DOCSLIB.ORG

The Jordan-Form Proof Made Easy∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Jordan-Form Proof Made Easy∗ THE JORDAN-FORM PROOF MADE EASY∗ LEO LIVSHITSy , GORDON MACDONALDz, BEN MATHESy, AND HEYDAR RADJAVIx Abstract. A derivation of the Jordan Canonical Form for linear transformations acting on finite dimensional vector spaces over C is given. The proof is constructive and elementary, using only basic concepts from introductory linear algebra and relying on repeated application of similarities. Key words. Jordan Canonical Form, block decomposition, similarity AMS subject classifications. 15A21, 00-02 1. Introduction. The Jordan Canonical Form (JCF) is undoubtably the most useful representation for illuminating the structure of a single linear transformation acting on a finite-dimensional vector space over C (or a general algebraically closed field.) Theorem 1.1. [The Jordan Canonical Form Theorem] Any linear transforma- tion T : Cn ! Cn has a block matrix (with respect to a direct-sum decomposition of Cn) of the form J1 0 0 · · · 0 2 0 J2 0 · · · 0 3 0 0 J · · · 0 6 3 7 6 . 7 6 . .. 0 7 6 7 6 0 0 0 0 J 7 4 p 5 where each Ji (called a Jordan block) has a matrix representation (with respect to some basis) of the form λ 1 0 · · · 0 2 0 λ 1 · · · 0 3 . 6 0 0 λ .. 0 7 : 6 7 6 . 7 6 . .. 1 7 6 7 6 0 0 0 0 λ 7 4 5 Of course, the Jordan Canonical Form Theorem tells you more: that the above decomposition is unique (up to permuting the blocks) while the sizes and number of blocks can be given in terms of generalized multiplicities of eigenvalues. ∗Received by the editors on ????. Accepted for publication on ????. Handling Editor: ???. yDepartment of Mathematics,Colby College, Waterville, Maine 04901, USA. (l [email protected], [email protected]). zDepartment of Mathematics and Computer Science, University of Prince Edward Island, Char- lottetown, P.E.I., C1A 4P3, Canada ([email protected]). Supported by a grant from NSERC Canada. xDepartment of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J3, Canada ([email protected]). Supported by a grant from NSERC Canada. 1 With the JCF theorem in hand, all the mysteries of single linear transformations are exposed and many of the other theorems of Linear Algebra (for example the Rank- Nullity Theorem and the Cayley-Hamilton Theorem) become immediately obvious. The JCF also has many practical applications. The one to which most students of mathematics are exposed is that of linear systems of differential equations with constant coefficients. With the JCF of the coefficient matrix in hand, solving such systems is a routine exercise. Unfortunately, the proof of the JCF is a mystery to many students. In many linear algebra courses, the JCF is given short shrift, since the traditional proofs re- quire not only knowledge of minimal polynomials and block decompositions but also advanced knowledge regarding nilpotents, generalized eigenvalues, esoteric subspaces (usually called admissible subspaces) and polynomial ideals (usually called stuffers or conductors). The proof does not vary much from linear algebra text to linear algebra text and the exposition in Hoffman and Kunze's text [4] is representative. A number of papers have given alternative proofs of JCF. For example: Fletcher and Sorensen [1], Galperin and Waksman [2] and V¨aliaho [5]). Of these proofs, ours is most similar in flavor to that of Fletcher and Sorenson, in that our proof is given in terms of matrices and that we take an algorithmic approach. But our proof is more elementary and relies on repeated similarities to gradually bring the matrix representation to the required form. The proof presented here requires only basic facility with row reduction, matrix representations, block decompositions, kernels and ranges of transformations, similar- ity, subspaces, minimal polynomials and eigenvalues. In short, the usual contents of a one semester introductory course in Linear Algebra. As far as pedagogy is concerned, this makes the JCF a natural culmination of such a course, bringing all the basic concepts together and ending the course with a bang. We have chosen to take a comprehensive approach, and included all the details of the proof. This makes the article longer, but, we hope, also makes it more accessible to students with only the aforementioned introductory course