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Linear Algebra

by: Maggie Casper

Linear Algebra MATH 206

Maggie Casper

GPA 3.86


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This 8 page Class Notes was uploaded by Maggie Casper on Thursday October 29, 2015. The Class Notes belongs to MATH 206 at Wellesley College taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/230919/math-206-wellesley-college in Mathematics (M) at Wellesley College.

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Date Created: 10/29/15
Fall Semester 07 08 Akila Weempana LECTURE 6 MATRIX ALGEBRA I INTRODUCTION 0 The rst few lectures in the class reviewed material that you had already covered even though the applications may have been new Starting with this lecture we start covering new material at least for some of you drawn from linear algebra and differential equations Since Math 206 is at least a co requisite for the course you should already have started covering basic matrix operations such as addition multiplication inversion etc so we will move quickly through that section If you have taken Math 206 already this material will be extremely basic Today s class will cover the most basic features of linear algebra what matrices are elemen tary operations such as transposition addition multiplication and determinants of matrices We will also de ne the inverse of a matrix and how to calculate it II WHAT IS A MATRIX Matrix 0 Most mathematics textbooks de ne a matrix as a rectangular array77 containing numbers or variables Although hard to describe in words a matrix can be visualized extremely easily The following are all matrices 45 m 4568 961902 l1357l l l 143 9 3 4 A matrix is typically described in terms of its dimensions An m x n pronounced m by n matrix is a matrix with m x 71 entries represented in m rows and n columns So the 5 matrices above are 2 x 4 3 x 2 2 x 2 1 x 3 and 3 x 1 matrices respectively The entries of a matrix are usually referenced according to their location within the matrix So in general if we have an m x 71 matrix known as X we will refer to the element in row i and column j of X as Xij The elements that lie on the diagonal line between the rst and the last entries ie those elements for which i j are known as the diagonal elements and the others as the offdiagonal elements 0 A matrix in which the number of rows equals the number of columns is called a square matrix Identity Matrix 0 The identity matrix denoted I is a special matrix It is a square matrix with with each diagonal entry being 1 and each non diagonal entry being 0 c We use the notation In to refer to the dimension ofthe matrix since number of rows number of columns in a square matrix we only need 1 number to know its dimensions Example Symmetric Matrix 0 A symmetric matrix is another special matrix A matrix A is said to be symmetric if AH A Vij In other words when we transpose the rows and columns of A the resulting matrix denoted A is equivalent to A 0 Given the de nition we can see that a matrix that is symmetric must also be square The converse is not true not all square matrices are symmetric Example 1 0 3 1 0 3 A 0 7 5 is symmetric since A 0 7 5 3 5 8 3 5 8 0 7 i i i 0 3 but B7 3 5 J is NOT symmetric s1nce B 7 7 5 Vector o A matrix with a single row or column is sometimes referred to as a vector A matrix with a single column is called a column vector and a matrix with a single row is called a row vector A 1 x 1 matrix ie a matrix with only 1 entry is called a scalar III BASIC MATRIX OPERATIONS Transposition o The transpose of an m x a matrix is an n x m matrix constructed by switching rows and columns of the original matrix That is to say elements that were on the rst column of the original matrix now make up the rst row of the transposed matrix The transpose of a given matrix X is denoted as X Example o The transposes of the 5 matrices given in the above section are 4 1 5 3 4 1 5 m1 3 65 13ml mail 4 742 8 7 c We can now simplify the de nition of a symmetric matrix A as A is symmetric iff A A Matrix Addition 0 Matrix addition is a simple operation The result of adding two matrices together is a third matrix Whose elements in any given row and column are obtained by adding elements in that same row and column in the rst matrix to the elements in the same row and column of the second matrix 0 As a result matrix