DOCSLIB.ORG

Algebra Us02cmth21 Matrix Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebra Us02cmth21 Matrix Theory ALGEBRA US02CMTH21 MATRIX THEORY Shreya Chauhan April 21, 2021 1 1 Special Types of Matrices Zero Divisors: If A and B are nonzero matrices such that AB = 0 then A and B are called zero divisors. 2 3 2 3 2 3 1 0 0 0 0 0 e.g. A = 4 5 B = 4 5 Then AB = 4 5 0 0 1 0 0 0 Theorem: The product of two matrices can be a zero matrix though none of them is a zero matrix. 2 3 2 3 0 c −b a2 ab ac 6 7 6 7 6 7 6 2 7 Let A = 6 −c 0 a 7 And B = 6 ab b bc 7 4 5 4 5 b −a 0 ac bc c2 Then 2 3 abc − abc b2c − b2c bc2 − bc2 6 7 6 2 2 2 2 7 AB = 6 −a c + a c −abc + abc −ac + ac 7 4 5 ba2 − a2b ab2 − ab2 abc − abc 2 3 0 0 0 6 7 6 7 = 6 0 0 0 7 4 5 0 0 0 while A 6= 0 B 6= 0 Important Result: 1 AB = AC does not imply B = C. i.e. cancellation laws does not hold for matrix multiplication. 2 3 2 3 2 3 1 0 0 0 0 0 e.g. Let A = 4 5 B = 4 5 C = 4 5 0 0 1 0 0 1 Then 2 3 0 0 AB = 4 5 0 0 And 2 3 0 0 AC = 4 5 0 0 But B 6= C. Idempotent Matrix: A square matrix A is said to be idempotent if A2 = A. For example, 2 3 1 0 0 6 7 6 7 A = 6 0 1 0 7. 4 5 0 0 1 Then A2 = A. Hence A is an idempotent matrix. A square matrix A is said to be idempotent of period p if p is the least positive integer such that Ap+1 = A. 2 For example, 2 3 −1 0 A = 4 5 0 −1 Then 2 3 1 0 A2 = 4 5 0 1 And 2 3 2 3 1 0 −1 0 A3 = 4 5 4 5 0 1 0 −1 2+1 So A = A. ) A is an idempotent of period 2. Nilpotent Matrix: A square matrix A is said to be a nilpotent matrix if Ak = 0 where k is a positive integer. If however k is the least integer for which Ak = 0 then k is called the index of nilpotent matrix A. For example, 2 3 0 0 1 6 7 6 7 A = 6 0 0 1 7 4 5 0 0 0 Then 3 2 3 0 0 0 6 7 2 6 7 A = 6 0 0 0 7 4 5 0 0 0 ) A is a nilpotent matrix index 2. Involuntary Matrix: A square matrix A is said to be an innvolutary matrix if A2 = I. Since I2 = I always, hence unit matrix I is involuntary matrix. 2 3 −1 0 Let A = 4 5. 0 −1 2 3 1 0 Then A2 = 4 5. 0 1 Hence A is also involuntary matrix. Orthogonal Matrix: A square matrix A is said to be orthogonal if AA0 = I = A0A. For example, 2 3 0 1 A = 4 5 1 0 Then 2 3 0 1 A0 = 4 5 1 0 4 Then 2 3 2 3 0 1 0 1 AA0 = 4 5 4 5 1 0 1 0 2 3 1 0 = 4 5 0 1 Similarly we can show that A0A = I. Hence A is an orthogonal matrix. Unitary Matrix: A square matrix A is said to be unitary if AθA = I = AAθ. For example, 2 3 0 i A = 4 5 i 0 Then 2 3 0 −i Aθ = 4 5 −i 0 Then 2 3 2 3 0 i 0 −i AAθ = 4 5 4 5 i 0 −i 0 2 3 −i2 0 = 4 5 0 −i2 2 3 1 0 = 4 5 0 1 5 Similarly we can show that AθA = I. Hence A is a unitary matrix. Theorem 1: If A and B are two idempotent matrices and if AB = BA = 0 then A + B will also be an idempotent matrix. Proof: Let A and B be two idempotent matrices. Then A2 = A and B2 = B, and suppose AB = BA = 0. Now (A + B)2 = (A + B)(A + B) = A(A + B) + B(A + B)(* DistributiveLaw) 2 2 = A + AB + BA + B (* DistributiveLaw) 2 2 = A + B (* A = A and B = B) Hence A + B is an idempotent matrix. Theorem 2: If A and B are two idempotent