in linear algebra. In this paper, V and W are finite-dimensional vector spaces over C (the field of complex numbers) and T : V ! W, is a linear transformation mapping V to W (although all definitions and results hold for linear transformations acting on finite-dimensional vector spaces over any algebraically closed field). Some standard terminology and notation that will prove useful is: • L(V; W) will denote the space of all linear transformations (also called op- erators) from the vector space V to the vector space W. We also write L(V) for L(V; W) • dim(V) denotes the dimension of V • The kernel (or nullspace) of a linear transformation T 2 L(V; W) is ker(T ) = fv 2 V : T(v) = 0g • The minimal polynomial of T is mT (x) (the unique monic polynomial of minimal degree such that mT (T )= 0.) • A direct sum decomposition V = V1 ⊕V2 ⊕· · ·⊕Vn of a vector space V is an n ordered list of subspaces fVjgj=1of V such that Vj \(V1 +· · ·+Vj−1) = f0g for all 2 ≤ j ≤ n (this is one way of saying that the subspaces are independent) 2 n and such that for all v 2 V there exist vi2 Vi such that v = i=1 vi. • A block matrix of a linear transformation T with respect toPa direct sum decomposition such as the one above is a matrix of linear transformations Tij 2 L(Vj; Vi) such that, with usual definition of matrix multiplication and n identification of a vector v = i=1 vi (where vi2 Vi) with P v1 2 v2 3 v3 6 7 ; 6 . 7 6 . 7 6 7 6 v 7 4 n 5 T11 T12 T13 · · · T1n v1 2 T21 T22 T23 · · · T2n 3 2 v2 3 T T T · · · T n v3 T (v) = 6 31 32 33 3 7 6 7 6 . 7 6 . 7 6 . .. 7 6 . 7 6 7 6 7 6 T T T · · · T 7 6 vn 7 4 n1 n2 n3 nn 5 4 5 • A linear transformation T : V ! V is said to be nilpotent if its only eigen- value is the zero eigenvalue. By consideration of the minimal polynomial it is clear that T n = 0 for some n 2 N. The smallest value of n such that T n = 0 is called the index (or index of nilpotency) of T . 2. Primary decomposition. Before getting into the technicalities of the proof of the JCF, a special case of the Primary Decomposition Theorem allows us to obtain a block diagonal matrix for a linear transformation T where the eigenvalue structure of each block is as simple as possible; there is only one eigenvalue for each block. Theorem 2.1. Let T 2 L(Cn) (or L(Fn) for F an algebraically closed field) with distinct eigenvalues λ1; λ2; : : : ; λm . Then there is a direct sum decomposition V = Kλ1 ⊕ Kλ2 ⊕ · · · ⊕ Kλn with respect to which T has block matrix λ1I + N1 0 · · · 0 2 0 λ2I + N2 · · · 0 3 . 6 . .. 0 7 6 7 6 0 0 · · · λ I + N 7 4 n n 5 where each Nj is nilpotent. We prove this using a sequence of Lemmas. Lemma 2.2. Let N be a nilpotent r × r matrix, M an invertible s × s matrix and F an arbitrary r × s matrix. Then the matrix equation X − NXM = F is solvable. Proof. Let k be such that N k 6= 0 and N k+1 = 0 (Of course, k can be 0). Check that X = F + NF M + N 2F M 2 + · · · + N kF M k 3 is a solution. The solution is actually unique, but we don't need that fact here. Lemma 2.3. If λ is not an eigenvalue of the s × s matrix A and N is a nilpotent r × r matrix, then the two matrices λI + N E λI + N 0 and 0 A 0 A are similar (for any choice of the r × s matrix E.) Proof. It is sufficient to show there exists matrix X such that λI + N E I X I X λI + N 0 = 0 A 0 I 0 I 0 A or equivalently, X(A − λI) − NX = E. Since A − λI is invertible by hypothesis, we can rewrite this as X − NX(A − λI)−1 = E(A − λI)−1; which is solvable by Lemma 2.2. Lemma 2.4. Every operator T 2 L(Cn) (or L(Fn) for F an algebraically closed field) with eigenvalue λ has a matrix of the form λI + N 0 0 A where N is nilpotent and λ is not an eigenvalue of A. Proof. Use induction on n, the dimension of the space. Clearly the Lemma is true if n = 1. Assuming it is true for all dimensions less than n, choose a basis fe1; e2; : : : ; eng such that T (e1) = λe1. Then T has block matrix λ E : 0 B If λ is not an eigenvalue of B, then Lemma 2.3 gives the result. If λ is an eigenvalue of B, since B acts on a space of dimension less than n, by the induction hypothesis we have that T has a matrix of the form λ E1 E2 2 0 λI 0 3 0 0 A 4 5 where λ is not an eigenvalue of A. Again by Lemma 2.3, this matrix is similar to λ E 0 1 λI + N 0 0 λI 0 = 2 3 0 A 0 0 A 4 5 4 where 0 E N = 1 : 0 0 Theorem 2.1 now follows by applying Lemma 2.4 inductively.
