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PRIMUS Problems, Resources, and Issues in Mathematics Undergraduate Studies ISSN: 1051-1970 (Print) 1935-4053 (Online) Journal homepage: https://www.tandfonline.com/loi/upri20 Block Matrices in Linear Algebra Stephan Ramon Garcia & Roger A. Horn To cite this article: Stephan Ramon Garcia & Roger A. Horn (2020) Block Matrices in Linear Algebra, PRIMUS, 30:3, 285-306, DOI: 10.1080/10511970.2019.1567214 To link to this article: https://doi.org/10.1080/10511970.2019.1567214 Accepted author version posted online: 05 Feb 2019. Published online: 13 May 2019. Submit your article to this journal Article views: 86 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=upri20 PRIMUS, 30(3): 285–306, 2020 Copyright # Taylor & Francis Group, LLC ISSN: 1051-1970 print / 1935-4053 online DOI: 10.1080/10511970.2019.1567214 Block Matrices in Linear Algebra Stephan Ramon Garcia and Roger A. Horn Abstract: Linear algebra is best done with block matrices. As evidence in sup- port of this thesis, we present numerous examples suitable for classroom presentation. Keywords: Matrix, matrix multiplication, block matrix, Kronecker product, rank, eigenvalues 1. INTRODUCTION This paper is addressed to instructors of a first course in linear algebra, who need not be specialists in the field. We aim to convince the reader that linear algebra is best done with block matrices. In particular, flexible thinking about the process of matrix multiplication can reveal concise proofs of important theorems and expose new results. Viewing linear algebra from a block-matrix perspective gives an instructor access to use- ful techniques, exercises, and examples. Many of the techniques, proofs, and examples presented here are familiar to specialists in linear algebra or operator theory. We think that everyone who teaches undergraduate linear algebra should be aware of them. A popular current textbook says that block matrices “appear in most modern applications of linear algebra because the notation high- lights essential structures in matrix analysis …”[5, p. 119]. The use of block matrices in linear algebra instruction aligns math- ematics pedagogy better with topics in advanced courses in pure Address all correspondence to Stephan Ramon Garcia, Department of Mathematics, Pomona College, Claremont, CA 91711, USA. Email: [email protected] 286 Garcia and Horn mathematics, computer science, data science, statistics, and other fields. For example, block-matrix techniques are standard fare in modern algo- rithms [3]. Textbooks such as [2–7] make use of block matrices. We take the reader on a tour of block-matrix methods and applica- tions. In Section 2, we use right-column partitions to explain several standard first-course results. In Section 3, we use left-column partitions to introduce the full-rank factorization, prove the invariance of the num- ber of elements in a basis, and establish the equality of row and column rank. Instructors of a first linear algebra course will be familiar with these topics, but perhaps not with a block matrix/column partition approach to them. Section 4 concerns block-column matrices. Applications include justification of a matrix-inversion algorithm and a proof of the unique- ness of the reduced row echelon form. Block-row and block-column matrices are used in Section 5 to obtain inequalities for the rank of sums and products of matrices, along with algebraic characterizations of matri- ces that share the same column space or null space. The preceding mater- ial culminates in Section 6, in which we consider block matrices of several types and prove that the geometric multiplicity of an eigenvalue is at most its algebraic multiplicity. We also obtain a variety of determinan- tal results that are suitable for presentation in class. We conclude in Section 7 with Kronecker products and several applications. Notation: We frame our discussion for complex matrices. However, all of our numerical examples involve only real matrices, which may be pre- ferred by some first-course instructors. We use MmÂn to denote the set of all m  n complex matrices; Mn denotes the set of all n  n complex matrices. Boldface letters, such as a; b; c; denote column vectors; n m e1; e2; :::; en is the standard basis of C : We regard elements of C as col- umn vectors; that is, m  1 matrices. If A 2 MmÂn; then each column of T A belongs to MmÂ1: The transpose of a matrix A is denoted by A : The null space and column space of a matrix A are denoted by null A and col A; respectively. The trace and determinant of a square matrix A are denoted by tr A and det A; respectively. 