Eigenvalues and Eigenvectors - Revision history

Khanh at 20:16, 5 November 2021

2021-11-05T20:16:55Z

Khanh: /* Eigenvector-eigenvalue identity */

2021-11-04T23:30:40Z

Eigenvector-eigenvalue identity

Khanh: /* Eigenvalues and eigenvectors of matrices */

2021-11-04T23:28:36Z

Eigenvalues and eigenvectors of matrices

Khanh at 23:22, 4 November 2021

2021-11-04T23:22:58Z

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Khanh: /* Dynamic equations */

2021-11-04T22:37:17Z

Dynamic equations

Khanh: /* Eigenvectors */

2021-11-04T21:59:32Z

Eigenvectors

Khanh at 21:53, 4 November 2021

2021-11-04T21:53:06Z

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Lila: /* Resources */

2021-09-24T20:41:09Z

Resources

Lila at 20:26, 24 September 2021

2021-09-24T20:26:23Z

Lila at 20:25, 24 September 2021

2021-09-24T20:25:18Z

@@ Line 114: / Line 114: @@
 If it occurs that {{mvar|v}} and {{mvar|w}} are scalar multiples, that is if
 :<math>A \mathbf v = \mathbf w = \lambda \mathbf v,</math>   ('''1''')
-then {{math|'''v'''}} is an '''eigenvector''' of the linear transformation {{mvar|A}} and the scale factor {{mvar|λ}} is the '''eigenvalue''' corresponding to that eigenvector. Equation ({{EquationNote|1}}) is the '''eigenvalue equation''' for the matrix {{mvar|A}}.
+then {{math|'''v'''}} is an '''eigenvector''' of the linear transformation {{mvar|A}} and the scale factor {{mvar|λ}} is the '''eigenvalue''' corresponding to that eigenvector. Equation ('''1''') is the '''eigenvalue equation''' for the matrix {{mvar|A}}.
 Equation ('''1''') can be stated equivalently as
@@ Line 159: / Line 159: @@
 Let ''λ''<sub>''i''</sub> be an eigenvalue of an ''n'' by ''n'' matrix ''A''. The '''algebraic multiplicity''' ''μ''<sub>''A''</sub>(''λ''<sub>''i''</sub>) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer ''k'' such that (''λ'' − ''λ''<sub>''i''</sub>)<sup>''k''</sup> divides evenly that polynomial.
-Suppose a matrix ''A'' has dimension ''n'' and ''d'' ≤ ''n'' distinct eigenvalues. Whereas Equation ({{EquationNote|4}}) factors the characteristic polynomial of ''A'' into the product of ''n'' linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of ''d'' terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
+Suppose a matrix ''A'' has dimension ''n'' and ''d'' ≤ ''n'' distinct eigenvalues. Whereas Equation ('''4''') factors the characteristic polynomial of ''A'' into the product of ''n'' linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of ''d'' terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
 :<math>|A - \lambda I| = (\lambda_1 - \lambda)^{\mu_A(\lambda_1)}(\lambda_2 - \lambda)^{\mu_A(\lambda_2)} \cdots (\lambda_d - \lambda)^{\mu_A(\lambda_d)}.</math>
-If ''d'' = ''n'' then the right-hand side is the product of ''n'' linear terms and this is the same as Equation ({{EquationNote|4}}). The size of each eigenvalue's algebraic multiplicity is related to the dimension ''n'' as
+If ''d'' = ''n'' then the right-hand side is the product of ''n'' linear terms and this is the same as Equation ('''4'''). The size of each eigenvalue's algebraic multiplicity is related to the dimension ''n'' as
 :<math>\begin{align}
 &\leq \mu_A(\lambda_i) \leq n, \\
@@ Line 172: / Line 172: @@
 ===Eigenspaces, geometric multiplicity, and the eigenbasis for matrices===
-Given a particular eigenvalue ''λ'' of the ''n'' by ''n'' matrix ''A'', define the set ''E'' to be all vectors '''v''' that satisfy Equation ({{EquationNote|2}}),
+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'''),
 :<math>E = \left\{\mathbf{v} : \left(A - \lambda I\right) \mathbf{v} = \mathbf{0}\right\}.</math>
@@ Line 214: / Line 214: @@
 === Left and right eigenvectors ===
-Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a '''right eigenvector''', namely a ''column'' vector that ''right'' multiplies the <math>n \times n</math> matrix <math>A</math> in the defining equation, Equation ({{EquationNote|1}}),
+Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a '''right eigenvector''', namely a ''column'' vector that ''right'' multiplies the <math>n \times n</math> matrix <math>A</math> in the defining equation, Equation ('''1'''),
 :<math>A \mathbf v = \lambda \mathbf v.</math>
@@ Line 223: / Line 223: @@
 :<math>A^\textsf{T} \mathbf u^\textsf{T} = \kappa \mathbf u^\textsf{T}.</math>
-Comparing this equation to Equation ({{EquationNote|1}}), it follows immediately that a left eigenvector of <math>A</math> is the same as the transpose of a right eigenvector of <math>A^\textsf{T}</math>, with the same eigenvalue. Furthermore, since the characteristic polynomial of <math>A^\textsf{T}</math> is the same as the characteristic polynomial of <math>A</math>, the eigenvalues of the left eigenvectors of <math>A</math> are the same as the eigenvalues of the right eigenvectors of <math>A^\textsf{T}</math>.
+Comparing this equation to Equation ('''1'''), it follows immediately that a left eigenvector of <math>A</math> is the same as the transpose of a right eigenvector of <math>A^\textsf{T}</math>, with the same eigenvalue. Furthermore, since the characteristic polynomial of <math>A^\textsf{T}</math> is the same as the characteristic polynomial of <math>A</math>, the eigenvalues of the left eigenvectors of <math>A</math> are the same as the eigenvalues of the right eigenvectors of <math>A^\textsf{T}</math>.
 ===Diagonalization and the eigendecomposition===
@@ Line 293: / Line 293: @@
 is an eigenvector of ''A'' corresponding to ''λ'' = 1, as is any scalar multiple of this vector.
-For ''λ''=3, Equation ({{EquationNote|2}}) becomes
+For ''λ''=3, Equation ('''2''') becomes
 :<math>\begin{align}
    (A - 3I)\mathbf{v}_{\lambda=3} &=
@@ Line 487: / Line 487: @@
 We say that a nonzero vector '''v''' ∈ ''V'' is an '''eigenvector''' of ''T'' if and only if there exists a scalar ''λ'' ∈ ''K'' such that
-{{NumBlk|:
- |<math>T(\mathbf{v}) = \lambda \mathbf{v}.</math>
+:<math>T(\mathbf{v}) = \lambda \mathbf{v}.</math>    ('''5''')
 This equation is called the eigenvalue equation for ''T'', and the scalar ''λ'' is the '''eigenvalue''' of ''T'' corresponding to the eigenvector '''v'''. ''T''('''v''') is the result of applying the transformation ''T'' to the vector '''v''', while ''λ'''''v''' is the product of the scalar ''λ'' with '''v'''.
@@ Line 524: / Line 522: @@
 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.
-Consider again the eigenvalue equation, Equation ({{EquationNote|5}}). Define an '''eigenvalue''' to be any scalar ''λ'' ∈ ''K'' such that there exists a nonzero vector '''v''' ∈ ''V'' satisfying Equation ({{EquationNote|5}}). It is important that this version of the definition of an eigenvalue specify that the vector be nonzero, otherwise by this definition the zero vector would allow any scalar in ''K'' to be an eigenvalue. Define an '''eigenvector''' '''v''' associated with the eigenvalue ''λ'' to be any vector that, given ''λ'', satisfies Equation ({{EquationNote|5}}). Given the eigenvalue, the zero vector is among the vectors that satisfy Equation ({{EquationNote|5}}), so the zero vector is included among the eigenvectors by this alternate definition.
+Consider again the eigenvalue equation, Equation ('''5'''). Define an '''eigenvalue''' to be any scalar ''λ'' ∈ ''K'' such that there exists a nonzero vector '''v''' ∈ ''V'' satisfying Equation ('''5'''). It is important that this version of the definition of an eigenvalue specify that the vector be nonzero, otherwise by this definition the zero vector would allow any scalar in ''K'' to be an eigenvalue. Define an '''eigenvector''' '''v''' associated with the eigenvalue ''λ'' to be any vector that, given ''λ'', satisfies Equation ('''5'''). Given the eigenvalue, the zero vector is among the vectors that satisfy Equation ('''5'''), so the zero vector is included among the eigenvectors by this alternate definition.
 ===Spectral theory===

