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

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BUSINESS MATHEMATICS

Matrices

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

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MATRICES A matrix is a rectangular array of numbers or variables Notation ▪ we often use bold non-italic capital letters to refer to them

▪ 𝐑 = 3 2 −2 12.5 −12.7 , 𝐁 = 𝑏1,1 𝑏1,2 ⋯ 𝑏1,𝑜 𝑏2,1 𝑏2,2 ⋯ 𝑏2,𝑜 ⋯ ⋯ ⋯ ⋯ 𝑏𝑛,1 𝑏𝑛,2 ⋯ 𝑏𝑛,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

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MATRICES Some notes on notation ▪ brackets:

• r

▪ indexing elements: 𝑏𝑗𝑘 or 𝑏𝑗,𝑘 ▪ We define 𝐁 = 𝑏𝑗𝑘 𝑛×𝑜 as the matrix of order 𝑛 × 𝑜 matrix with elements 𝑏𝑗𝑘, 𝑗 = 1, … 𝑛, 𝑘 = 1, … , 𝑜: 𝐁 = 𝑏11 𝑏12 ⋯ 𝑏1𝑜 𝑏21 𝑏22 ⋯ 𝑏2𝑜 ⋯ ⋯ ⋯ ⋯ 𝑏𝑛1 𝑏𝑛2 ⋯ 𝑏𝑛𝑜

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MATRICES In 𝐑 = 3 2 −2 12.5 −12.7 the element 𝑟2,1 refers to ▪ the cell at row 2 and column 1 ▪ so to −2 ▪ while 𝑟1,2 is in row 1 and column 2 and has value 2 The order of 𝐑 is 3 × 2, not 2 × 3 Conventions: ▪ element 𝑏row index,column index ▪ order 𝑛row × 𝑜column ▪ when no ambiguity you write 𝑏𝑗𝑘 instead of 𝑏𝑗,𝑘

Notice the order

• f the indices

A column is vertical ...

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EXERCISE 1 Given is 𝐚 = 𝑨𝑗𝑘 = 1 −3 5 4 2 . Find 𝑨1,2 + 𝑨2,2.

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

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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 = 𝑏𝑗𝑘 + 𝑐𝑗𝑘) ▪ multiplication (𝑑𝐁, through 𝑑𝑏 𝑗𝑘 = 𝑑 × 𝑏𝑗𝑘) ▪ negative matrix (−𝐁, through −𝑏 𝑗𝑘 = −𝑏𝑗𝑘) ▪ subtraction (𝐁 − 𝐂, through 𝑏 − 𝑐 𝑗𝑘 = 𝑏𝑗𝑘 − 𝑐𝑗𝑘) ▪ equality (𝐁 = 𝐂, through 𝑏𝑗𝑘 = 𝑐𝑗𝑘) But what about the inner product? ▪ not available for matrices ▪ instead: matrix multiplication

The matrices 𝐁 and 𝐂 must be of equal order

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

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MATRIX MULTIPLICATION Let 𝐁 and 𝐂 be two matrices, of order 𝑛 × 𝑞 respectively 𝑞 × 𝑜 We define the matrix product 𝐁𝐂 as 𝐁𝐂 𝑗𝑘 = ෍

𝑙=1 𝑞

𝑏𝑗𝑙𝑐𝑙𝑘 , 𝑗 = 1, … , 𝑛, 𝑘 = 1, … , 𝑜 ▪ 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

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MATRIX MULTIPLICATION Illustration:

▪ 𝐁𝐂 1,2 = σ𝑙=1

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𝑏1,𝑙𝑐𝑙,2 = 𝑏1,1𝑐1,2 + 𝑏1,2𝑐2,2 ▪ 𝐁𝐂 3,3 = σ𝑙=1

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𝑏3,𝑙𝑐𝑙,3 = 𝑏3,1𝑐1,3 + 𝑏3,2𝑐2,3

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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 ?

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

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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 3 6 and 𝐂 = 2 −8 −1 4 ▪ 𝐁𝐂 = 0 0 , but 𝐂𝐁 = −22 −44 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

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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 6 , 𝐂 = 3 −4 −2 3 , and 𝐃 = 1 4 −1 −1 ▪ 𝐁𝐂 = 𝐁𝐃 = −1 2 −3 6

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MATRIX MULTIPLICATION What about powers of a matrix? Let us define 𝐁2 = 𝐁𝐁 for any square matrix 𝐁 ▪ why square? Likewise 𝐁3 = 𝐁𝐁𝐁, etc. ▪ what is 𝐁𝐁𝐁: is it 𝐁 𝐁𝐁 or is it 𝐁𝐁 𝐁? More in general 𝐁𝑜 = ቊ 𝐁 𝑜 = 1 𝐁𝐁𝑜−1 𝑜 = 2,3, … And what do you think of 𝐁0?

mind the difference between “a square matrix” and “a squared matrix”

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EXERCISE 4 Given is 𝐁 = 2 −1 3 and 𝐂 = 0 2 1 −3 . Find 𝐁𝟑 ⋅ 𝐂.

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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 𝐁

▪ 𝐁

We will soon introduce a sort

• f division by a matrix:

matrix inversion

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MATRIX TRANSPOSITION Consider 𝐁 = 𝐁𝑛×𝑜 = 𝑏1,1 𝑏1,2 ⋯ 𝑏1,𝑜 𝑏2,1 𝑏2,2 ⋯ 𝑏2,𝑜 ⋯ ⋯ ⋯ ⋯ 𝑏𝑛,1 𝑏𝑛,2 ⋯ 𝑏𝑛,𝑜 The transpose of 𝐁, denoted by 𝐁′ is given by 𝐁′ = 𝑏1,1 𝑏2,1 ⋯ 𝑏𝑛,1 𝑏1,2 𝑏2,2 ⋯ 𝑏𝑛,2 ⋯ ⋯ ⋯ ⋯ 𝑏1,𝑜 𝑏2,𝑜 ⋯ 𝑏𝑛,𝑜 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′

“reflection in the diagonal”

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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′

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SYMMETRIC MATRICES Definition The matrix 𝐁 is symmetric if and only if 𝐁 = 𝐁′ ▪ so if and only if 𝑏𝑗𝑘 = 𝑏𝑘𝑗 for all 𝑗, 𝑘 ▪ note: only a square matrix can be symmetric Example: 𝐁 = 1 3 3 6 is symmetric If 𝐁 = 1 2 −1 3 6 5 then 𝐁𝐁′ is symmetric ▪ 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?)

You can see this for the example without even doing a calculation!

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

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OLD EXAM QUESTION 22 October 2014, Q1d

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OLD EXAM QUESTION 10 December 2014, Q1f

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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