Matrices
Matrices are a very classical tabular form to represent data, for example in accounting. They are built from columns that are juxtaposed and can be split horizontally into a stack of rows. The novelty in mathematics is that matrices that are built from nu
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Summary. Matrices are a very classical tabular form to represent data, for example in accounting. They are built from columns that are juxtaposed and can be split horizontally into a stack of rows. The novelty in mathematics is that matrices that are built from numbers can be used to perform calculations that are of general benefit to mathematics. –Σ–
Fig. 26.1. A matrix built from columns in Greek temples. For every column j, we have its building blocks aij , referring to row i. The third dimension of depth is not dealt with in this book, but there are also three-dimensional matrices in mathematics.
© Springer International Publishing Switzerland 2016 G. Mazzola et al., Cool Math for Hot Music, Computational Music Science, DOI 10.1007/978-3-319-42937-3_26
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26 Matrices
Matrices are the backbone of algebra. They are indispensable for most concrete calculations, but they also share the structure of a category (we shall discuss categories in Chapter 29) in a particularly transparent way. Moreover, matrices provide us with examples of important algebraic structures of modules (we shall discuss modules in Chapter 27). Finally, matrices also provide us with core examples of non-commutative rings.
26.1 Generalities on Matrices For natural numbers n ≥ 1, we denote by [1, n] the set {1, 2, . . . n} of the first n positive natural numbers. Definition 67 Suppose we are given two positive natural numbers m, n and a commutative ring R. A m × n-matrix with coefficients in R is a set map M : [1, m] × [1, n] → R. The images M (i, j) are denoted with indices, M (i, j) = Mi,j . Matrices are typically represented in matrix form with m rows and n columns: ⎛ ⎞ M11 M12 . . . M1n ⎜ ⎟ ⎜ M21 M22 . . . M2n ⎟ ⎟. M = (Mij ) = ⎜ ⎜ ⎟ ⎝ ... ⎠ Mm1 Mm2 . . . Mmn The set of m × n-matrices with coefficients in R is denoted by Mm,n (R). The matrix transposition is a bijection ∼
?t : Mm,n (R) → Mn,m (R) : M → M t with (M t )ji = Mij ; we have (M t )t = M . For an element λ ∈ R and a matrix M ∈ Mm,n (R), we have its scalar-multiplied λM that has the coefficients (λM )ij = λMij . The identity matrix En of rank n is the matrix in Mn,n (R) with (En )ij = δij , where δii = 1R for all indices i, and zero otherwise (δij is called the Kronecker delta). There are several core algebraic operations on matrices. To begin with, if M, N ∈ Mm,n (R), then we define their sum M + N by (M + N )ij = Mij + Nij . This turns Mm,n (R) into a commutative group, and we have the isomorphism ∼ of groups Mm,n (R) → Rmn , the cartesian product of mn copies of the additive group of R. The product of matrices is slightly more involved: If M ∈ Mm,n (R) and N ∈ Mn,l (R), then we define their product M N ∈ Mm,l (R) by
26.1 Generalities on Matrices
(M N )ik =
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Mij Njk .
j
This means that the coefficient (M N )ik at row i and column k is the sum of the products Mij Njk of the ith-row coefficients of M with the corresponding kth column coefficients of N . To show the number of rows and columns of a matrix M ∈ Mm,n (R), we also write it as a symbol of a function (and you will
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