matrices and if they com- mute then AB is idempotent matrix. Proof: Let A and B be two idempotent matrices. Then A2 = A and B2 = B. Assume the A and B commutes. i.e. AB = BA 6 Now (AB)2 = (AB)(AB) = A(BA)B (* AssociativeLaw) = A(AB)B (* AB = BA) = (AA)(BB)(* AssociativeLaw) = A2B2 = AB Hence AB is idempotent matrix. Theorem 3: If A and B are two n-rowed orthogonal matrices then AB and BA are also orthogonal matrices. Proof: Let A and B be two orthogonal matrices. Then AA0 = A0A = I And BB0 = B0B = I Now (AB)(AB)0 = (AB)(B0A0) 0 0 = A(BB )A (* AssociativeLaw) = A(I)A0 = (AI)A0 = AA0 = I 7 Similarly (AB)0(AB) = (B0A0)(AB) 0 0 = B (A A)B (* AssociativeLaw) = B0(I)B = (B0I)B = B0B = I So we get (AB)(AB)0 = I = (AB)0(AB). Hence AB is an orthogonal matrix. Similarly we can show that BA is an orthogonal matrix. Theorem 4: If A and B are two n-rowed unitary matrices then AB and BA are also unitary matrices. Proof: Let A and B be two unitary matrices. Then AAθ = AθA = I And BBθ = BθB = I Now (AB)(AB)θ = (AB)(BθAθ) θ θ = A(BB )A (* AssociativeLaw) = A(I)Aθ = (AI)Aθ = AAθ = I 8 Similarly (AB)θ(AB) = (BθAθ)(AB) θ θ = B (A A)B (* AssociativeLaw) = Bθ(I)B = (BθI)B = BθB = I So we get (AB)(AB)θ = I = (AB)θ(AB). Hence AB is an unitary matrix. Similarly we can show that BA is an unitary matrix. 2 3 2 −2 −4 6 7 6 7 Example 1: Show that the matrix A = 6 −1 3 4 7 is an idem- 4 5 1 −2 −3 potent matrix. Solution: We have 2 3 2 −2 −4 6 7 6 7 A = 6 −1 3 4 7 4 5 1 −2 −3 9 Then 2 3 2 3 2 −2 −4 2 −2 −4 6 7 6 7 2 6 7 6 7 A = 6 −1 3 4 7 6 −1 3 4 7 4 5 4 5 1 −2 −3 1 −2 −3 2 3 4 + 2 − 4 −4 − 6 + 8 −8 − 8 + 12 6 7 6 7 = 6 −2 − 3 + 4 2 + 9 − 8 4 + 12 − 12 7 4 5 2 + 2 − 3 −2 − 6 + 6 −4 − 8 + 9 2 3 2 −2 −4 6 7 6 7 = 6 −1 3 4 7 4 5 1 −2 −3 = A 2 ) A = A. Hence A is idempotent matrix. 2 3 1 1 3 6 7 6 7 Example 2: Show that 6 5 2 6 7 is a nilpotent matrix of index 3. 4 5 −2 −1 −3 10 2 3 1 1 3 6 7 6 7 Solution: Let A = 6 5 2 6 7 4 5 −2 −1 −3 Then 2 3 2 3 1 1 3 1 1 3 6 7 6 7 2 6 7 6 7 A = 6 5 2 6 7 6 5 2 6 7 4 5 4 5 −2 −1 −3 −2 −1 −3 2 3 1 + 5 − 6 1 + 2 − 3 3 + 6 − 9 6 7 6 7 = 6 5 + 10 − 12 5 + 4 − 6 15 + 12 − 18 7 4 5 −2 − 5 + 6 −2 − 3 + 3 −6 − 6 + 9 2 3 0 0 0 6 7 6 7 = 6 3 3 −3 7 4 5 −1 −1 −3 11 Then A3 = A2A 2 3 2 3 0 0 0 1 1 3 6 7 6 7 6 7 6 7 = 6 3 3 −3 7 6 5 2 6 7 4 5 4 5 −1 −1 −3 −2 −1 −3 2 3 0 0 0 6 7 6 7 = 6 0 0 0 7 4 5 0 0 0 Thus A3 = 0. Hence A is a nilpotent matrix of index 3. 2 3 1 1 + i Example 3: Prove that the matrix p1 4 5 is unitary. 3 1 − i −1 2 3 1 1 + i Solution: Let A = p1 6 7 3 4 5 1 − i −1 2 3 1 1 − i Then A0 = p1 4 5 3 1 + i −1 2 3 1 1 + i Then Aθ = A0 = p1 4 5 3 1 − i −1 12 Then 2 3 2 3 θ 1 1 1 + i 1 1 1 + i AA = p 4 5 p 4 5 3 1 − i −1 3 1 − i −1 2 3 1 1 + (1 − i)2 1 + i − (1 + i) = 4 5 3 (1 − i) − (1 + i) 1 − i2 + 1 2 3 1 3 0 = 4 5 3 0 3 2 3 1 0 = 4 5 0 1 = I Thus AAθ = I. Similarly we get AθA = I Hence A is a unitary matrix. 2 3 −5 −8 0 6 7 6 7 Example 4: Show that A = 6 3 5 0 7 is involuntary.
Load more

Recommended publications
  • Stat 5102 Notes: Regression
