations really easy. The word "eigen" comes from German and means "own", while it is the Dutch word for "characteristic", and so this chapter could also be called "Characteristic values and characteristic vectors". /Filter /FlateDecode v x�31�31R0P0bcKC�C�B.cC � �I$�r9yr�+r�{ E��=}J�JS�����|�h��X.O�����'�����������?���������o`�������GG����� �xl� >> , the fabric is said to be linear.[48]. [14] Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability. Therefore. E }� gC. ξ sin − x n. 1 C C = x is solved by the following eigenvalues and eigenvectors: = d1 ;1and x = e1= (1 ;0 ;0 ;:::;0 )T, = d2 ;2and x = e2= (0 ;1 ;0 ;:::;0 )T, .. . Assume P exists as in (iii), and de ne vi = Pei. Wikipedia gives you a formula that can be used. is 4 or less. , Calculator of eigenvalues and eigenvectors. endobj x�m�1j�@E�Q!��GМ ��� �"�T)L*�e���^�f /Length 1325 The generalized eigenvalue problem is to determine the solution to the equation Av = λBv, where A and B are n-by-n matrices, v is a column vector of length n, and λ is a scalar. Problem 379; Hint. Diagonalization is a process of converting a n x n square matrix into a diagonal matrix having eigenvalues of first matrix as its non-zero elements. a 60 0 obj �6�� ���-�m�k_X~Vt�]-O�dtv6 Proposition Let be a triangular matrix. /Length 167 I The first principal eigenvector of the graph is also referred to merely as the principal eigenvector. ) /Filter /FlateDecode n The main eigenfunction article gives other examples. [6][7] Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization. Consider the matrix A = [ a − b b a], where a and b are real numbers and b ≠ 0. v . with eigenvalue different products.[e]. Eigenvalues, returned as a diagonal matrix with the eigenvalues of A on the main diagonal or the eigenvalues of the pair, (A,B), with ... To calculate the eigenvectors of a sparse matrix, or to calculate the eigenvalues of a sparse matrix that is not real and symmetric, use the eigs function. {\displaystyle (A-\lambda I)v=0} /Length 138 ] {\displaystyle n\times n} 65 0 obj 1 {\displaystyle (A-\xi I)V=V(D-\xi I)} A The roots of this polynomial, and hence the eigenvalues, are 2 and 3. endobj The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors. In this case the eigenfunction is itself a function of its associated eigenvalue. In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A The corresponding eigenvalue, often denoted by Since the zero vector 0 has no direction this would make no sense for the zero vector. << << H �\�. λ , The eigenvectors are the columns of the "v" matrix. λ γ The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. A matrix \(M\) is diagonalizable if there exists an invertible matrix \(P\) and a diagonal matrix \(D\) such that \[ D=P^{-1}MP. /Length 211 λ > {\displaystyle n\times n} {\displaystyle H} stream / I (Note the diagonal matrix … 1 ± The following are properties of this matrix and its eigenvalues: Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. Define an eigenvector v associated with the eigenvalue λ to be any vector that, given λ, satisfies Equation (5). . ( Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T. When T admits an eigenbasis, T is diagonalizable. {\displaystyle i} − is then the largest eigenvalue of the next generation matrix. D Vc�B-�(��vHIfs�v*W���C�� 1 0 0 0 0 4 0 0 0 0 6 0 0 0 0 2 It’s not hard to see that adding in the - lambda term to each element on the diag and setting equal to zero would reveal the eigenvalues to be just values on the diag. >> >> contains a factor represents the eigenvalue. 72 0 obj 2 In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. orthonormal eigenvectors Let have eigenvalues with (is strictly less than if some of … xڕ�+�@������й l�]�GB (A�m����0[0�0�����/:��;n[v}�]�Y:���ݻ�=Ш$�b���4&S��|��Ɍc�d&��\l��0���܀��:�HRg�hݐ!�"E�� tU|��7~4��kC��5HCv�$S���_��! [ 1) If a "×"matrix !has "linearly independent eigenvectors $then !is diagonalizable, i.e., !=676<8 where the columns of 6are the linearly independent normalized eigenvectors $of ! 1 The row vector is called a left eigenvector of . A can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. x�u�!