2021년 2월 17일 (수) 07:49에 Pythagoras0님의 편집

2021-02-17T07:49:20Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-26T12:06:35Z

메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2020-12-22T06:03:06Z

노트: 새 문단

새 문서

== 노트 ==

===위키데이터===
* ID : [https://www.wikidata.org/wiki/Q21406831 Q21406831]
===말뭉치===
# So for example, choosing y=2 yeilds the vector <3,2> which is thus an eigenvector that has eigenvalue k=3.<ref name="ref_b3ab6998">[http://sites.science.oregonstate.edu/math/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/eigen/eigen.html Eigenvalues and Eigenvectors]</ref>
# From the examples above we can infer a property of eigenvectors and eigenvalues: eigenvectors from distinct eigenvalues are linearly independent.<ref name="ref_b3ab6998" />
# There is a very important class of matrices called symmetric matrices that have quite nice properties concerning eigenvalues and eigenvectors.<ref name="ref_b3ab6998" />
# We must choose values of s and t that yield two orthogonal vectors (the third comes from the eigenvalue k=8).<ref name="ref_b3ab6998" />
# This page is a brief introduction to eigenvalue/eigenvector problems (don't worry if you haven't heard of the latter).<ref name="ref_0f0205b4">[https://lpsa.swarthmore.edu/MtrxVibe/EigMat/MatrixEigen.html Eigenvalues and Eigenvectors]</ref>
# For each eigenvalue there will be an eigenvector for which the eigenvalue equation is true.<ref name="ref_0f0205b4" />
# Going through the same procedure for the second eigenvalue: Again, the choice of +1 and -2 for the eigenvector was arbitrary; only their ratio is important.<ref name="ref_0f0205b4" />
# Eigenvectors and eigenvalues live in the heart of the data science field.<ref name="ref_a0d289db">[https://medium.com/fintechexplained/what-are-eigenvalues-and-eigenvectors-a-must-know-concept-for-machine-learning-80d0fd330e47 What are Eigenvalues and Eigenvectors?]</ref>
# This article will aim to explain what eigenvectors and eigenvalues are, how they are calculated and how we can use them.<ref name="ref_a0d289db" />
# Eigenvalues and eigenvectors form the basics of computing and mathematics.<ref name="ref_a0d289db" />
# I will then illustrate how eigenvectors and eigenvalues are calculated.<ref name="ref_a0d289db" />
# If the eigenvalue is negative, the direction is reversed.<ref name="ref_44493f54">[https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Eigenvalues and eigenvectors]</ref>
# Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations.<ref name="ref_44493f54" />
# Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue.<ref name="ref_44493f54" />
# referred to as the eigenvalue equation or eigenequation.<ref name="ref_44493f54" />
# If all eigenvalues are different, then plugging these back in gives independent equations for the components of each corresponding eigenvector, and the system is said to be nondegenerate.<ref name="ref_b4e0db9f">[https://mathworld.wolfram.com/Eigenvalue.html Eigenvalue -- from Wolfram MathWorld]</ref>
# If the eigenvalues are -fold degenerate, then the system is said to be degenerate and the eigenvectors are not linearly independent.<ref name="ref_b4e0db9f" />
# This vignette uses an example of a \(3 \times 3\) matrix to illustrate some properties of eigenvalues and eigenvectors.<ref name="ref_4deeebde">[https://cran.r-project.org/web/packages/matlib/vignettes/eigen-ex1.html Eigenvalues and Eigenvectors: Properties]</ref>
# this is the eigenvalue associated with that eigenvector.<ref name="ref_93b37600">[https://www.khanacademy.org/math/linear-algebra/alternate-bases/eigen-everything/v/linear-algebra-introduction-to-eigenvalues-and-eigenvectors Introduction to eigenvalues and eigenvectors (video)]</ref>
# And it's corresponding eigenvalue is 1.<ref name="ref_93b37600" />
# And it's corresponding eigenvalue is minus 1.<ref name="ref_93b37600" />
# So in this case, this would be an eigenvector of A, and this would be the eigenvalue associated with the eigenvector.