    Stat 5102 Notes: Regression Charles J. Geyer April 27, 2007 In these notes we do not use the “upper case letter means random, lower case letter means nonrandom” convention. Lower case normal weight letters (like x and β) indicate scalars (real variables). Lowercase bold weight letters (like x and β) indicate vectors. Upper case bold weight letters (like X) indicate matrices. 1 The Model The general linear model has the form p X yi = βjxij + ei (1.1) j=1 where i indexes individuals and j indexes different predictor variables. Ex- plicit use of (1.1) makes theory impossibly messy. We rewrite it as a vector equation y = Xβ + e, (1.2) where y is a vector whose components are yi, where X is a matrix whose components are xij, where β is a vector whose components are βj, and where e is a vector whose components are ei. Note that y and e have dimension n, but β has dimension p. The matrix X is called the design matrix or model matrix and has dimension n × p. As always in regression theory, we treat the predictor variables as non- random. So X is a nonrandom matrix, β is a nonrandom vector of unknown parameters. The only random quantities in (1.2) are e and y. As always in regression theory the errors ei are independent and identi- cally distributed mean zero normal. This is written as a vector equation e ∼ Normal(0, σ2I), where σ2 is another unknown parameter (the error variance) and I is the identity matrix. This implies y ∼ Normal(µ, σ2I), 1 where µ = Xβ.
    [Show full text]
  • DECOMPOSITION of SINGULAR MATRICES INTO IDEMPOTENTS 11 Us Show How to Construct Ai+1, Bi+1, Ci+1, Di+1
    DECOMPOSITION OF SINGULAR MATRICES INTO IDEMPOTENTS ADEL ALAHMADI, S. K. JAIN, AND ANDRE LEROY Abstract. In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempo- tents and apply these factorizations for proving our main results. We generalize works due to Laffey ([12]) and Rao ([3]) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems (cf. [3], Theorems 5 and 7 and [4]). We also consider singular matrices over B´ezoutdomains as to when such a matrix is a product of idempotent matrices. 1. Introduction and definitions It was shown by Howie [10] that every mapping from a finite set X to itself with image of cardinality ≤ cardX − 1 is a product of idempotent mappings. Erd¨os[7] showed that every singular square matrix over a field can be expressed as a product of idempotent matrices and this was generalized by several authors to certain classes of rings, in particular, to division rings and euclidean domains [12]. Turning to singular elements let us mention two results: Rao [3] characterized, via continued fractions, singular matrices over a commutative PID that can be decomposed as a product of idempotent matrices and Hannah-O'Meara [9] showed, among other results, that for a right self-injective regular ring R, an element a is a product of idempotents if and only if Rr:ann(a) = l:ann(a)R= R(1 − a)R. The purpose of this paper is to provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and to apply these factorizations for proving our main results.