Load more

Recommended publications
  • Solving the Multi-Way Matching Problem by Permutation Synchronization
    Solving the multi-way matching problem by permutation synchronization Deepti Pachauriy Risi Kondorx Vikas Singhy yUniversity of Wisconsin Madison xThe University of Chicago [email protected] [email protected] [email protected] http://pages.cs.wisc.edu/∼pachauri/perm-sync Abstract The problem of matching not just two, but m different sets of objects to each other arises in many contexts, including finding the correspondence between feature points across multiple images in computer vision. At present it is usually solved by matching the sets pairwise, in series. In contrast, we propose a new method, Permutation Synchronization, which finds all the matchings jointly, in one shot, via a relaxation to eigenvector decomposition. The resulting algorithm is both computationally efficient, and, as we demonstrate with theoretical arguments as well as experimental results, much more stable to noise than previous methods. 1 Introduction 0 Finding the correct bijection between two sets of objects X = fx1; x2; : : : ; xng and X = 0 0 0 fx1; x2; : : : ; xng is a fundametal problem in computer science, arising in a wide range of con- texts [1]. In this paper, we consider its generalization to matching not just two, but m different sets X1;X2;:::;Xm. Our primary motivation and running example is the classic problem of matching landmarks (feature points) across many images of the same object in computer vision, which is a key ingredient of image registration [2], recognition [3, 4], stereo [5], shape matching [6, 7], and structure from motion (SFM) [8, 9]. However, our approach is fully general and equally applicable to problems such as matching multiple graphs [10, 11].
    [Show full text]
  • Arxiv:1912.06217V3 [Math.NA] 27 Feb 2021
    ROUNDING ERROR ANALYSIS OF MIXED PRECISION BLOCK HOUSEHOLDER QR ALGORITHMS∗ L. MINAH YANG† , ALYSON FOX‡, AND GEOFFREY SANDERS‡ Abstract. Although mixed precision arithmetic has recently garnered interest for training dense neural networks, many other applications could benefit from the speed-ups and lower storage cost if applied appropriately. The growing interest in employing mixed precision computations motivates the need for rounding error analysis that properly handles behavior from mixed precision arithmetic. We develop mixed precision variants of existing Householder QR algorithms and show error analyses supported by numerical experiments. 1. Introduction. The accuracy of a numerical algorithm depends on several factors, including numerical stability and well-conditionedness of the problem, both of which may be sensitive to rounding errors, the difference between exact and finite precision arithmetic. Low precision floats use fewer bits than high precision floats to represent the real numbers and naturally incur larger rounding errors. Therefore, error attributed to round-off may have a larger influence over the total error and some standard algorithms may yield insufficient accuracy when using low precision storage and arithmetic. However, many applications exist that would benefit from the use of low precision arithmetic and storage that are less sensitive to floating-point round-off error, such as training dense neural networks [20] or clustering or ranking graph algorithms [25]. As a step towards that goal, we investigate the use of mixed precision arithmetic for the QR factorization, a widely used linear algebra routine. Many computing applications today require solutions quickly and often under low size, weight, and power constraints, such as in sensor formation, where low precision computation offers the ability to solve many problems with improvement in all four parameters.
    [Show full text]
  • On Multivariate Interpolation
    On Multivariate Interpolation Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A. [email protected] http://www.math.umn.edu/∼olver Abstract. A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of non-commutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established. † Supported in part by NSF Grant DMS 11–08894. April 6, 2016 1 1. Introduction. Interpolation theory for functions of a single variable has a long and distinguished his- tory, dating back to Newton’s fundamental interpolation formula and the classical calculus of finite differences, [7, 47, 58, 64]. Standard numerical approximations to derivatives and many numerical integration methods for differential equations are based on the finite dif- ference calculus. However, historically, no comparable calculus was developed for functions of more than one variable. If one looks up multivariate interpolation in the classical books, one is essentially restricted to rectangular, or, slightly more generally, separable grids, over which the formulae are a simple adaptation of the univariate divided difference calculus. See [19] for historical details. Starting with G. Birkhoff, [2] (who was, coincidentally, my thesis advisor), recent years have seen a renewed level of interest in multivariate interpolation among both pure and applied researchers; see [18] for a fairly recent survey containing an extensive bibli- ography. De Boor and Ron, [8, 12, 13], and Sauer and Xu, [61, 10, 65], have systemati- cally studied the polynomial case.
