An engineering problem can be reduced to a form [A]{x} = {y} and then to an eigenvalue problem [A]{V} = λ {V}, whose solution i.e. det(A - λ*I) = 0 gives you eigenvalues (λ) and eigenvectors {V}. What do eigenvalues and eigenvectors physically mean ????
In mathematical sense the system matrix [A] can be thought of as a transformation matrix whose effect varies with directions and is maximum in certain direction. These maxima are called eigenvalues and the corresponding directions are known as eigenvectors. Let us explain it with a couple of examples:
1. A cantilever beam is given an initial deflection and then released. Its vibration is an eigenvalue problem and the eigenvalues are the natural frequencies of vibration and the eigenvectors are the mode shapes of the vibration.
2. Google search is an eigenvalue problem. When you search a 'keyword'. Your search engine goes to millions of websites and an eigenvalue problem is formulated in which the system matrix, P is called Markov Transition Matrix. The eigenvalue problem is given as [P]{W} = 1 {W}. Since each P is probability value, the eigenvalue λ = 1. The Pij is the conditional probability that we are on the i-th webpage, given that we gave search at the j-th webpage. Pij for each webpage i in the web is calculated based on the incoming link from neighboring webpage j, Lij and number of outgoing links from the neighboring webpage, j, Nj . So equation for weight at webpage i becomes:
Wi = LijWj / Nj = Pij Wj Where Pij = Lij / Nj and weight of j-th webpage, Wj.
Suppose our world wide web has only 3 webpages, Then our equations will be:
W1 = P11 W1 + P12 W2 + P13 W3
W2 = P21 W1 + P22 W2 + P23 W3
W3 = P31 W1 + P32 W2 + P33 W3
Which becomes {W}3x1 = [P]3x3 {W}3x1 => [P]3x3 {W}3x1 = λ {W}3x1 …… (1)
For this problem λ = 1 and let us say our solution for the eigenvalue problem (1) is {W}3x1 = {0.65, 0.5, 0.95}T, then since third value is the largest, means your web search will display the websites in following order: Third, First, Second.
Principal stresses in mechanics are the eigenvalues of the stress tensor:
https://en.wikipedia.org/wiki/Stress_(mechanics)
Eigenvalues give the displacement of an atom or a molecule from its equilibrium position and the direction of displacement is given by eigen vectors (after applying force)..... Rest is to observe and analyze the dynamics of the system...
Edit: You might have seen a long steel/iron rod attached for protection alongside the stairs (particularly in India). Now hit it, not too hard, with your hand for 2-3 times and, touch gently and feel the vibrations. This will make you understand the statement given above.
Edit: Understand the above statement fro Guitar also.
Used in a theory of systems in differential eqs. In physics, eigen vectors are inertia tensor and eigen values are moment of inertiaț.
An engineering problem can be reduced to a form [A]{x} = {y} and then to an eigenvalue problem [A]{V} = λ {V}, whose solution i.e. det(A - λ*I) = 0 gives you eigenvalues (λ) and eigenvectors {V}. What do eigenvalues and eigenvectors physically mean ????
In mathematical sense the system matrix [A] can be thought of as a transformation matrix whose effect varies with directions and is maximum in certain direction. These maxima are called eigenvalues and the corresponding directions are known as eigenvectors. Let us explain it with a couple of examples:
1. A cantilever beam is given an initial deflection and then released. Its vibration is an eigenvalue problem and the eigenvalues are the natural frequencies of vibration and the eigenvectors are the mode shapes of the vibration.
2. Google search is an eigenvalue problem. When you search a 'keyword'. Your search engine goes to millions of websites and an eigenvalue problem is formulated in which the system matrix, P is called Markov Transition Matrix. The eigenvalue problem is given as [P]{W} = 1 {W}. Since each P is probability value, the eigenvalue λ = 1. The Pij is the conditional probability that we are on the i-th webpage, given that we gave search at the j-th webpage. Pij for each webpage i in the web is calculated based on the incoming link from neighboring webpage j, Lij and number of outgoing links from the neighboring webpage, j, Nj . So equation for weight at webpage i becomes:
Wi = LijWj / Nj = Pij Wj Where Pij = Lij / Nj and weight of j-th webpage, Wj.
Suppose our world wide web has only 3 webpages, Then our equations will be:
W1 = P11 W1 + P12 W2 + P13 W3
W2 = P21 W1 + P22 W2 + P23 W3
W3 = P31 W1 + P32 W2 + P33 W3
Which becomes {W}3x1 = [P]3x3 {W}3x1 => [P]3x3 {W}3x1 = λ {W}3x1 …… (1)
For this problem λ = 1 and let us say our solution for the eigenvalue problem (1) is {W}3x1 = {0.65, 0.5, 0.95}T, then since third value is the largest, means your web search will display the websites in following order: Third, First, Second.
Eigen-values and eigen-vectors are associated with linear models, which are usual in engineering as a first approximation. Three emblematic cases:
The linear description of material deformation leads to the strain tensor. The principal strains and the principal directions of strain are its eigen-values and eigen-vectors.
The linear description of body forces leads to the stress tensor. The principal stresses and principal directions of stresses are its eigen-values and eigen-vectors.
In the linear modelizations of small vibrations, the eigen-values are de natural frequencies and the eigen-vectors describe the natural modes of vibration.
Eigen-values is useful in analyzing the stability of a dynamic system in the neighborhood of its equilibrium point
Eigen values and eigen vectors find wide range application varying from image processing to google page rank search engine operation etc
Eigen values and eigen vectors are largely used in linear system control for stability analysis, system decomposition, system analysis and design using state represention. Linear system control is based essentialy on linear algebra.
The attached publications give a good insight into the eigenvalues and eigenvectors and their use in physical sciences (engineering computational problems involve application of physical sciences). I would like to add that there is a huge upsurge in modeling of engineering problems which involve solution of differential equations. Matrix methods are mostly widely used for the solution of differential equations using numerical methods. The latter are well suited for solving non-linear problems as well as those with large number of variables (which have be considered in realistic modeling of actual systems). Determination of eigenvalues and eigenvectors has become an essential step in arriving at the final solution to the problem studied.
In structural engineering eigenvalues are usually used to determine the response of stuctures under random or stochastic processes. For example response of a building subjected to earthquake or wind-storm
The attached article will be helpful to those interested in computing eigenvalues and eigenvectors for industrial sized matrices (which are commonly encountered in many engineering problems).
So many applications: stress, strain, moment of inertia, rigid rotation, vibration (modal analysis), stability and many other
I recommend you a book in this subject: Eigenvalue and Eigenvector Problems in Applied Mechanics. Springer 2019.
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