    [Show full text]
  • Mathematics Study Guide
    Mathematics Study Guide Matthew Chesnes The London School of Economics September 28, 2001 1 Arithmetic of N-Tuples • Vectors specificed by direction and length. The length of a vector is called its magnitude p 2 2 or “norm.” For example, x = (x1, x2). Thus, the norm of x is: ||x|| = x1 + x2. pPn 2 • Generally for a vector, ~x = (x1, x2, x3, ..., xn), ||x|| = i=1 xi . • Vector Order: consider two vectors, ~x, ~y. Then, ~x> ~y iff xi ≥ yi ∀ i and xi > yi for some i. ~x>> ~y iff xi > yi ∀ i. • Convex Sets: A set is convex if whenever is contains x0 and x00, it also contains the line segment, (1 − α)x0 + αx00. 2 2 Vector Space Formulations in Economics • We expect a consumer to have a complete (preference) order over all consumption bundles x in his consumption set. If he prefers x to x0, we write, x x0. If he’s indifferent between x and x0, we write, x ∼ x0. Finally, if he weakly prefers x to x0, we write x x0. • The set X, {x ∈ X : x xˆ ∀ xˆ ∈ X}, is a convex set. It is all bundles of goods that make the consumer at least as well off as with his current bundle. 3 3 Complex Numbers • Define complex numbers as ordered pairs such that the first element in the vector is the real part of the number and the second is complex. Thus, the real number -1 is denoted by (-1,0). A complex number, 2+3i, can be expressed (2,3). • Define multiplication on the complex numbers as, Z · Z0 = (a, b) · (c, d) = (ac − bd, ad + bc).
    [Show full text]
  • Block Diagonalization
    126 (2001) MATHEMATICA BOHEMICA No. 1, 237–246 BLOCK DIAGONALIZATION J. J. Koliha, Melbourne (Received June 15, 1999) Abstract. We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution dou- ble commutes with the given matrix. Applications include a new characterization of an eigenprojection and of the Drazin inverse of a given matrix. Keywords: eigenprojection, resolutions of the unit matrix, block diagonalization MSC 2000 : 15A21, 15A27, 15A18, 15A09 1. Introduction and preliminaries In this paper we are concerned with a block diagonalization of a given matrix A; by definition, A is block diagonalizable if it is similar to a matrix of the form A1 0 ... 0 0 A2 ... 0 (1.1) =diag(A1,...,Am). ... ... 00... Am Then the spectrum σ(A)ofA is the union of the spectra σ(A1),...,σ(Am), which in general need not be disjoint. The ultimate such diagonalization is the Jordan form of the matrix; however, it is often advantageous to utilize a coarser diagonalization, easier to construct, and customized to a particular distribution of the eigenvalues. The most useful diagonalizations are the ones for which the sets σ(Ai)arepairwise disjoint; it is the aim of this paper to give a full characterization of these diagonal- izations. ∈ n×n For any matrix A we denote its kernel and image by ker A and im A, respectively. A matrix E is idempotent (or a projection matrix )ifE2 = E,and 237 nilpotent if Ep = 0 for some positive integer p.
    [Show full text]
  • Chapter 2 a Short Review of Matrix Algebra
    Chapter 2 A short review of matrix algebra 2.1 Vectors and vector spaces Definition 2.1.1. A vector a of dimension n is a collection of n elements typically written as ⎛ ⎞ ⎜ a1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ a2 ⎟ a = ⎜ ⎟ =(ai)n. ⎜ . ⎟ ⎝ . ⎠ an Vectors of length 2 (two-dimensional vectors) can be thought of points in 33 BIOS 2083 Linear Models Abdus S. Wahed the plane (See figures). Chapter 2 34 BIOS 2083 Linear Models Abdus S. Wahed Figure 2.1: Vectors in two and three dimensional spaces (-1.5,2) (1, 1) (1, -2) x1 (2.5, 1.5, 0.95) x2 (0, 1.5, 0.95) x3 Chapter 2 35 BIOS 2083 Linear Models Abdus S. Wahed • A vector with all elements equal to zero is known as a zero vector and is denoted by 0. • A vector whose elements are stacked vertically is known as column vector whereas a vector whose elements are stacked horizontally will be referred to as row vector. (Unless otherwise mentioned, all vectors will be referred to as column vectors). • A row vector representation of a column vector is known as its trans- T pose. We will use⎛ the⎞ notation ‘ ’or‘ ’ to indicate a transpose. For ⎜ a1 ⎟ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎟ T instance, if a = ⎜ ⎟ and b =(a1 a2 ... an), then we write b = a ⎜ . ⎟ ⎝ . ⎠ an or a = bT . • Vectors of same dimension are conformable to algebraic operations such as additions and subtractions. Sum of two or more vectors of dimension n results in another n-dimensional vector with elements as the sum of the corresponding elements of summand vectors.