    [Show full text]
  • Arxiv:1709.00765V4 [Cond-Mat.Stat-Mech] 14 Oct 2019
    Number of hidden states needed to physically implement a given conditional distribution Jeremy A. Owen,1 Artemy Kolchinsky,2 and David H. Wolpert2, 3, 4, ∗ 1Physics of Living Systems Group, Department of Physics, Massachusetts Institute of Technology, 400 Tech Square, Cambridge, MA 02139. 2Santa Fe Institute 3Arizona State University 4http://davidwolpert.weebly.com† (Dated: October 15, 2019) We consider the problem of how to construct a physical process over a finite state space X that applies some desired conditional distribution P to initial states to produce final states. This problem arises often in the thermodynamics of computation and nonequilibrium statistical physics more generally (e.g., when designing processes to implement some desired computation, feedback controller, or Maxwell demon). It was previously known that some conditional distributions cannot be implemented using any master equation that involves just the states in X. However, here we show that any conditional distribution P can in fact be implemented—if additional “hidden” states not in X are available. Moreover, we show that is always possible to implement P in a thermodynamically reversible manner. We then investigate a novel cost of the physical resources needed to implement a given distribution P : the minimal number of hidden states needed to do so. We calculate this cost exactly for the special case where P represents a single-valued function, and provide an upper bound for the general case, in terms of the nonnegative rank of P . These results show that having access to one extra binary degree of freedom, thus doubling the total number of states, is sufficient to implement any P with a master equation in a thermodynamically reversible way, if there are no constraints on the allowed form of the master equation.
    [Show full text]
  • On the Asymptotic Existence of Partial Complex Hadamard Matrices And
    Discrete Applied Mathematics 102 (2000) 37–45 View metadata, citation and similar papers at core.ac.uk brought to you by CORE On the asymptotic existence of partial complex Hadamard provided by Elsevier - Publisher Connector matrices and related combinatorial objects Warwick de Launey Center for Communications Research, 4320 Westerra Court, San Diego, California, CA 92121, USA Received 1 September 1998; accepted 30 April 1999 Abstract A proof of an asymptotic form of the original Goldbach conjecture for odd integers was published in 1937. In 1990, a theorem reÿning that result was published. In this paper, we describe some implications of that theorem in combinatorial design theory. In particular, we show that the existence of Paley’s conference matrices implies that for any suciently large integer k there is (at least) about one third of a complex Hadamard matrix of order 2k. This implies that, for any ¿0, the well known bounds for (a) the number of codewords in moderately high distance binary block codes, (b) the number of constraints of two-level orthogonal arrays of strengths 2 and 3 and (c) the number of mutually orthogonal F-squares with certain parameters are asymptotically correct to within a factor of 1 (1 − ) for cases (a) and (b) and to within a 3 factor of 1 (1 − ) for case (c). The methods yield constructive algorithms which are polynomial 9 in the size of the object constructed. An ecient heuristic is described for constructing (for any speciÿed order) about one half of a Hadamard matrix. ? 2000 Elsevier Science B.V. All rights reserved.
    [Show full text]
  • Determinants of Commuting-Block Matrices by Istvan Kovacs, Daniel S
    Determinants of Commuting-Block Matrices by Istvan Kovacs, Daniel S. Silver*, and Susan G. Williams* Let R beacommutative ring, and Matn(R) the ring of n × n matrices over R.We (i,j) can regard a k × k matrix M =(A ) over Matn(R)asablock matrix,amatrix that has been partitioned into k2 submatrices (blocks)overR, each of size n × n. When M is regarded in this way, we denote its determinant by |M|.Wewill use the symbol D(M) for the determinant of M viewed as a k × k matrix over Matn(R). It is important to realize that D(M)isann × n matrix. Theorem 1. Let R be acommutative ring. Assume that M is a k × k block matrix of (i,j) blocks A ∈ Matn(R) that commute pairwise. Then | | | | (1,π(1)) (2,π(2)) ··· (k,π(k)) (1) M = D(M) = (sgn π)A A A . π∈Sk Here Sk is the symmetric group on k symbols; the summation is the usual one that appears in the definition of determinant. Theorem 1 is well known in the case k =2;the proof is often left as an exercise in linear algebra texts (see [4, page 164], for example). The general result is implicit in [3], but it is not widely known. We present a short, elementary proof using mathematical induction on k.Wesketch a second proof when the ring R has no zero divisors, a proof that is based on [3] and avoids induction by using the fact that commuting matrices over an algebraically closed field can be simultaneously triangularized.