�PD��h��H/ �����H� � (@" ���8J���[email protected]!ϼٙ��`4��d�Oŏ%�!�3������������4'�R��ۑ�b5ؙl��q9�"S G E [10][28] By the definition of eigenvalues and eigenvectors, γT(λ) ≥ 1 because every eigenvalue has at least one eigenvector. ψ x�32�36V0P0bCS3c�C�B. The relative values of x 3 , then. In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. . ( This website uses cookies to ensure you get the best experience. /Length 221 a stiffness matrix. The Mona Lisa example pictured here provides a simple illustration. If endobj i {\displaystyle v_{2}} 1 stream Eigenvectors, and Diagonal-ization Math 240 Eigenvalues and Eigenvectors Diagonalization Example Example If Ais the matrix A= 1 1 3 5 ; then the vector v = (1;3) is an eigenvector for Abecause Av = 1 1 3 5 1 3 = 4 12 = 4v: The corresponding eigenvalue is = 4. The values of λ that satisfy the equation are the generalized eigenvalues. Display decimals, number of significant digits: Clean. for use in the solution equation, A similar procedure is used for solving a differential equation of the form. /Length 88 A variation is to instead multiply the vector by ц For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it. ) v "(��`0J ��Փ+� J�tV {\displaystyle k} {\displaystyle \psi _{E}} d ͪ����j�tu�tU��(l��@(�'��f�=Ş:�4oH�P��� �M�����g����YhW More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling. �\�. n A = endstream t {\displaystyle 1/{\sqrt {\deg(v_{i})}}} 16.2.1 Prescription for diagonalization of a matrix To “diagonalize” a matrix: I Take a given N N matrix A I Construct a matrix S that has the eigenvectors of A as its columns I Then the matrix (S 1AS) is diagonal and has the eigenvalues of A as its diagonal elements. x�eα V For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. It is a key element of the denition that an eigenvector can never be the zero vector. Ax x= ⇒ −=λ λ ( )IA x0 Let . {\displaystyle {\begin{bmatrix}0&1&2\end{bmatrix}}^{\textsf {T}}} endstream The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. n {\displaystyle v_{1}} << endstream Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Clean Cells or Share Insert in. E %PDF-1.5 T ] {\displaystyle A} ) is a scalar and [43] Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the determinant, which for an {\displaystyle V} x γ / If is an eigenvector of the transpose, it satisfies By transposing both sides of the equation, we get. E endstream {\displaystyle A} The picture is more complicated, but as in the 2 by 2 case, our best insights come from finding the matrix's eigenvectors: that is, those vectors whose direction the transformation leaves unchanged. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. {\displaystyle D} Furthermore, damped vibration, governed by. within the space of square integrable functions. endstream − D Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. Any nonzero vector with v1 = −v2 solves this equation. {\displaystyle E_{1}} A ξ The eigenvectors associated to solve Since we can choose, for example, Moreover, so we can choose, as an eigenvector associated to, the following vector: Therefore, the diagonal matrix of eigenvalues is and the invertible matrix of eigenvectors is The diagonalization is not unique ( (Erste Mitteilung)", Earliest Known Uses of Some of the Words of Mathematics (E), Lemma for linear independence of eigenvectors, "Eigenvalue, eigenfunction, eigenvector, and related terms", "Eigenvalue computation in the 20th century", 10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C, "Neutrinos Lead to Unexpected Discovery in Basic Math", Learn how and when to remove this template message, Eigen Values and Eigen Vectors Numerical Examples, Introduction to Eigen Vectors and Eigen Values, Eigenvectors and eigenvalues | Essence of linear algebra, chapter 10, Same Eigen Vector Examination as above in a Flash demo with sound, Numerical solution of eigenvalue problems, Java applet about eigenvectors in the real plane, Wolfram Language functionality for Eigenvalues, Eigenvectors and Eigensystems, https://en.wikipedia.org/w/index.php?title=Eigenvalues_and_eigenvectors&oldid=991578900, All