    [Show full text]
  • Table of Symbols
    Table of Symbols Symbol Meaning Reference ∅ Empty set Page 10 ∈ Member symbol Page 10 ⊆ Subset symbol Page 10 ⊂ Proper subset symbol Page 10 ∩ Intersection symbol Page 10 ∪ Union symbol Page 10 ⊗ Tensor symbol Page 136 ≈ Approximate equality sign Page 79 −−→ PQ Displacement vector Page 147 | z | Absolute value of complex z Page 13 | A | determinant of matrix A Page 115 || u || Norm of vector u Page 212 || u ||p p-norm of vector u Page 306 u · v Standard inner product Page 216 u, v Inner product Page 312 Acof Cofactor matrix of A Page 122 adj A Adjoint of matrix A Page 122 A∗ Conjugate (Hermitian) transpose of matrix A Page 91 AT Transpose of matrix A Page 91 C(A) Column space of matrix A Page 183 cond(A) Condition number of matrix A Page 344 C[a, b] Function space Page 151 C Complex numbers a + bi Page 12 Cn Standard complex vector space Page 149 compv u Component Page 224 z Complex conjugate of z Page 13 δij Kronecker delta Page 65 dim V Dimension of space V Page 194 det A Determinant of A Page 115 domain(T ) Domain of operator T Page 189 diag{λ1,λ2,...,λn} Diagonal matrix with λ1,λ2,...,λn along diagonal Page 103 Eij Elementary operation switch ith and jth rows Page 25 356 Table of Symbols Symbol Meaning Reference Ei(c) Elementary operation multiply ith row by c Page 25 Eij (d) Elementary operation add d times jth row to ith row Page 25 Eλ(A) Eigenspace Page 254 Hv Householder matrix Page 237 I,In Identity matrix, n × n identity Page 65 idV Identity function for V Page 156 (z) Imaginary part of z Page 12 ker(T ) Kernel of operator T Page 188 Mij (A) Minor of A Page 117 M(A) Matrix of minors of A Page 122 max{a1,a2,...,am} Maximum value Page 40 min{a1,a2,...,am} Minimum value Page 40 N (A) Null space of matrix A Page 184 N Natural numbers 1, 2,..
    [Show full text]
  • Econometrics Lecture 1: Review of Matrix Algebra
    Econometrics Lecture 1: Review of Matrix Algebra R. G. Pierse 1 Introduction A matrix is a rectangular array of numbers. If the matrix has n rows and m columns it is said to be an n × m matrix. This is called the dimension of the matrix. A matrix with a single column ( n × 1) is called a column vector.A matrix with a single row ( 1 × m) is called a row vector. A matrix with only one row and one column ( a single number) is called a scalar. The standard convention for denoting a matrix is to use a capital letter in bold typeface as in A, B, C. A column vector is denoted with a lowercase letter in bold typeface as in a, b, c. A row vector is denoted with a lowercase letter in bold typeface, followed by a prime, as in a0, b0, c0. A scalar is generally denoted with a lowercase letter is normal typeface as in a, b, c. An n × m matrix A can be written out explicitly in terms of its elements as in: 2 3 a11 a12 ··· a1j a1m 6 a21 a22 ··· a2j a2m 7 6 7 6 . .. 7 6 . 7 A = 6 7 : 6 ai1 ai2 aij aim 7 6 7 4 5 an1 an2 anj anm Each element has two subscripts: the first is the row index and the second the column index so that aij refers to the element in the ith row and jth column of A. 2 Matrix Operations The standard operations of addition, subtraction and multiplication can be defined for two matrices as long as the dimensions of the matrices satisfy appropriate conditions to ensure that the operation makes sense.