    [Show full text]
  • Triangular Factorization
    Chapter 1 Triangular Factorization This chapter deals with the factorization of arbitrary matrices into products of triangular matrices. Since the solution of a linear n n system can be easily obtained once the matrix is factored into the product× of triangular matrices, we will concentrate on the factorization of square matrices. Specifically, we will show that an arbitrary n n matrix A has the factorization P A = LU where P is an n n permutation matrix,× L is an n n unit lower triangular matrix, and U is an n ×n upper triangular matrix. In connection× with this factorization we will discuss pivoting,× i.e., row interchange, strategies. We will also explore circumstances for which A may be factored in the forms A = LU or A = LLT . Our results for a square system will be given for a matrix with real elements but can easily be generalized for complex matrices. The corresponding results for a general m n matrix will be accumulated in Section 1.4. In the general case an arbitrary m× n matrix A has the factorization P A = LU where P is an m m permutation× matrix, L is an m m unit lower triangular matrix, and U is an×m n matrix having row echelon structure.× × 1.1 Permutation matrices and Gauss transformations We begin by defining permutation matrices and examining the effect of premulti- plying or postmultiplying a given matrix by such matrices. We then define Gauss transformations and show how they can be used to introduce zeros into a vector. Definition 1.1 An m m permutation matrix is a matrix whose columns con- sist of a rearrangement of× the m unit vectors e(j), j = 1,...,m, in RI m, i.e., a rearrangement of the columns (or rows) of the m m identity matrix.
    [Show full text]
  • Think Global, Act Local When Estimating a Sparse Precision
    Think Global, Act Local When Estimating a Sparse Precision Matrix by Peter Alexander Lee A.B., Harvard University (2007) Submitted to the Sloan School of Management in partial fulfillment of the requirements for the degree of Master of Science in Operations Research at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2016 ○c Peter Alexander Lee, MMXVI. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Author................................................................ Sloan School of Management May 12, 2016 Certified by. Cynthia Rudin Associate Professor Thesis Supervisor Accepted by . Patrick Jaillet Dugald C. Jackson Professor, Department of Electrical Engineering and Computer Science and Co-Director, Operations Research Center 2 Think Global, Act Local When Estimating a Sparse Precision Matrix by Peter Alexander Lee Submitted to the Sloan School of Management on May 12, 2016, in partial fulfillment of the requirements for the degree of Master of Science in Operations Research Abstract Substantial progress has been made in the estimation of sparse high dimensional pre- cision matrices from scant datasets. This is important because precision matrices underpin common tasks such as regression, discriminant analysis, and portfolio opti- mization. However, few good algorithms for this task exist outside the space of L1 penalized optimization approaches like GLASSO. This thesis introduces LGM, a new algorithm for the estimation of sparse high dimensional precision matrices. Using the framework of probabilistic graphical models, the algorithm performs robust covari- ance estimation to generate potentials for small cliques and fuses the local structures to form a sparse yet globally robust model of the entire distribution.
    [Show full text]
  • Irreducibility in Algebraic Groups and Regular Unipotent Elements
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 1, January 2013, Pages 13–28 S 0002-9939(2012)11898-2 Article electronically published on August 16, 2012 IRREDUCIBILITY IN ALGEBRAIC GROUPS AND REGULAR UNIPOTENT ELEMENTS DONNA TESTERMAN AND ALEXANDRE ZALESSKI (Communicated by Pham Huu Tiep) Abstract. We study (connected) reductive subgroups G of a reductive alge- braic group H,whereG contains a regular unipotent element of H.Themain result states that G cannot lie in a proper parabolic subgroup of H. This result is new even in the classical case H =SL(n, F ), the special linear group over an algebraically closed field, where a regular unipotent element is one whose Jor- dan normal form consists of a single block. In previous work, Saxl and Seitz (1997) determined the maximal closed positive-dimensional (not necessarily connected) subgroups of simple algebraic groups containing regular unipotent elements. Combining their work with our main result, we classify all reductive subgroups of a simple algebraic group H which contain a regular unipotent element. 1. Introduction Let H be a reductive linear algebraic group defined over an algebraically closed field F . Throughout this text ‘reductive’ will mean ‘connected reductive’. A unipo- tent element u ∈ H is said to be regular if the dimension of its centralizer CH (u) coincides with the rank of H (or, equivalently, u is contained in a unique Borel subgroup of H). Regular unipotent elements of a reductive algebraic group exist in all characteristics (see [22]) and form a single conjugacy class. These play an important role in the general theory of algebraic groups.
