What s Eigenanalysis? Matrix eigenanalysis is a computational theory for the matrix equation

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1 Eigenanalysis What s Eigenanalysis? Fourier s Eigenanalysis Model is a Replacement Process Powers and Fourier s Model Differential Equations and Fourier s Model Fourier s Model Illustrated What is Eigenanalysis? What s an Eigenvalue? What s an Eigenvector? Data Conversion Example History of Fourier s Model Determining Equations. How to Compute Eigenpairs. Independence of Eigenvectors.

2 What s Eigenanalysis? Matrix eigenanalysis is a computational theory for the matrix equation y = Ax.

3 Fourier s Eigenanalysis Model For exposition purposes, we assume A is a 3 3 matrix. (1) x = c 1 v 1 + c 2 v 2 + c 3 v 3 implies y = Ax = c 1 λ 1 v 1 + c 2 λ 2 v 2 + c 3 λ 3 v 3. Eigenanalysis Notation The scale factors λ 1, λ 2, λ 3 and independent vectors v 1, v 2, v 3 depend only on A. Symbols c 1, c 2, c 3 stand for arbitrary numbers. This implies variable x exhausts all possible fixed vectors in R 3.

4 Fourier s Model is a Replacement Process A (c 1 v 1 + c 2 v 2 + c 3 v 3 ) = c 1 λ 1 v 1 + c 2 λ 2 v 2 + c 3 λ 3 v 3. To compute Ax from x = c 1 v 1 + c 2 v 2 + c 3 v 3, replace each vector v i by its scaled version λ i v i. Fourier s model is said to hold provided there exist scale factors and independent vectors satisfying (1). Fourier s model is known to fail for certain matrices A.

5 Powers and Fourier s Model Equation (1) applies to compute powers A n of a matrix A using only the basic vector space toolkit. To illustrate, only the vector toolkit for R 3 is used in computing A 5 x = x 1 λ 5 1 v 1 + x 2 λ 5 2 v 2 + x 3 λ 5 3 v 3. This calculation does not depend upon finding previous powers A 2, A 3, A 4 as would be the case by using matrix multiply. Details for A 3 (x) Let x = x 1 v 1 + x 2 v 2 + x 3 v 3. Then A 3 (x) = A 2 (A(x)) = A 2 (x 1 λ 1 v 1 + x 2 λ 2 v 2 + x 3 λ 3 v 3 ) = A(A(x 1 λ 1 v 1 + x 2 λ 2 v 2 + x 3 λ 3 v 3 )) = A(x 1 λ 2 1 v 1 + x 2 λ 2 2 v 2 + x 3 λ 2 3 v 3) = x 1 λ 3 1 v 1 + x 2 λ 3 2 v 2 + x 3 λ 3 3 v 3

6 Differential Equations and Fourier s Model Systems of differential equations can be solved using Fourier s model, giving a compact and elegant formula for the general solution. An example: x 1 = x 1 + 3x 2, x 2 = 2x 2 x 3, x 3 = 5x 3. The general solution is given by the formula [Fourier s theorem, proved later] x x 2 = c 1 e t + c 2 e 2t 1 + c 3 e 5t 2, x 3 14 which is related to Fourier s model by the symbolic formulas x() = c 1 v 1 + c 2 v 2 + c 3 v 3 undergoes replacements v i e λ it v i to obtain x(t) = c 1 e λ 1t v 1 + c 2 e λ 2t v 2 + c 3 e λ 3t v 3.

7 Fourier s model illustrated Let A = λ 1 = 1, λ 2 = 2, λ 3 = 5, v 1 = 1 Then Fourier s model holds (details later) and x = c 1 1 Ax = c 1 (1), v2 = c c2 (2), v3 = + c c3 ( 5). implies Eigenanalysis might be called the method of simplifying coordinates. The nomenclature is justified, because Fourier s model computes y = Ax by scaling independent vectors v 1, v 2, v 3, which is a triad or coordinate system.

8 What is Eigenanalysis? The subject of eigenanalysis discovers a coordinate system v 1, v 2, v 3 and scale factors λ 1, λ 2, λ 3 such that Fourier s model holds. Fourier s model simplifies the matrix equation y = Ax, through the formula A(c 1 v 1 + c 2 v 2 + c 3 v 3 ) = c 1 λ 1 v 1 + c 2 λ 2 v 2 + c 3 λ 3 v 3.

9 What s an Eigenvalue? It is a scale factor. An eigenvalue is also called a proper value or a hidden value or a characteristic value. Symbols λ 1, λ 2, λ 3 used in Fourier s model are eigenvalues. The eigenvalues of a model are scale factors. Think of them as a system of units hidden in the matrix A. What s an Eigenvector? Symbols v 1, v 2, v 3 in Fourier s model are called eigenvectors, or proper vectors or hidden vectors or characteristic vectors. They are assumed independent. The eigenvectors of a model are independent directions of application for the scale factors (eigenvalues). Think of each eigenpair (λ, v) as a coordinate axis v where the action of matrix A is to move λ units along v.

10 Data Conversion Example Let x in R 3 be a data set variable with coordinates x 1, x 2, x 3 recorded respectively in units of meters, millimeters and centimeters. We consider the problem of conversion of the mixed-unit x-data into proper MKS units (meters-kilogram-second) y-data via the equations y 1 = x 1, (2) y 2 =.1x 2, y 3 =.1x 3. Equations (2) are a model for changing units. Scaling factors λ 1 = 1, λ 2 =.1, λ 3 =.1 are the eigenvalues of the model.

11 Data Conversion Example Continued Problem (2) can be represented as y = Ax, where the diagonal matrix A is given by λ 1 A = λ 2, λ 1 = 1, λ 2 = 1 λ 3 1, λ 3 = 1 1. Fourier s model for this matrix A is 1 1 A c 1 + c c 3 = c 1 λ 1 + c 2 λ c 3 λ The eigenvectors v 1, v 2, v 3 of the model are the columns of the identity matrix.