<ref name="ref_93b37600" />
# Taking the determinant of the terms within the parenthesis (Equation 3) and solving the resulting system of linear equations will provide the eigenvalues.<ref name="ref_a0ecac94">[https://deepai.org/machine-learning-glossary-and-terms/eigenvalue Eigenvalue]</ref>
# In most undergraduate linear algebra courses, eigenvalues (and their cousins, the eigenvectors) play a prominent role.<ref name="ref_8cf0135b">[https://www.maa.org/press/periodicals/convergence/math-origins-eigenvectors-and-eigenvalues Math Origins: Eigenvectors and Eigenvalues]</ref>
# In today's language, we would say that Cauchy's research program was to show that a symmetric matrix has real eigenvalues.<ref name="ref_8cf0135b" />
# In "Sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des mouvements des planétes" (1829), Cauchy used the Lagrange multiplier method to begin his eigenvalue problem.<ref name="ref_8cf0135b" />
# \times (n-1)\) minors within the eigenvalue matrix whose determinants would be complex conjugates of each other.<ref name="ref_8cf0135b" />
# So, how do we go about finding the eigenvalues and eigenvectors for a matrix?<ref name="ref_3b96cd28">[https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx Review : Eigenvalues & Eigenvectors]</ref>
# Knowing this will allow us to find the eigenvalues for a matrix.<ref name="ref_3b96cd28" />
# Once we have the eigenvalues we can then go back and determine the eigenvectors for each eigenvalue.<ref name="ref_3b96cd28" />
# To find eigenvalues of a matrix all we need to do is solve a polynomial.<ref name="ref_3b96cd28" />
# The symbol ψ (psi) represents an eigenfunction (proper or characteristic function) belonging to that eigenvalue.<ref name="ref_b3be5402">[https://www.britannica.com/science/eigenvalue Eigenvalue | mathematics]</ref>
# a Hamiltonian, or energy, operator and the eigenvalues are energy values, but operators corresponding to other dynamical variables such as total angular momentum are also used.<ref name="ref_b3be5402" />
# Experimental measurements of the proper dynamical variable will yield eigenvalues.<ref name="ref_b3be5402" />
# Eigenvalues and eigenvectors can be found all over mathematics, and especially applied mathematics.<ref name="ref_76e346ab">[https://towardsdatascience.com/eigenvalues-and-eigenvectors-89483fb56d56 Eigenvalues and Eigenvectors]</ref>
# I knew that the solution to the PCA problem was the eigenvalue decomposition of the Sample Variance-Covariance Matrix.<ref name="ref_76e346ab" />
# With a 2x2 matrix, we can solve for eigenvalues by hand.<ref name="ref_76e346ab" />
# But HOW do you compute eigenvalues for large matrices?<ref name="ref_76e346ab" />
# Suppose F has distinct eigenvalues with negative real parts and that the dominant eigenvalue is real.<ref name="ref_e7efc8f8">[https://www.x-mol.com/paperRedirect/5539006 Eigenvalues of the covariance matrix as early warning signals for critical transitions in ecological systems]</ref>
# This is because the dynamics along the direction of the eigenvector corresponding to the dominant eigenvalue become slower as the dominant eigenvalue of the Jacobian matrix approaches zero.<ref name="ref_e7efc8f8" />
# Because the other eigenvalues are not approaching zero at the same rate as the dominant eigenvalue, the variance of the dynamics along that direction increases at a much higher rate.<ref name="ref_e7efc8f8" />
# For simplicity the example we give in the following sections only has real eigenvalues.<ref name="ref_e7efc8f8" />
# In this section, we define eigenvalues and eigenvectors.<ref name="ref_8188aa01">[https://textbooks.math.gatech.edu/ila/eigenvectors.html Eigenvalues and Eigenvectors]</ref>
# We will find the eigenvalues and eigenvectors of A without doing any computations.