    [Show full text]
  • Mata Glossary of Common Terms
    Title stata.com Glossary Description Commonly used terms are defined here. Mata glossary arguments The values a function receives are called the function’s arguments. For instance, in lud(A, L, U), A, L, and U are the arguments. array An array is any indexed object that holds other objects as elements. Vectors are examples of 1- dimensional arrays. Vector v is an array, and v[1] is its first element. Matrices are 2-dimensional arrays. Matrix X is an array, and X[1; 1] is its first element. In theory, one can have 3-dimensional, 4-dimensional, and higher arrays, although Mata does not directly provide them. See[ M-2] Subscripts for more information on arrays in Mata. Arrays are usually indexed by sequential integers, but in associative arrays, the indices are strings that have no natural ordering. Associative arrays can be 1-dimensional, 2-dimensional, or higher. If A were an associative array, then A[\first"] might be one of its elements. See[ M-5] asarray( ) for associative arrays in Mata. binary operator A binary operator is an operator applied to two arguments. In 2-3, the minus sign is a binary operator, as opposed to the minus sign in -9, which is a unary operator. broad type Two matrices are said to be of the same broad type if the elements in each are numeric, are string, or are pointers. Mata provides two numeric types, real and complex. The term broad type is used to mask the distinction within numeric and is often used when discussing operators or functions.
    [Show full text]
  • The Use of the Vec-Permutation Matrix in Spatial Matrix Population Models Christine M
    Ecological Modelling 188 (2005) 15–21 The use of the vec-permutation matrix in spatial matrix population models Christine M. Hunter∗, Hal Caswell Biology Department MS-34, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA Abstract Matrix models for a metapopulation can be formulated in two ways: in terms of the stage distribution within each spatial patch, or in terms of the spatial distribution within each stage. In either case, the entries of the projection matrix combine demographic and dispersal information in potentially complicated ways. We show how to construct such models from a simple block-diagonal formulation of the demographic and dispersal processes, using a special permutation matrix called the vec-permutation matrix. This formulation makes it easy to calculate and interpret the sensitivity and elasticity of λ to changes in stage- and patch-specific demographic and dispersal parameters. © 2005 Christine M. Hunter and Hal Caswell. Published by Elsevier B.V. All rights reserved. Keywords: Matrix population model; Spatial model; Metapopulation; Multiregional model; Dispersal; Movement; Sensitivity; Elasticity 1. Introduction If each local population is described by a stage- classified matrix population model (see Caswell, A spatial population model describes a finite set of 2001), a spatial matrix model describing the metapop- discrete local populations coupled by the movement ulation can be written as or dispersal of individuals. Human demographers call these multiregional populations (Rogers, 1968, 1995), n(t + 1) = An(t) (1) ecologists call them multisite populations (Lebreton, 1996) or metapopulations (Hanski, 1999). The dynam- The population vector, n, includes the densities of ics of metapopulations are determined by the patterns each stage in each local population (which we will of dispersal among the local populations and the de- refer to here as a patch).
    [Show full text]
  • 9. Properties of Matrices Block Matrices
    9. Properties of Matrices Block Matrices It is often convenient to partition a matrix M into smaller matrices called blocks, like so: 01 2 3 11 ! B C B4 5 6 0C A B M = B C = @7 8 9 1A C D 0 1 2 0 01 2 31 011 B C B C Here A = @4 5 6A, B = @0A, C = 0 1 2 , D = (0). 7 8 9 1 • The blocks of a block matrix must fit together to form a rectangle. So ! ! B A C B makes sense, but does not. D C D A • There are many ways to cut up an n × n matrix into blocks. Often context or the entries of the matrix will suggest a useful way to divide the matrix into blocks. For example, if there are large blocks of zeros in a matrix, or blocks that look like an identity matrix, it can be useful to partition the matrix accordingly. • Matrix operations on block matrices can be carried out by treating the blocks as matrix entries. In the example above, ! ! A B A B M 2 = C D C D ! A2 + BC AB + BD = CA + DC CB + D2 1 Computing the individual blocks, we get: 0 30 37 44 1 2 B C A + BC = @ 66 81 96 A 102 127 152 0 4 1 B C AB + BD = @10A 16 0181 B C CA + DC = @21A 24 CB + D2 = (2) Assembling these pieces into a block matrix gives: 0 30 37 44 4 1 B C B 66 81 96 10C B C @102 127 152 16A 4 10 16 2 This is exactly M 2.
    [Show full text]