MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with - PowerPoint PPT Presentation

Matrices BUSINESS MATHEMATICS 1

CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question Further study 2

MATRICES A matrix is a rectangular array of numbers or variables Notation ▪ we often use bold non-italic capital letters to refer to them 𝑏 1,1 𝑏 1,2 ⋯ 𝑏 1,𝑜 3 2 𝑏 2,1 𝑏 2,2 ⋯ 𝑏 2,𝑜 , 𝐁 = ▪ 𝐑 = −2 0 ⋯ ⋯ ⋯ ⋯ 12.5 −12.7 𝑏 𝑛,1 𝑏 𝑛,2 ⋯ 𝑏 𝑛,𝑜 Terminology ▪ these matrices consist of 6 respectively 𝑛𝑜 elements ▪ the order (or size is) 3 × 2 respectively 𝑛 × 𝑜 ▪ when 𝑛 = 𝑜 , the matrix is a square matrix ▪ when 𝑛 ≠ 𝑜 , the matrix is rectangular 3

MATRICES Some notes on notation ▪ brackets: or ▪ indexing elements: 𝑏 𝑗𝑘 or 𝑏 𝑗,𝑘 ▪ We define 𝐁 = 𝑏 𝑗𝑘 𝑛×𝑜 as the matrix of order 𝑛 × 𝑜 matrix with elements 𝑏 𝑗𝑘 , 𝑗 = 1, … 𝑛 , 𝑘 = 1, … , 𝑜 : 𝑏 11 𝑏 12 ⋯ 𝑏 1𝑜 𝑏 21 𝑏 22 ⋯ 𝑏 2𝑜 𝐁 = ⋯ ⋯ ⋯ ⋯ 𝑏 𝑛1 𝑏 𝑛2 ⋯ 𝑏 𝑛𝑜 4

MATRICES Notice the order 3 2 of the indices In 𝐑 = the element 𝑟 2,1 refers to −2 0 12.5 −12.7 ▪ the cell at row 2 and column 1 A column is vertical ... ▪ so to −2 ▪ while 𝑟 1,2 is in row 1 and column 2 and has value 2 The order of 𝐑 is 3 × 2 , not 2 × 3 Conventions: ▪ element 𝑏 row index,column index ▪ order 𝑛 row × 𝑜 column ▪ when no ambiguity you write 𝑏 𝑗𝑘 instead of 𝑏 𝑗,𝑘 5

EXERCISE 1 Given is 𝐚 = 𝑨 𝑗𝑘 = 1 0 −3 . Find 𝑨 1,2 + 𝑨 2,2 . 5 4 2 6

SPECIAL MATRICES 0 0 ⋯ 0 0 0 ⋯ 0 Zero matrix: 𝟏 = ⋯ ⋯ ⋯ ⋯ 0 0 ⋯ 0 ▪ for a matrix of any order 1 0 ⋯ 0 0 1 ⋯ 0 Identity matrix: 𝐉 = ⋯ ⋯ ⋯ ⋯ 0 0 ⋯ 1 ▪ for a square matrix (so, 𝑛 = 𝑜 ) 8

OPERATIONS WITH MATRICES We can define some basic operations with matrices, similar to the basic operations with vectors ▪ addition ( 𝐁 + 𝐂 , through 𝑏 + 𝑐 𝑗𝑘 = 𝑏 𝑗𝑘 + 𝑐 𝑗𝑘 ) ▪ multiplication ( 𝑑𝐁 , through 𝑑𝑏 𝑗𝑘 = 𝑑 × 𝑏 𝑗𝑘 ) ▪ negative matrix ( −𝐁 , through −𝑏 𝑗𝑘 = −𝑏 𝑗𝑘 ) ▪ subtraction ( 𝐁 − 𝐂 , through 𝑏 − 𝑐 𝑗𝑘 = 𝑏 𝑗𝑘 − 𝑐 𝑗𝑘 ) ▪ equality ( 𝐁 = 𝐂 , through 𝑏 𝑗𝑘 = 𝑐 𝑗𝑘 ) But what about the inner product? The matrices 𝐁 ▪ not available for matrices and 𝐂 must be of equal order ▪ instead: matrix multiplication 9