addition is legal only when performed with two matrices of identical dimensions ie a 5 x 6 matrix can only be added to another 5 x 6 matrix Example 4 1 5 1 1 2 i 5 2 7 1 3 7 0 3 75 T 1 6 2 0 Matrix subtraction is very similar The result of A B where A and B are both m x n matrices is another m x 71 matrix C where Cij aij 7 bij Example 4 1 5 1 1 2 7 3 0 3 1 3 7 0 3 75 T 1 0 12 0 Rules of Addition AB BA ABC ABC Scalar Multiplication c Any matrix can be multiplied by a scalar If A is an m x 71 matrix and q is a scalar the matrix that results from multiplying A by the scalar q is a m x 71 matrix C where each element is obtained by multiplying the corresponding element A by q ie Cij q x A 5 415 7 20525 137 51535 Matrix Multiplication 0 Matrix multiplication is by far the most complicated of the basic operations When we multiply matrix A by matrix B denoted AB to obtain matrix C a given element is obtained by multiplying the ith row of A by the jth column of B Therefore multiplication of two matrices is legal only when the ith row of the rst matrix has the same number of elements as the jth column of the second matrix in other words the number of columns in A has to equal the number of rows in B o More generally an m x 71 matrix A can be multiplied by an n x p matrix B where p is any positive integer The resulting matrix C is of dimension m x p with elements de ned as follows CH 221 Alk X B1 for i 1 m j 1p A simple example illustrates the procedure much more clearly than the notation Suppose we have a 3 x 2 matrix A and a 2 x 3 matrix B Note that according to the rules of multiplication we can do either AB which will be a 3 x 3 matrix or BA which will be a 2 x 2 matrix in this case Note If B was a 2 x 5 matrix for example only AB would be de ned BA would not be de ned since the number of columns in B 5 is greater than the number of rows in A 3 Example 1 1 4 1 5 Suppose A 1 3 and B Then 2 1 3 7 75 11 5412 AB 13 1710 26 275 3713725 11 415 15718 BA 1371 35 18725i 0 Rules of Multiplication where A is m x n D is n x p and E is p x q AB 7A BA ADE ADE A1 A ImA A AD DA 0 Rules of Vector Multiplication If x is an n x 1 column vector then a z m is a scalar equal 221 mil 2 11 129021 1nn1 2 1 1 1 1 1 90219612 9622 zznmnz b mm is a symmetric matrix of dimension n mm V L n11n n22n 717 Determinant o The determinant of a square matrix is a unique number associated with that matrix The determinant of a matrix is usually denoted by writing the name of the matrix surrounded by two vertical bars eg the determinant of A is How is the determinant of a matrix calculated For a 1 x 1 matrix a 2 x 2 matrix or a 3 x 3 matrix the process is fairly simple For higher dimension matrices the calculation is more involved 0 The determinant of a 1 x 1 matrix a scalar is the scalar itself 0 The determinant of a 2 x 2 matrix of the form A Z I i d J is lAl 7 ad7bc a b c o The determinant of a 3 x 3 matrix of the form A d e f is g h i lAl aei bfg cdh 7 659 bdi afh Although this seems hard to remember there is a simple trick that enables us to calculate the determinant of a 3 x 3 matrix The trick requires you to write down two copies of the matrix side by side then add the products of the three leftmost diagonals and subtract the products of the three rightmost diagonals o The determinant of an nx 71 matrix is found using a procedure known as a Laplace expansion The Laplace expansion is a recursive process ie the determinant of an n x 71 matrix is expressed as a function of determinants of several 71 7 1 x n 7 1 matrices each of which in turn are expressed as functions of several 71 7 2 x n 7 2 matrices etc Since we know how to calculate the determinant of a 3 x 3 matrix or a 2 x 2 matrix we can always build back up to the top 0 In order to write down the Laplace expansion formula we need to de ne a few other terms The minor of a matrix is the determinant of a submatrix formed from a given matrix In particular the ijth minor of a matrix denoted as is the determinant of a matrix obtained by omitting the ith row and jth column of the original matrix 2 7 0 1 i i i l 5 6 4 8 l i i i 0 Consider the follow1ng matrix A 0 0 g 0 Some of the minors of this matrix are l 1 73 1 4 6 4 8 2 7 1 7 0 1 anl 0 9 0 432 mm 0 0 0 OandlM41l 6 4 8 7450 73 1 4 1 73 4 0 9 0 The cofactor of a matrix is another determinant of a submatrix formed from the original The ijth cofactor