Entry

Math: Matrix: Eigenvector: Eigenvalue: Property: Can you give some properties of eigenvalues?

Jan 21st, 2006 18:36
Knud van Eeden,

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--- Knud van Eeden --- 17 January 2021 - 00:31 am --------------------
Math: Matrix: Eigenvector: Eigenvalue: Property: Can you give some 
properties of eigenvalues?
Properties of the found eigenvalues:
===
-Different to each other
 If the eigenvalues are real and different to each other, then it can 
be proved that
 the the N linear equations are linearly independent
 (use the Wronskian determinant to verify this linear independence.
  If this gives non-zero this means independence)
  [book: Edwards, Charles Henry / Penney, David E. - differential 
equations and boundary value problems / Computing and modeling - p. 
278 'Distinct real eigenvalues']
===
-Perpendicularity
 The eigenvectors for different to each other eigenvalues are 
perpendicular to each other
 (this is simular to saying that the N linear equations are linearly 
independent)
  [book: Bronshtein, I. N. / Semendyayev, K. A. - handbook of 
mathematics - page 150 'Eigenvectors belonging to distinct eigenvalues 
of a symmetric matrix are mutually orthogonal']
  [book: Golab, Stanislav - tensor calculus - p. 125 'The eigenvalues 
of a tensor']
  [book: Kibble, T. W. B. - classical mechanics - p. 321]
  [book: Slater, John C. / Frank, Nathaniel H. - mechanics - p. 
275 '... show that the vectors so found are orthogonal...']
===
-Symmetry
 All the eigenvalues for a symmetric matrix are real
 ---
  [book: Bronshtein, I. N. / Semendyayev, K. A. - handbook of 
mathematics - page 150 'All the eigenvalues for a symmetric matrix are 
real']
 ---
 Note:
  As any inertia tensor is symmetric (because the products of inertia,
  e.g. xy versus yx, xz versus zx, occur in symmetric positions) thus
  all the eigenvalues are real in that case
===
-Sum
 The sum of the eigenvalues equals the trace (=sum of the values on 
the diagonal of the given determinant) of the given matrix A.
 (you can use this property e.g. to check the found values of your
  eigenvalues, and or to find the last eigenvalues if you all the
  others but one).
  ---
  e.g. if given a 2 x 2 matrix A then
   lambda1 + lambda2 = (trace of the given 2 x 2 matrix A)
  ---
  e.g. if given a 3 x 3 matrix A then
   lambda1 + lambda2 + lambda3 = (trace of the given 3 x 3 matrix A)
  ...
  e.g. if given a N x N matrix A then
   lambda1 + lambda2 + lambda3 + ... + lambdaN = (trace of the given N 
x N matrix A)
  ---
  [book: Bronshtein, I. N. / Semendyayev, K. A. - handbook of 
mathematics - p. 150 'Eigenvalues and eigenvectors']
  [book: Strang, Gilbert - introduction to applied mathematics - p. 
51 'Eigenvalues and dynamical systems']
  [video lecture: Strang, Gilbert - Eigenvectors and eigenvalues]
===
-Product
 The product of the eigenvalues equals the determinant of the given 
matrix A
  ---
  e.g. if given a 2 x 2 matrix A then
   lambda1 . lambda2 = (determinant of the given 2 x 2 matrix A)
  ---
  e.g. if given a 3 x 3 matrix A then
   lambda1 . lambda2 . lambda3 = (determinant of the given 3 x 3 
matrix A)
  ---
  e.g. if given a N x N matrix A then
   lambda1 . lambda2 . lambda3 . ... . lambdaN = (determinant of the 
given N x N matrix A)
  ---
  [book: Bronshtein, I. N. / Semendyayev, K. A. - handbook of 
mathematics - p. 150 'Eigenvalues and eigenvectors']
  [book: Strang, Gilbert - introduction to applied mathematics - p. 
51 'Eigenvalues and dynamical systems']
---
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Internet: see also:
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Math: Transformation: Eigenvector: Eigenvalue: Link: Can you give an 
overview of links?
http://www.faqts.com/knowledge_base/view.phtml/aid/39001/fid/1856
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