2. RIGHT-COLUMN PARTITIONS If 12 452 A ¼ and B ¼ ; (1) 34 671 then the entries of AB are dot products of rows of A with columns of B: Block Matrices in Linear Algebra 287 1 Á 4 þ 2 Á 61Á 5 þ 2 Á 71Á 2 þ 2 Á 1 16 19 4 AB ¼ ¼ : (2) 3 Á 4 þ 4 Á 63Á 5 þ 4 Á 73Á 2 þ 4 Á 1 36 43 10 However, there are other ways to organize these computations. We examine right-column partitions in this section. If A 2 MmÂr and B ¼½b1 b2 ::: bn2MrÂn; then the jth column of AB is Abj: That is, AB ¼ ½Ab1 Ab2 ::: Abn : (3) An intentional approach to column partitions can facilitate proofs of important results from elementary linear algebra. Example 4. If A and B are the matrices from equation (1), then B ¼ ½b1 b2 b3; in which 4 5 2 b ¼ ; b ¼ ; and b ¼ : 1 6 2 7 3 1 Partitioned matrix multiplication yields the expected answer (2): "#"#"#"#"#"#"# 12 4 12 5 12 2 ½Ab1 Ab2 Ab3 ¼ 34 6 34 7 34 1 "#"#"#"# "# 16 19 4 16 19 4 ¼ ¼ 36 43 10 36 43 10 ¼ AB: Example 5. Matrix-vector equations can be bundled together. For example, suppose that x1; x2; :::; xk are eigenvectors of A 2 Mn for the eigenvalue k and let X ¼½x1 x2 ::: xk2MnÂk: Then AX ¼ ½Ax1 Ax2 ::: Axk ¼ ½kx1 kx2 ::: kxk ¼ kX: This observation can be used to prove that the geometric multiplicity of an eigenvalue is at most its algebraic multiplicity; see Example 36. The following example provides a short proof of an important impli- cation in “the invertible matrix theorem,” which is in the core of a first course in linear algebra. Example 6 (Universal consistency yields right inverse). If Ax ¼ b is con- n n sistent for each b 2 C ; there are bi 2 C such that Abi ¼ ei for i ¼ 1; 2; :::; n: Then 288 Garcia and Horn A½b1 b2 ::: bn ¼ ½Ab1 Ab2 ::: Abn ¼ ½e1 e2 ::: en ¼ I; so AB ¼ I for B ¼½b1 b2 ::: bn: In the preceding example, we obtained a right inverse for a square matrix A. The fact that a right inverse for A is also a left inverse is nontri- vial; it can fail for linear transformations if the underlying vector space is not finite dimensional [2, P.2.7]. Here is an explanation that is based on column partitions. Example 7 (One-sided inverses are two-sided inverses). If A; B 2 Mn and AB ¼ I, then AðBxÞ¼x for all x 2 Cn and hence col A ¼ Cn: The Dimension Theorem [2, Cor. 2.5.4] ensures that null A ¼f0g: Partition IÀBA ¼½x1 x2 ::: xn according to its columns. Then ½Ax1 Ax2 ::: Axn ¼ A½x1 x2 ::: xn ¼ AIðÞÀBA ¼ AÀðÞAB A ¼ AÀIA ¼ 0; so each xi ¼ 0 since null A ¼f0g: Thus, IÀBA ¼ 0 and hence BA ¼ I. Although it cannot be recommended as a practical numerical algo- rithm, Cramer’s rule is an important concept. Why does it work? Example 8 (Cramer’s rule). Let A ¼½a1 a2 ::: an2Mn be invertible, n let b 2 C ; and let Ai 2 Mn be the matrix obtained by replacing the ith T n column of A with b: Then there is a unique x ¼½x1 x2 ::: xn 2 C such that Ax ¼ b: Cofactor expansion along the ith row of A reveals that (ith columns underlined) x ¼ det½e ::: e xe ::: e i ÂÃ1 iÀ1 iþ1 n À1 À1 À1 À1 À1 ¼ det A a1 ::: A aiÀ1 A b A aiþ1 ::: A an ÂÃÀÁ À1 À1 ¼ det A a1 ::: aiÀ1 baiþ1 ::: an ¼ det A Ai det A ¼ i : det A 3. LEFT-COLUMN PARTITIONS We have gotten some mileage out of partitioning the matrix on the right- hand side of a product. If we partition the matrix on the left-hand side of a product, other opportunities emerge. If A ¼½a1 a2 ::: an2MmÂn T n and x ¼½x1 x2 ::: xn 2 C ; then Block Matrices in Linear Algebra 289 Ax ¼ x1a1 þ x2a2 þÁÁÁþxnan: (9) That is, Ax is a linear combination of the columns of A. The next example illustrates that relationships between geometric objects, such as vectors and subspaces, can often be framed algebraically. Example 10 (Geometry and matrix algebra). Let A 2 MmÂn and B 2 MmÂk: We claim that col B col A () there exists an X 2 MnÂk such that AX ¼ B; moreover, if the columns of A are linearly independent, then X is unique. If each column of B ¼½b1 b2 ::: bk2MmÂk is a linear combination n of the columns of A 2 MmÂn; then (9) ensures that there are xi 2 C such that bi ¼ Axi for each i; if the columns of A are linearly independent, then the xi are uniquely determined. Let X ¼½x1 x2 ::: xk2MnÂk: Then B ¼ ½b1 b2 ::: bk ¼½Ax1 Ax2 ::: Axk ¼A½x1 x2 ::: xk ¼ AX: Conversely, if AX ¼ B, then (9) indicates that each column of B lies in col A: The following example uses Example 10 to show that any two bases for the same subspace of Cn have the same number of elements [1], [2, P.3.38].
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