12 Summary The eigenvalues of a model are scale factors, normally represented by symbols λ 1, λ 2, λ 3,.... The eigenvectors of a model are independent directions of application for the scale factors (eigenvalues). They are normally represented by symbols v 1, v 2, v 3,....

13 History of Fourier s Model The subject of eigenanalysis was popularized by J. B. Fourier in his 1822 publication on the theory of heat, Théorie analytique de la chaleur. His ideas can be summarized as follows for the n n matrix equation y = Ax. The vector y = Ax is obtained from eigenvalues λ 1, λ 2,..., λ n and eigenvectors v 1, v 2,..., v n by replacing the eigenvectors by their scaled versions λ 1 v 1, λ 2 v 2,..., λ n v n : x = c 1 v 1 + c 2 v c n v n implies y = x 1 λ 1 v 1 + x 2 λ 2 v c n λ n v n.

14 Determining Equations The eigenvalues and eigenvectors are determined by homogeneous matrix vector equations. In Fourier s model A(c 1 v 1 + c 2 v 2 + c 3 v 3 ) = c 1 λ 1 v 1 + c 2 λ 2 v 2 + c 3 λ 3 v 3 choose c 1 = 1, c 2 = c 3 =. The equation reduces to Av 1 = λ 1 v 1. Similarly, taking c 1 = c 2 =, c 2 = 1 implies Av 2 = λ 2 v 2. Finally, taking c 1 = c 2 =, c 3 = 1 implies Av 3 = λ 3 v 3. This proves the following fundamental result. Theorem 1 (Determining Equations in Fourier s Model) Assume Fourier s model holds. Then the eigenvalues and eigenvectors are determined by the three equations Av 1 = λ 1 v 1, Av 2 = λ 2 v 2, Av 3 = λ 3 v 3.

15 Determining Equations and Linear Algebra The three relations of the theorem can be distilled into one homogeneous matrix vector equation Av = λv. Write it as Ax λx =, then replace λx by λix to obtain the standard form a (A λi)v =, v. Let B = A λi. The equation Bv = has a nonzero solution v if and only if there are infinitely many solutions. Because the matrix is square, infinitely many solutions occurs if and only if rref(b) has a row of zeros. Determinant theory gives a more concise statement: det(b) = if and only if Bv = has infinitely many solutions. This proves the following result. a Identity I is required to factor out the matrix A λi. It is wrong to factor out A λ, because A is 3 3 and λ is 1 1, incompatible sizes for matrix addition.

16 College Algebra and Eigenanalysis Theorem 2 (Characteristic Equation) If Fourier s model holds, then the eigenvalues λ 1, λ 2, λ 3 are roots λ of the polynomial equation det(a λi) =. The equation det(a λi) = is called the characteristic equation. The characteristic polynomial is the polynomial on the left, det(a λi), normally obtained by cofactor expansion or the triangular rule.

17 Eigenvectors and Frame Sequences Theorem 3 (Finding Eigenvectors of A) For each root λ of the characteristic equation, write the frame sequence for B = A λi with last frame rref(b), followed by solving for the general solution v of the homogeneous equation Bv =. Solution v uses invented symbols t 1, t 2,.... The vector basis answers t1 v, t2 v,... are independent eigenvectors of A paired to eigenvalue λ. Proof: The equation Av = λv is equivalent to Bv =. Because det(b) =, then this system has infinitely many solutions, which implies the frame sequence starting at B ends with rref(b) having at least one row of zeros. The general solution then has one or more free variables which are assigned invented symbols t 1, t 2,..., and then the vector basis is obtained by from the corresponding list of partial derivatives. Each basis element is a nonzero solution of Av = λv. By construction, the basis elements (eigenvectors for λ) are collectively independent. The proof is complete.

18 Eigenpairs of a Matrix Definition 1 (Eigenpair) An eigenpair is an eigenvalue λ together with a matching eigenvector v satisfying the equation Av = λv. The pairing implies that scale factor λ is applied to direction v. An applied view of an eigenpair is a coordinate axis v and a unit system along this axis. The action of the matrix A is to move λ units along this axis. A 3 3 matrix A for which Fourier s model holds has eigenvalues λ 1, λ 2, λ 3 and corresponding eigenvectors v 1, v 2, v 3. The eigenpairs of A are (λ 1, v 1 ), (λ 2, v 2 ), (λ 3, v 3 ).

19 Eigenvectors are Independent Theorem 4 (Independence of Eigenvectors) If (λ 1, v 1 ) and (λ 2, v 2 ) are two eigenpairs of A and λ 1 λ 2, then v 1, v 2 are independent. More generally, if (λ 1, v 1 ),..., (λ k, v k ) are eigenpairs of A corresponding to distinct eigenvalues λ 1,..., λ k, then v 1,..., v k are independent. Theorem 5 (Distinct Eigenvalues) If an n n matrix A has n distinct eigenvalues, then its eigenpairs (λ 1, v 1 ),..., (λ n, v n ) produce independent eigenvectors v 1,..., v n. Therefore, Fourier s model holds: ( n ) n A c i v i = c i (λ i v i ). i=1 i=1

9.1 Eigenanalysis I Eigenanalysis II Advanced Topics in Linear Algebra Kepler s laws

9.1 Eigenanalysis I Eigenanalysis II Advanced Topics in Linear Algebra Kepler s laws Chapter 9 Eigenanalysis Contents 9. Eigenanalysis I.................. 49 9.2 Eigenanalysis II................. 5 9.3 Advanced Topics in Linear Algebra..... 522 9.4 Kepler s laws................... 537