    [Show full text]
  • Zero-Sum Triangles for Involutory, Idempotent, Nilpotent and Unipotent Matrices
    Zero-Sum Triangles for Involutory, Idempotent, Nilpotent and Unipotent Matrices Pengwei Hao1, Chao Zhang2, Huahan Hao3 Abstract: In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for the recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent and unipotent properties, especially nilpotent and unipotent matrices of index 2. With the zero-sum rule we also give the conditions for the special matrices and the generic methods for the generation of those special matrices. The generated integer triangles are mostly newly discovered, and more combinatorial identities can be found with them. Keywords: zero-sum rule, triangles of numbers, generation of special matrices, involutory, idempotent, nilpotent and unipotent matrices 1. Introduction Zero-sum was coined in early 1940s in the field of game theory and economic theory, and the term “zero-sum game” is used to describe a situation where the sum of gains made by one person or group is lost in equal amounts by another person or group, so that the net outcome is neutral, or the sum of losses and gains is zero [1]. It was later also frequently used in psychology and political science. In this work, we apply the zero-sum rule to three adjacent cells to construct integer triangles. Pascal's triangle is named after the French mathematician Blaise Pascal, but the history can be traced back in almost 2,000 years and is referred to as the Staircase of Mount Meru in India, the Khayyam triangle in Persia (Iran), Yang Hui's triangle in China, Apianus's Triangle in Germany, and Tartaglia's triangle in Italy [2].
    [Show full text]
  • On the Use of Idempotent Matrices in the Treatment of Linear Restrictions in Regression Analysis
    ... ·.,p.. --r ON THE USE OF IDEMPOTENT MATRICES IN THE TREATMENT OF LINEAR RESTRICTIONS IN REGRESSION ANALYSIS Johns. Chipman M. M. RaJd Technical Report No. 10 University of 1-finnesota and Carnegie Institute of Technology ~ . --• I, University of Minnesota Minneapolis, Minnesota Reproduction in whole or part is pennitted for any purpose of the United States Government. !/ Work done in part under contract Nonr 2582(00), Task NR 042-200 of the Office of Naval Research. ... on the Use of Idempotent Ma.trices in the Treatment of Linear Restrictions in Regression .Analysis1 by John s. Chipman and M. M. Rao University of Minnesota and Carnegie Institute of Technology o. Introduction~ Summary The purpose of the present paper is to present a unified treatment of the problem of estimation and hypothesis testing in linear regression analy­ sis. In sections 2 and 3 below, we consider the estimation of regression coefficients subject to a set of linear restrictions, according to the Markov criterion of best linear unbiasedness and the least squares criterion, respectively. In section 4 we consider the testing of a set of linear ; restrictions on the coefficients of a regression equation. In section 5 we take up the general linear hypothesis, that is, the testing of one set of linear restrictions subject to another set being true. In the course of the treatment of the above problems, idempotent matrices play a natural and central role; accordingly, section 1 is devoted to the derivation of the principal theorems. Since idempotent transformations may be viewed geometrically as projections in linear spaces, secti.on 6 is devoted to the geometric interpretation of the results.