@@ Line 462: / Line 462: @@
 For a Hermitian matrix, the norm squared of the ''j''th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix,
 :<math>|v_{i,j}|^2 = \frac{\prod_{k}{(\lambda_i-\lambda_k(M_j))}}{\prod_{k \neq i}{(\lambda_i-\lambda_k)}},</math>
-where <math display="inline">M_j</math> is the [[submatrix]] formed by removing the ''j''th row and column from the original matrix.
+where <math display="inline">M_j</math> is the submatrix formed by removing the ''j''th row and column from the original matrix.
 ==Eigenvalues and eigenfunctions of differential operators==

@@ Line 113: / Line 113: @@
 If it occurs that {{mvar|v}} and {{mvar|w}} are scalar multiples, that is if
-{{NumBlk|:|<math>A \mathbf v = \mathbf w = \lambda \mathbf v,</math>|{{EquationRef|1}}}}
+:<math>A \mathbf v = \mathbf w = \lambda \mathbf v,</math>   ('''1''')
 then {{math|'''v'''}} is an '''eigenvector''' of the linear transformation {{mvar|A}} and the scale factor {{mvar|λ}} is the '''eigenvalue''' corresponding to that eigenvector. Equation ({{EquationNote|1}}) is the '''eigenvalue equation''' for the matrix {{mvar|A}}.
-Equation ({{EquationNote|1}}) can be stated equivalently as
+Equation ('''1''') can be stated equivalently as
-{{NumBlk|:
+:<math>\left(A - \lambda I \right) \mathbf v = \mathbf 0,</math>   ('''2''')
 where {{mvar|I}} is the {{mvar|n}} by {{mvar|n}} identity matrix and '''0''' is the zero vector.
@@ Line 126: / Line 123: @@
 ===Eigenvalues and the characteristic polynomial===
-Equation ({{EquationNote|2}}) has a nonzero solution ''v'' if and only if the determinant of the matrix (''A'' − ''λI'') is zero. Therefore, the eigenvalues of ''A'' are values of ''λ'' that satisfy the equation
+Equation ('''2''') has a nonzero solution ''v'' if and only if the determinant of the matrix (''A'' − ''λI'') is zero. Therefore, the eigenvalues of ''A'' are values of ''λ'' that satisfy the equation
-{{NumBlk|:
+:<math>|A-\lambda I| = 0</math>   ('''3''')
-Using Leibniz' rule for the determinant, the left-hand side of Equation ({{EquationNote|3}}) is a polynomial function of the variable ''λ'' and the degree of this polynomial is ''n'', the order of the matrix ''A''. Its coefficients depend on the entries of ''A'', except that its term of degree ''n'' is always (−1)<sup>''n''</sup>''λ''<sup>''n''</sup>. This polynomial is called the ''characteristic polynomial'' of ''A''. Equation ({{EquationNote|3}}) is called the ''characteristic equation'' or the ''secular equation'' of ''A''.
+Using Leibniz' rule for the determinant, the left-hand side of Equation ('''3''') is a polynomial function of the variable ''λ'' and the degree of this polynomial is ''n'', the order of the matrix ''A''. Its coefficients depend on the entries of ''A'', except that its term of degree ''n'' is always (−1)<sup>''n''</sup>''λ''<sup>''n''</sup>. This polynomial is called the ''characteristic polynomial'' of ''A''. Equation ('''3''') is called the ''characteristic equation'' or the ''secular equation'' of ''A''.
 The fundamental theorem of algebra implies that the characteristic polynomial of an ''n''-by-''n'' matrix ''A'', being a polynomial of degree ''n'', can be factored into the product of ''n'' linear terms,
-{{NumBlk|:
+:<math>|A - \lambda I| = (\lambda_1 - \lambda )(\lambda_2 - \lambda) \cdots (\lambda_n - \lambda),</math>   ('''4''')
- |<math>|A - \lambda I| = (\lambda_1 - \lambda )(\lambda_2 - \lambda) \cdots (\lambda_n - \lambda),</math>
 where each ''λ''<sub>''i''</sub> may be real but in general is a complex number. The numbers ''λ''<sub>1</sub>, ''λ''<sub>2</sub>, …, ''λ''<sub>''n''</sub>, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of ''A''.