of a matrix is de ned as 10171 71 7 lMijl that is to say the co factor is similar to the minor except that it may have a different sign If the sum of i and j is an even number then the sign is positive if the sum is an odd number the sign is negative So 10111 1411117 10121 1411217 10231 1412317 10241 1M24l etc c We are nally ready to write down the formula for the Laplace expansion of a matrix A If A is an n x 71 matrix then 71 lAl Zaileijlfor anyi1n j1 o In other words we can nd the determinant of a matrix by picking any row then multiplying each element of the row by the appropriate cofactor 0 Which should you choose Well its always easier to pick a row with a lot of zeros since that minimizes the number of calculations one has to do So in our case the best thing to do would be to use the expansion of the 3rd row lAl 0l031l0l032l9l033l0l034l 9 033 9 M33 2 7 I 9 5 6 8 I 73 4 lAl 781 c As you can see this is a very tedious calculation even for a 4 x 4 matrix Fortunately we will rarely have to calculate determinants by hand using computers for numerical matrices and other analytical short cuts for algebraic matrices Nevertheless you should have some idea about how determinants are calculated using co factors 0 Properties of the Determinant There are several important properties that relate to determinants 7 The determinant of A is equal to the determinant of its transpose lAl lA l 7 The determinant of a diagonal matrix ie one whose off diagonal elements are ALL zero is equal to the product of the element of the diagonals 7 A is invertible iff lAl 7 0 7 If any row of a matrix is a linear combination of one or more of the other rows of the matrix then the matrix has a determinant of zero The same holds true for columns So matrices with linearly dependent rows or columns are not invertible 7 If any row of a matrix consists only of zeros then the matrix has a determinant of zero IV CALCULATING THE INVERSE OF A MATRIX c Any square matrix that has an inverse is said to be a nonsingular matrix A matrix that has no inverse is said to be a singular matrix The inverse of a n dimensional square matrix A is denoted A l if it exists The inverse matrix has the property that AA 1 In and A lA In 0 If a matrix has an inverse matrix then the matrix must be a square matrix ie only square matrices have inverses This does not mean that ALL square matrices have inverses it just means that matrices that are not square do not have an inverse matrix 37 757 717 47 Consider the matrix A 11L 3 J 7 the inverse of this matrix is A 1 since 1 0 71 7 71 7 7 AA 7A A7A701 Properties of the Inverse 7 If A 1 is the inverse of a matrix A7 then A lr1 A 7 If A 1 is the inverse of a matrix A7 then A 1 A 1 7 If A and B are both square matrices with inverse matrices A 1 and B l Then the matrix C AB has an inverse de ned as Cquot1 Calculating the inverse of a matrix is a chore First7 we have to de ne something called the adjoint of the matrix 0 The adjoint of an n x 71 matrix A is another 71 x 71 matrix de ned as 10111 10121 39 10ml adjA C where C C l Cizz 11011 1021 10411 0 In other words the adjoint of an n x 71 matrix A is the transpose of an n x 71 matrix C whose ijth element is the ijth cofactor of A Given the adjoint of an n x 71 matrix A matrix A that is invertilole7 we can then calculate the inverse as 1 A 1 iadjm lAl Example 7 6 2 1 Suppose A 5 8 We can calculate lAl 79 so we know that the inverse exists 1 73 4 o The matrix of cofactors of the matrix A can be calculated as 6 8 58 5 6 10111 73 4 487101217 14 712710131 173 721 7 1 21 2 7 10211 7 73 4 73110221 14 7102317 173 13 71 21 27 10311 6 8 50103217 5 8 71110331 5 6 723 c We can then write down the adjoint of A as the transpose of the above matrix of cofactors 48 712 721 48 731 50 134 731 7 13 712 7 711 50 711723 721 13 723 0 Therefore 1 1 48 731 50 7489 319 7509 A 1jadjAig 712 7 711 129 779 119 l 721 13 723 219 7139 239 0 I ll leave it up to you verify that this is right In general calculating the inverse of a 3 x 3 matrix is a lot of work and a 4 x 4 matrix is a monumental task to invert Fortunately7 you will rarely have to this by hand There are plenty of excellent software packages like Mathematica and Matlab7 that will calculate the determinant for you7 even if the matrix consists of symbolic expressions instead of numbers


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