<ref name="ref_8188aa01" />
# The vector Av has the same length as v , but the opposite direction, so the associated eigenvalue is − 1.<ref name="ref_8188aa01" />
# This means that w is an eigenvector with eigenvalue 1.<ref name="ref_8188aa01" />
# By default eig does not always return the eigenvalues and eigenvectors in sorted order.<ref name="ref_c4433b40">[https://www.mathworks.com/help/matlab/ref/eig.html Eigenvalues and eigenvectors]</ref>
# Extract the eigenvalues from the diagonal of D using diag(D) , then sort the resulting vector in ascending order.<ref name="ref_c4433b40" />
# Both (V,D) and (Vs,Ds) produce the eigenvalue decomposition of A .<ref name="ref_c4433b40" />
# Another important use of eigenvalues and eigenvectors is diagonalisation, and it is to this that we now turn.<ref name="ref_2bebf794">[http://wwwf.imperial.ac.uk/metric/metric_public/matrices/eigenvalues_and_eigenvectors/eigenvalues2.html Eigenvalues and eigenvectors of 3 by 3 matrices]</ref>
# In structural design optimization, the eigenvalues may appear either as objective function or as constraint functions.<ref name="ref_fa09d9b8">[https://link.springer.com/chapter/10.1007/978-1-4612-1962-0_8 Eigenvalues in Optimum Structural Design]</ref>
# Free vibration frequencies and load magnitudes in stability analysis are computed by solving large and sparse generalized symmetric eigenvalue problems.<ref name="ref_fa09d9b8" />
# Eigenvalue constraints can therefore be represented using matrix inequalities as opposed to directly referring to the eigenvalues themselves.<ref name="ref_fa09d9b8" />
# An overview of different structural design problems where eigenvalues appear as either constraints or objective function is given.<ref name="ref_fa09d9b8" />
# we are going to have p eigenvalues, \(\lambda _ { 1 , } \lambda _ { 2 } \dots \lambda _ { p }\).<ref name="ref_ec5547e6">[https://online.stat.psu.edu/stat505/lesson/4/4.5 4.5 - Eigenvalues and Eigenvectors]</ref>
# we obtain the desired eigenvalues.<ref name="ref_ec5547e6" />
# In general, we will have p solutions and so there are p eigenvalues, not necessarily all unique.<ref name="ref_ec5547e6" />
# Finding the eigenvalues and eigenvectors of a linear operator is one of the most important problems in Linear Algebra.<ref name="ref_ea7b623a">[https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/07%3A_Eigenvalues_and_Eigenvectors/7.02%3A_Eigenvalues 7.2: Eigenvalues]</ref>
# (As an example, quantum mechanics is based upon understanding the eigenvalues and eigenvectors of operators on specifically defined vector spaces.<ref name="ref_ea7b623a" />
# The projection map \(P:\mathbb{R}^3 \to \mathbb{R}^3\) defined by \(P(x,y,z)=(x,y,0)\) has eigenvalues \(0\) and \(1\).<ref name="ref_ea7b623a" />
# Let \(T\in \mathcal{L}(V,V)\), and let \(\lambda\in \mathbb{F}\) be an eigenvalue of \(T\).<ref name="ref_ea7b623a" />
# It is often convenient to solve eigenvalue problems like using matrices.<ref name="ref_cf190fdc">[https://quantummechanics.ucsd.edu/ph130a/130_notes/node248.html Eigenvalue Problems with Matrices]</ref>
# In Section 12, we developed the idea of eigenvalues and eigenvectors in the case of linear transformations \(\Re^{2}\rightarrow \Re^{2}\).<ref name="ref_c5d21ce9">[https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/12%3A_Eigenvalues_and_Eigenvectors/12.02%3A_The_Eigenvalue-Eigenvector_Equation 12.2: The Eigenvalue-Eigenvector Equation]</ref>
# These eigenvalues could be real or complex or zero, and they need not all be different.<ref name="ref_c5d21ce9" />
# To find the eigenvectors associated to each eigenvalue, we solve the homogeneous system \((M-\lambda_{i}I)X=0\) for each \(i\).<ref name="ref_c5d21ce9" />
# So the multiplicity two eigenvalue has two independent eigenvectors, \(\begin{pmatrix}-1\\1\\0\end{pmatrix}\) and \(\begin{pmatrix}1\\0\\1\end{pmatrix}\) that determine an invariant plane.<ref name="ref_c5d21ce9" />
# Those lines are eigenspaces, and each has an associated eigenvalue.