EXERCISE 2 Given is 𝐁 = 2 −1 and 𝐂 = 0 2 −3 . Find 2𝐁 − 𝐂 . 3 0 1 10

MATRIX MULTIPLICATION Let 𝐁 and 𝐂 be two matrices, of order 𝑛 × 𝑞 respectively 𝑞 × 𝑜 We define the matrix product 𝐁𝐂 as 𝑞 𝐁𝐂 𝑗𝑘 = ෍ 𝑏 𝑗𝑙 𝑐 𝑙𝑘 , 𝑗 = 1, … , 𝑛, 𝑘 = 1, … , 𝑜 𝑙=1 ▪ alternatively written as 𝐁 ⋅ 𝐂 ▪ but do not write 𝐁 × 𝐂 Notice: the result of multiplication of two matrices is a matrix ▪ different for the inner product of two vectors 12

MATRIX MULTIPLICATION Illustration: 2 ▪ 𝐁𝐂 1,2 = σ 𝑙=1 𝑏 1,𝑙 𝑐 𝑙,2 = 𝑏 1,1 𝑐 1,2 + 𝑏 1,2 𝑐 2,2 2 ▪ 𝐁𝐂 3,3 = σ 𝑙=1 𝑏 3,𝑙 𝑐 𝑙,3 = 𝑏 3,1 𝑐 1,3 + 𝑏 3,2 𝑐 2,3 13

MATRIX MULTIPLICATION Notice the orders of the matrices: 𝐁 𝑛×𝑞 𝐂 𝑞×𝑜 = 𝐁𝐂 𝑛×𝑜 ▪ so #columns in 𝐁 should match #rows in 𝐂 ▪ and #rows in 𝐁𝐂 is #rows in 𝐁 ▪ and #columns in 𝐁𝐂 is #columns in 𝐂 Consequences: given a matrix 𝐁 or order 3 × 3 and a matrix 𝐂 of order 3 × 2 ▪ 𝐁𝐂 exists and is of order 3 × 2 ▪ 𝐂𝐁 does not exist ▪ what about 𝐁𝐁 ? and 𝐂𝐂 ? and 𝐁𝐂 𝐁𝐂 ? 14

EXERCISE 3 Given is 𝐁 = 2 −1 and 𝐂 = 0 2 −3 . Find 𝐁 ⋅ 𝐂 . 3 0 1 15

MATRIX MULTIPLICATION It follows that (with suitable 𝐁 , 𝐂 , and 𝐃 ) ▪ 𝐁 𝐂 + 𝐃 = 𝐁𝐂 + 𝐁𝐃 (distributive property) 𝐁𝐂 𝐃 = 𝐁 𝐂𝐃 = 𝐁𝐂𝐃 (associative property) ▪ But not that ▪ 𝐁𝐂 = 𝐂𝐁 (commutative property) ▪ example: take 𝐁 = 1 2 2 −8 6 and 𝐂 = 3 −1 4 ▪ 𝐁𝐂 = 0 0 0 , but 𝐂𝐁 = −22 −44 0 11 22 By the way, notice that in this example: ▪ 𝐁𝐂 = 𝟏 , while 𝐁 ≠ 𝟏 and 𝐂 ≠ 𝟏 ▪ while for numbers 𝑏𝑐 = 0 ⇔ 𝑏 = 0 or 𝑐 = 0 17

MATRIX MULTIPLICATION Some properties (for suitable 𝐁 , 𝐂 , and 𝐃 ): 𝐁𝟏 = 𝟏 and 𝟏𝐁 = 𝟏 𝐁𝐉 = 𝐁 and 𝐉𝐁 = 𝐁 𝐁𝐂 = 𝐁𝐃 ⇏ 𝐂 = 𝐃 ▪ example: 𝐁 = 1 2 3 −4 6 , 𝐂 = , and 𝐃 = 3 −2 3 1 4 −1 −1 ▪ 𝐁𝐂 = 𝐁𝐃 = −1 2 −3 6 18