Wikipedia articles written in American English, Articles with unsourced statements from March 2013, Articles with Russian-language sources (ru), Wikipedia external links cleanup from December 2019, Wikipedia spam cleanup from December 2019, Creative Commons Attribution-ShareAlike License, The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the, The direct sum of the eigenspaces of all of, In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751; published: 1760), The relevant passage of Segner's work was discussed briefly by. n {\displaystyle |\Psi _{E}\rangle } [14], Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,[12] and Alfred Clebsch found the corresponding result for skew-symmetric matrices. We work through two methods of finding the characteristic equation for λ, then use this to find two eigenvalues. The prefix eigen- is adopted from the German word eigen (cognate with the English word own) for "proper", "characteristic", "own". The geometric multiplicity γT(λ) of an eigenvalue λ is the dimension of the eigenspace associated with λ, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. where I is the n by n identity matrix and 0 is the zero vector. {\displaystyle n\times n} . y 1 γ G�(���8������3�?o��T/8�'��ٷΚn�d�s����JEEj�IFjmڤ� ?���b�dM����HU��� Bi�\z��ወ�/�M�#o�Q���A�; ~_/�L�7�1k�/��%�C��5GH���y"o����ߏhUю>�gz endobj denotes the conjugate transpose of + = μ >> deg {\displaystyle u} [13] Charles-François Sturm developed Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. 1 Since the matrix contains diagonal elements only, we sometimes write it in term of a vector. Sponsored Links. This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. where each λi may be real but in general is a complex number. >> μ For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961. >> {\displaystyle D-A} /Filter /FlateDecode k endobj << (c) Diagonalize the matrix A by finding a nonsingular matrix S and a diagonal matrix D such that S − 1 A S = D . << is the eigenvalue and , 1 70 0 obj 2 E ) is a fundamental number in the study of how infectious diseases spread. {\displaystyle \lambda I_{\gamma _{A}(\lambda )}} Thanks to all of you who support me on Patreon. The The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. . . 1 Set P to be the square matrix of order n for which the column vectors are the eigenvectors Cj. stream PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). >> A matrix whose elements above the main diagonal are all zero is called a lower triangular matrix, while a matrix whose elements below the main diagonal are all zero is called an upper triangular matrix. endobj 1 >> V {\displaystyle n} >> where the eigenvector v is an n by 1 matrix. [ ��8V���� ˳�� However, if the entries of A are all algebraic numbers, which include the rationals, the eigenvalues are complex algebraic numbers. An example of a 2-by-2 diagonal matrix is [ 3 0 0 2 ] {\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}, while an example of a 3-by-3 diagonal matrix is [ 6 0 0 0 7 0 0 0 4 ] {\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&7&0\\0&0&4\end{smallm… C While the definition of an eigenvector used in this article excludes the zero vector, it is possible to define eigenvalues and eigenvectors such that the zero vector is an eigenvector.[42]. 50 0 obj This orthogonal decomposition is called principal component analysis (PCA) in statistics. /Length 137 << 1 ] , k , which is a negative number whenever θ is not an integer multiple of 180°. , consider how the definition of geometric multiplicity implies the existence of i is an eigenvector of A corresponding to λ = 1, as is any scalar multiple of this vector. which is the union of the zero vector with the set of all eigenvectors associated with λ. E is called the eigenspace or characteristic space of T associated with λ. λ ƥi| D . (i9w�7�%U���q ��:����� �D � rx��'���ѐ��t��+s�ǵ�C+�� A − is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics: where /Length 82 i Proposition An orthonormal matrix P has the property that P−1 = PT. = ) λ v ) 1 {\displaystyle \gamma _{A}(\lambda )} The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. A