@@ Line 584: / Line 584: @@
 ==Dynamic equations==
-The simplest [[difference equation]]s have the form
+The simplest difference equations have the form
 : <math>x_t = a_1 x_{t-1} + a_2 x_{t-2} + \cdots + a_k x_{t-k}.</math>
@@ Line 593: / Line 593: @@
 : <math>x_t = c_1\lambda_1^t + \cdots + c_k\lambda_k^t.</math>
-A similar procedure is used for solving a [[differential equation]] of the form
+A similar procedure is used for solving a differential equation of the form
 : <math>\frac{d^k x}{dt^k} + a_{k-1}\frac{d^{k-1}x}{dt^{k-1}} + \cdots + a_1\frac{dx}{dt} + a_0 x = 0.</math>

@@ Line 620: / Line 620: @@
 :<math>
    \left\{ \begin{aligned} 4x + y &= 6x \\ 6x + 3y &= 6y\end{aligned} \right.
-   </math> {{spaces|4}} that is {{spaces|4}} <math>
+   </math>        that is       <math>
    \left\{ \begin{aligned} -2x + y &= 0 \\ 6x - 3y &= 0\end{aligned} \right.
 </math>

← Older revision		Revision as of 20:41, 24 September 2021
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	* [http://math.mit.edu/~gs/linearalgebra/ila0601.pdf Eigenvalues and Eigenvectors], MIT Math Department		* [http://math.mit.edu/~gs/linearalgebra/ila0601.pdf Eigenvalues and Eigenvectors], MIT Math Department
	* [https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Eigenvalues and Eigenvectors], Wikipedia		* [https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Eigenvalues and Eigenvectors], Wikipedia
		+	* [https://www.khanacademy.org/math/linear-algebra/alternate-bases/eigen-everything/v/linear-algebra-example-solving-for-the-eigenvalues-of-a-2x2-matrix Solving for Eigenvalues of 2x2 Matrix], Khan Academy
		+	* [https://www.khanacademy.org/math/linear-algebra/alternate-bases/eigen-everything/v/linear-algebra-eigenvalues-of-a-3x3-matrix Eigenvalues of a 3x3 Matrix], Khan Academy
		+	* [https://www.youtube.com/watch?v=IdsV0RaC9jM Finding Eigenvalues and Eigenvectors: 2x2 Matrix Example], patrickJMT
		+	* [https://metric.ma.ic.ac.uk/metric_public/matrices/eigenvalues_and_eigenvectors/eigenvalues2.html Eigenvalues and Eigenvectors of a 3x3 Matrix],

← Older revision		Revision as of 20:26, 24 September 2021
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		+	==Definition==
	In linear algebra, an eigenvector of a matrix is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by <math> \lambda </math>, is the factor by which the eigenvector is scaled. That is, given some eigenvector <math> v_i </math> of a square matrix <math>		In linear algebra, an eigenvector of a matrix is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by <math> \lambda </math>, is the factor by which the eigenvector is scaled. That is, given some eigenvector <math> v_i </math> of a square matrix <math>
	\mathbf{A}</math>, <math> \mathbf{A}v_i = \lambda_i v_i</math>, where <math> \lambda_i </math> is the corresponding eigenvalue of <math> v_i </math>. For example:		\mathbf{A}</math>, <math> \mathbf{A}v_i = \lambda_i v_i</math>, where <math> \lambda_i </math> is the corresponding eigenvalue of <math> v_i </math>. For example:

@@ Line 12: / Line 12: @@
 \end{bmatrix}</math>
-Then, <math> \mathbf{A}v_1 = \begin{bmatrix}
 \\
 \\