    [Show full text]
  • Elements of Matrix Algebra
    ELEMENTS OF MATRIX ALGEBRA CHUNG-MING KUAN Department of Finance National Taiwan University September 09, 2009 c Chung-Ming Kuan, 1996, 2001, 2009. E-mail: [email protected]; URL: homepage.ntu.edu.tw/∼ckuan. CONTENTS i Contents 1 Vector 1 1.1 Vector Operations . 1 1.2 Euclidean Inner Product and Norm . 2 1.3 Unit Vector . 2 1.4 Direction Cosine . 3 1.5 Statistical Applications . 5 Exercises ...................................... 6 2 Vector Space 8 2.1 The Dimension of a Vector Space . 8 2.2 The Sum and Direct Sum of Vector Spaces . 10 2.3 Orthogonal Basis Vectors . 11 2.4 Orthogonal Projection . 12 2.5 Statistical Applications . 14 Exercises . 15 3 Matrix 16 3.1 Basic Matrix Types . 16 3.2 Matrix Operations . 17 3.3 Scalar Functions of Matrix Elements . 18 3.4 Matrix Rank . 20 3.5 Matrix Inversion . 21 3.6 Statistical Applications . 23 Exercises . 24 4 Linear Transformation 26 4.1 Change of Basis . 26 4.2 Systems of Linear Equations . 28 4.3 Linear Transformation . 29 Exercises . 32 5 Special Matrices 34 5.1 Symmetric Matrix . 34 5.2 Skew-Symmetric Matrix . 34 5.3 Quadratic Form and Definite Matrix . 35 5.4 Differentiation Involving Vectors and Matrices . 36 c Chung-Ming Kuan, 2001, 2009 CONTENTS ii 5.5 Idempotent and Nilpotent Matrices . 37 5.6 Orthogonal Matrix . 38 5.7 Projection Matrix . 39 5.8 Partitioned Matrix . 40 5.9 Statistical Applications . 41 Exercises . 41 6 Eigenvalue and Eigenvector 43 6.1 Eigenvalue and Eigenvector . 43 6.2 Diagonalization . 44 6.3 Orthogonal Diagonalization .
    [Show full text]
  • (AA+F = AA+, and (A+A)T = A+A
    proceedings of the american mathematical society Volume 31, No. I, January 1972 THE GENERALIZED INVERSE OF A NONNEGATIVE MATRLX R. J. PLEMMONS AND R. E. CLINE1 Abstract. Necessary and sufficient conditions are given in order that a nonnegative matrix have a nonnegative Moore- Penrose generalized inverse. 1. Introduction. Let A be an arbitrary m x n real matrix. Then the Moore-Penrose generalized inverse of A is the unique n x m real matrix A+ satisfying the equations A = AA+A, A+ = A+AA+, (AA+f = AA+, and (A+A)T= A+A. The properties and applications of A+ are described in a number of papers including Penrose [7], [8], Ben-Israel and Charnes [1], Cline [2], and Greville [6]. The main value of the generalized inverse, both conceptually and practically, is that it provides a solution to the following least squares problem: Of all the vectors x which minimize \\b — Ax\\, which has the smallest \\x\\21 The solution is x = A+b. If A is nonnegative (written A ^ 0), that is, if the components of A are all nonnegative real numbers, then A+ is not necessarily nonnegative. In particular, if A 5: 0 is square and nonsingular, then A+ = A-1 ^ 0 if and only if A is monomial, i.e., A can be expressed as a product of a diagonal matrix and a permutation matrix, so that A~x = DAT for some diagonal matrix D with positive diagonal elements. The main purpose of this paper is to give necessary and sufficient conditions on A ^ 0 in order that A+ ^ 0.
    [Show full text]
  • Sign Idempotent Matrices and Generalized Inverses ∑
    ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol. 5, No. 3, 2010, pp. 233-240 Sign Idempotent Matrices and Generalized Inverses Jingyue Zhang 1, +, T. Z. Huang 1, Zhongshan Li 2 and Xingwen Zhu 1 1School of Mathematical Sciences, University of Electronic Science and Technology, 610054 2Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30302-4110, USA (Received December 14, 2009, accepted December 22, 2009) Abstract. A matrix whose entries consist of ,,0 is called a sign pattern matrix. Let QA denote the set of all real n n matrices B such that the signs of entries in B match the corresponding entries in A . For nonnegative sign patterns, sign idempotent patterns have been characterized. In this paper, we Firstly give an equi-valent proposition to characterize general sign idempotent matrices (sign idempotent). Next, we study properties of a class of matrices which can be generalized permutationally similar to specialized sign patterns. Finally, we consider the relationships among the allowance of idempotent, generalized inverses and the allowance of tripotent in symmetric sign idempotent patterns. Keywords: sign pattern matrix; symmetric sign pattern; sign idempotent; allowance of idempotent; generalized inverses 1. Introduction A matrix whose entries consist of ,,0 is called a sign pattern matrix. A subpattern of a sign pattern ma-trix A is a sign pattern matrix obtained by replacing some of the or entries in A with 0. The sign pattern In is the diagonal pattern of order n with + diagonal entries. In order to study conveniently, we also ues I to denote diagonal patterns with + diagonal entries.
    [Show full text]