MATRIX MULTIPLICATION What about powers of a matrix? Let us define 𝐁 2 = 𝐁𝐁 for any square matrix 𝐁 ▪ why square? mind the difference Likewise 𝐁 3 = 𝐁𝐁𝐁 , etc. between “a square matrix” and “a squared matrix” ▪ what is 𝐁𝐁𝐁 : is it 𝐁 𝐁𝐁 or is it 𝐁𝐁 𝐁 ? 𝐁 𝑜 = 1 More in general 𝐁 𝑜 = ቊ 𝐁𝐁 𝑜−1 𝑜 = 2,3, … And what do you think of 𝐁 0 ? 19

EXERCISE 4 Given is 𝐁 = 2 −1 and 𝐂 = 0 2 −3 . Find 𝐁 𝟑 ⋅ 𝐂 . 3 0 1 20

MORE OPERATIONS WITH MATRICES Moral 1: every operation must be explicitly defined Moral 2: mathematicians try to find a definition that ▪ is useful (why 𝐁𝐂 is useful will become clear later on) ▪ reduces to the similar operation for scalars Not all scalar operations have an extension to matrices Example: ▪ ln 𝐁 1 ▪ We will soon introduce a sort 𝐁 of division by a matrix: ▪ 𝐁 matrix inversion 22

MATRIX TRANSPOSITION 𝑏 1,1 𝑏 1,2 ⋯ 𝑏 1,𝑜 𝑏 2,1 𝑏 2,2 ⋯ 𝑏 2,𝑜 Consider 𝐁 = 𝐁 𝑛×𝑜 = ⋯ ⋯ ⋯ ⋯ 𝑏 𝑛,1 𝑏 𝑛,2 ⋯ 𝑏 𝑛,𝑜 The transpose of 𝐁 , denoted by 𝐁 ′ is given by 𝑏 1,1 𝑏 2,1 ⋯ 𝑏 𝑛,1 𝑏 1,2 𝑏 2,2 ⋯ 𝑏 𝑛,2 𝐁 ′ = ⋯ ⋯ ⋯ ⋯ “reflection in the 𝑏 1,𝑜 𝑏 2,𝑜 ⋯ 𝑏 𝑛,𝑜 diagonal” In words, 𝐁 ′ has 𝑜 rows and 𝑛 columns so 𝐁 ′ is a ( 𝑜 × 𝑛 )- matrix and row 𝑗 of 𝐁 is column 𝑗 of 𝐁 ′ 23

MATRIX TRANSPOSITION Some properties (for suitable 𝐁 , 𝐂 , and 𝐃 ): 𝐁 ′ ′ = 𝐁 ▪ 𝐁 + 𝐂 ′ = 𝐁 ′ + 𝐂 ′ ▪ 𝐁𝐂 ′ = 𝐂 ′ 𝐁 ′ and (therefore!) 𝐁𝐂𝐃 ′ = 𝐃 ′ 𝐂 ′ 𝐁 ′ ▪ 𝑑𝐁 ′ = 𝑑𝐁 ′ ▪ 24

SYMMETRIC MATRICES Definition The matrix 𝐁 is symmetric if and only if 𝐁 = 𝐁 ′ ▪ so if and only if 𝑏 𝑗𝑘 = 𝑏 𝑘𝑗 for all 𝑗 , 𝑘 You can see this for the ▪ note: only a square matrix can be symmetric example without even Example: 𝐁 = 1 3 doing a calculation! 6 is symmetric 3 If 𝐁 = 1 2 −1 then 𝐁𝐁 ′ is symmetric 3 6 5 ▪ note: 𝐁 ′ 𝐁 is symmetric too but in general 𝐁 ′ 𝐁 ≠ 𝐁𝐁 ′ In general 𝐁 ′ 𝐁 is symmetric for an arbitrary matrix 𝐁 ▪ and so is 𝐁𝐁 ′ (why?) 25

EXERCISE 5 Given is 𝐘 = 4 −2 5 −1 . Find 𝐘 ′ . 0 3 26

OLD EXAM QUESTION 22 October 2014, Q1d 28

OLD EXAM QUESTION 10 December 2014, Q1f 29

FURTHER STUDY Sydsæter et al. 5/E 9.2-9.3 Tutorial exercises week 3 matrices matrix addition, matrix multiplication, matrix transpose matrix multiplication is not commutative 30