I So eigenvalues and eigenvectors are the way to break up a square matrix and find this diagonal matrix lambda with the eigenvalues, lambda 1, lambda 2, to lambda n. That's the purpose. stream Display decimals, number of significant digits: Clean. {\displaystyle n} = /Filter /FlateDecode + A {\displaystyle k} >> endobj v 2 MATH 340: EIGENVECTORS, SYMMETRIC MATRICES, AND ORTHOGONALIZATION (iii) ) (ii): This is similar to the above implication. If Let λi be an eigenvalue of an n by n matrix A. T (iii) If λ i 6= λ j then the eigenvectors are orthogonal. . . This allows one to represent the Schrödinger equation in a matrix form. Therefore, the eigenvalues of A are values of λ that satisfy the equation. /Filter /FlateDecode If the entries of the matrix A are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. k th smallest eigenvalue of the Laplacian. More: Diagonal matrix Jordan decomposition Matrix exponential. H endobj >> 2 γ >> T �:3�^I)�i��K%�V�%%��[_|ס�P��ధaFΤ��z���Ѕ��56���@�p�t9�B��F+���G����8Aᰔ�j���=�}E���V ��-(&��i�s�U�O�#9�Pk݀�a��T���,#�J l��cOtb6� �Ne�g=M����x4����rƞ~1Ǵ$#�9}b` k The picture is more complicated, but as in the 2 by 2 case, our best insights come from finding the matrix's eigenvectors: that is, those vectors whose direction the transformation leaves unchanged. �i��T�X��ȧ|Dq�&Z��+N*;�(nh �����/\1�hgt3��{ q'db����\3�S1S��[Qe�(��-襡w���g� θ {\displaystyle H} , The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of λ �@�G,��2�M�F���Vb�����h9J��2Ų�h���)�����=��C�(�^L&!c���������O8�Po(�^��:[��r;�������6�h�ٌ������`f���mAp���`��AX�5��V ��P~����� ��pr,o��!�t�D�J+��s�e�I�3�����e1 {\displaystyle {\begin{bmatrix}b\\-3b\end{bmatrix}}} ) , {\displaystyle H|\Psi _{E}\rangle } , 0 d ��,���S|ś7�^�L����$�(�$�c�c}J���pƬ@��0F�U����~B�����i���"'2�\��hn���3w=p牡q���r%g��P���3��/�S]� ����z,]Z��k����m{W��� �(p�gc�, ] n %���� A = VΛV –1. 1 0 0 0 0 4 0 0 0 0 6 0 0 0 0 2 It’s not hard to see that adding in the - lambda term to each element on the diag and setting equal to zero would reveal the eigenvalues to be just values on the diag. , and in where T {\displaystyle A^{\textsf {T}}} 0 x�}˱ Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed n. To prove the inequality In particular, for λ = 0 the eigenfunction f(t) is a constant. × A {\displaystyle A} {\displaystyle R_{0}} λ << << The total geometric multiplicity γA is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. endobj >> AV = VΛ. , the These eigenvalues correspond to the eigenvectors, As in the previous example, the lower triangular matrix. endstream v T For example, the linear transformation could be a differential operator like ! A x�u�=N�@�����4>���z�EJg) H��@T��"Q��s4%Gp���0��;���7�7_*��y8�8=�w��da�)�6�_Z7�?8&��o���?��_o�9���3p�EM�X� � is its associated eigenvalue. {\displaystyle {\begin{bmatrix}0&-2&1\end{bmatrix}}^{\textsf {T}},} /Filter /FlateDecode endstream n Moreover, if P is the matrix with the columns C 1, C 2, ..., and C n the n eigenvectors of A, then the matrix P-1 AP is a diagonal matrix. {\displaystyle \mu _{A}(\lambda )\geq \gamma _{A}(\lambda )} :) https://www.patreon.com/patrickjmt !! This condition can be written as the equation. ( /Length 138 A Equation (3) is called the characteristic equation or the secular equation of A. stream Given a particular eigenvalue λ of the n by n matrix A, define the set E to be all vectors v that satisfy Equation (2). A << k . I y On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector �\�@Q.}O_����T. Similarly that the columns of this matrix are the corresponding eigenvectors. The diagonal elements of are the corresponding eigenvalues. [23][24] {\displaystyle (A-\mu I)^{-1}} ξ 1 − In other words, Which is not this matrix. Explicit algebraic formulas for the roots of a polynomial exist only if the degree << {\displaystyle V} Because the eigenspace E is a linear subspace, it is closed under addition. Any nonzero vector with v1 = v2 solves this equation. Right multiplying both sides of the equation by Q−1. d The terms "Eigenvalues" and "Eigenvect… i = >> ) , /Filter /FlateDecode Thus, the vectors vλ=1 and vλ=3 are eigenvectors of A associated with the eigenvalues λ=1 and λ=3, respectively. μ A second key concept in this chapter is the notion of eigenvector and eigenvalue. n λ ξ {\displaystyle E_{1}\geq E_{2}\geq E_{3}} then is the primary orientation/dip of clast, Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either. H , for any nonzero real number Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications. endobj ( {\displaystyle H} Essentially, the matrices A and Λ represent the same linear transformation expressed in two different bases. D {\displaystyle \mathbf {i} ^{2}=-1.