<ref name="ref_9993bab9">[https://setosa.io/ev/eigenvectors-and-eigenvalues/ Eigenvectors and Eigenvalues explained visually]</ref>
# So far we've only looked at systems with real eigenvalues.<ref name="ref_9993bab9" />
# The eigenvalues are plotted in the real/imaginary plane to the right.<ref name="ref_9993bab9" />
# To get more practice with applications of eigenvalues/vectors, also ceck out the excellent Differential Equations course.<ref name="ref_9993bab9" />
# Perhaps the most used type of matrix decomposition is the eigendecomposition that decomposes a matrix into eigenvectors and eigenvalues.<ref name="ref_9f3a8e35">[https://machinelearningmastery.com/introduction-to-eigendecomposition-eigenvalues-and-eigenvectors/ Gentle Introduction to Eigenvalues and Eigenvectors for Machine Learning]</ref>
# A matrix could have one eigenvector and eigenvalue for each dimension of the parent matrix.<ref name="ref_9f3a8e35" />
# Not all square matrices can be decomposed into eigenvectors and eigenvalues, and some can only be decomposed in a way that requires complex numbers.<ref name="ref_9f3a8e35" />
# However, we often want to decompose matrices into their eigenvalues and eigenvectors.<ref name="ref_9f3a8e35" />
# The roots of are called the eigenvalues of .<ref name="ref_cd9b807f">[http://www.netlib.org/utk/people/JackDongarra/etemplates/node22.html Eigenvalues and Eigenvectors]</ref>
# Proposition Let be a matrix and its eigenvalues.<ref name="ref_724bfa73">[https://www.statlect.com/matrix-algebra/properties-of-eigenvalues-and-eigenvectors Properties of eigenvalues and eigenvectors]</ref>
# Therefore, the eigenvalues of are Transposition does not change the eigenvalues and multiplication by doubles them.<ref name="ref_724bfa73" />
# This problem is further transformed to the eigenvalue problem.<ref name="ref_83ab36e6">[https://www.intechopen.com/books/applied-linear-algebra-in-action/eigenvalue-problems Eigenvalue Problems]</ref>
# The scalar λ is called an eigenvalue of A, and x is an eigenvector of A corresponding to λ.<ref name="ref_83ab36e6" />
# (The QR algorithm is used for determining all the eigenvalues of a matrix.<ref name="ref_83ab36e6" />
# The eigenvalue problem is related to the homogeneous system of linear equations, as we will see in the following discussion.<ref name="ref_83ab36e6" />
# It is not too difficult to compute eigenvalues and their corresponding eigenvectors when the matrix transformation at hand has a clear geometric interpretation.<ref name="ref_3a0395ac">[https://brilliant.org/wiki/eigenvalues-and-eigenvectors/ Eigenvalues and Eigenvectors]</ref>
# To determine the eigenvalues of a matrix A A A, one solves for the roots of p A ( x ) p_{A} (x) pA(x), and then checks if each root is an eigenvalue.<ref name="ref_3a0395ac" />
# A constrained non-homogeneous linear eigenvalue problem is introduced.<ref name="ref_b9726125">[https://royalsocietypublishing.org/doi/10.1098/rspa.2009.0426 A constrained eigenvalue problem and nodal and modal control of vibrating systems]</ref>
# It is shown that the problem may be transformed to a singular unsymmetric generalized eigenvalue problem.<ref name="ref_b9726125" />
# The problem is transformed to a singular generalized eigenvalue problem.<ref name="ref_b9726125" />
# Equation (2.4) is an unsymmetric generalized eigenvalue problem with singular M. Depending on the given data, K may become singular as well.<ref name="ref_b9726125" />
===소스===
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 * ID :  [https://www.wikidata.org/wiki/Q21406831 Q21406831]

← 이전 판		2020년 12월 26일 (토) 12:06 판
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		+	* ID : [https://www.wikidata.org/wiki/Q21406831 Q21406831]

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2021년 2월 17일 (수) 07:49에 Pythagoras0님의 편집

Pythagoras0: /* 메타데이터 */ 새 문단

Pythagoras0: /* 노트 */ 새 문단