}. {\displaystyle v_{\lambda _{3}}={\begin{bmatrix}1&\lambda _{3}&\lambda _{2}\end{bmatrix}}^{\textsf {T}}} [V,D,W] = eig(A,B) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'*B. I The generation time of an infection is the time, is the same as the transpose of a right eigenvector of V If V is nonsingular, this becomes the eigenvalue decomposition. stream For the origin and evolution of the terms eigenvalue, characteristic value, etc., see: Eigenvalues and Eigenvectors on the Ask Dr. det >> xڭϽ�0�3$���h�� bb���::ht�G�QxFҳE����w��z�7��� �4���SP(�,�Ad�>E���銉B�� B�6}jC���"�.I�H� �?�J ����K�W�̵����t8��\d=q�1l);�y��|�ey����P�&K7}� /Length 195 bU�hj5������)� r':� ��h����Ji���F�. The eigenvectors v of this transformation satisfy Equation (1), and the values of λ for which the determinant of the matrix (A − λI) equals zero are the eigenvalues. μ 59 0 obj Similarly, because E is a linear subspace, it is closed under scalar multiplication. , the fabric is said to be planar. As a brief example, which is described in more detail in the examples section later, consider the matrix, Taking the determinant of (A − λI), the characteristic polynomial of A is, Setting the characteristic polynomial equal to zero, it has roots at λ=1 and λ=3, which are the two eigenvalues of A. %E��\�N� th diagonal entry is ( Or we could say that the eigenspace for the eigenvalue 3 is the null space of this matrix. {\displaystyle {\begin{bmatrix}1&0&0\end{bmatrix}}^{\textsf {T}},} 73 0 obj Each point on the painting can be represented as a vector pointing from the center of the painting to that point. /Length 182 endstream λ A Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. endobj 64 0 obj On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ. = 3 x�32�3S0P0bc#3s�C�B.crAɹ\N�\�� stream For the complex conjugate pair of imaginary eigenvalues. / Because the columns of Q are linearly independent, Q is invertible. a matrix whose top left block is the diagonal matrix ;��"ɄԘ͗�e��%24�ͯ��&�V�y�%��+�h&���L��,��p�W?/֟��3)��Dx�Z-��b��7���������{�/��A�7��`�۞i]#�3�/�d�����j�PHÔ + (�Bd�s��� ��=��\��� , which means that the algebraic multiplicity of 35 0 obj [ 49 0 obj Similarly that the columns of this matrix are the corresponding eigenvectors. ) In that case, if is the basis of eigenvectors, and the eigenpairs are , then the construction of and proceeds as in the state above. endstream As a consequence, eigenvectors of different eigenvalues are always linearly independent. >> /Filter /FlateDecode and is therefore 1-dimensional. The spectrum of an operator always contains all its eigenvalues but is not limited to them. − As in the matrix case, in the equation above For other uses, see, Vectors that map to their scalar multiples, and the associated scalars, Eigenvalues and the characteristic polynomial, Eigenspaces, geometric multiplicity, and the eigenbasis for matrices, Diagonalization and the eigendecomposition, Three-dimensional matrix example with complex eigenvalues, Eigenvalues and eigenfunctions of differential operators, Eigenspaces, geometric multiplicity, and the eigenbasis, Associative algebras and representation theory, Cornell University Department of Mathematics (2016), University of Michigan Mathematics (2016), An extended version, showing all four quadrants, representation-theoretical concept of weight, criteria for determining the number of factors, "Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile", "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen.
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