A matrix over a field F is a rectangular array of elements from F. The symbol


 Cecily Bradford
 1 years ago
 Views:
Transcription
1
2 Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted by capital letters and the equation A [a ij means that the element in the i th row and j th column of the matrix A equals a ij It is also occasionally convenient to write a ij (A) ij For the present, all matrices will have rational entries, unless otherwise stated EXAMPLE The formula a ij /(i + j) for i, j 4 defines a 4 matrix A [a ij, namely A DEFINITION (Equality of matrices) Matrices A, B are said to be equal if A and B have the same size and corresponding elements are equal; ie, A and B M m n (F) and A [a ij, B [b ij, with a ij b ij for i m, j n DEFINITION (Addition of matrices) Let A [a ij and B [b ij be of the same size Then A + B is the matrix obtained by adding corresponding elements of A and B; that is A + B [a ij + [b ij [a ij + b ij
3 4 CHAPTER MATRICES DEFINITION (Scalar multiple of a matrix) Let A [a ij and t F (that is t is a scalar) Then ta is the matrix obtained by multiplying all elements of A by t; that is ta t[a ij [ta ij DEFINITION 4 (Additive inverse of a matrix) Let A [a ij Then A is the matrix obtained by replacing the elements of A by their additive inverses; that is A [a ij [ a ij DEFINITION 5 (Subtraction of matrices) Matrix subtraction is defined for two matrices A [a ij and B [b ij of the same size, in the usual way; that is A B [a ij [b ij [a ij b ij DEFINITION 6 (The zero matrix) For each m, n the matrix in M m n (F), all of whose elements are zero, is called the zero matrix (of size m n) and is denoted by the symbol 0 The matrix operations of addition, scalar multiplication, additive inverse and subtraction satisfy the usual laws of arithmetic (In what follows, s and t will be arbitrary scalars and A, B, C are matrices of the same size) (A + B) + C A + (B + C); A + B B + A; 0 + A A; 4 A + ( A) 0; 5 (s + t)a sa + ta, (s t)a sa ta; 6 t(a + B) ta + tb, t(a B) ta tb; 7 s(ta) (st)a; 8 A A, 0A 0, ( )A A; 9 ta 0 t 0 or A 0 Other similar properties will be used when needed
4 MATRIX ARITHMETIC 5 DEFINITION 7 (Matrix product) Let A [a ij be a matrix of size m n and B [b jk be a matrix of size n p; (that is the number of columns of A equals the number of rows of B) Then AB is the m p matrix C [c ik whose (i, k) th element is defined by the formula n c ik a ij b jk a i b k + + a in b nk EXAMPLE j [ [ [ [ [ [ [ [ [ ; [ [ [ 4 4 ; [ 4 [ [ ; [ [ [ Matrix multiplication obeys many of the familiar laws of arithmetic apart from the commutative law (AB)C A(BC) if A, B, C are m n, n p, p q, respectively; t(ab) (ta)b A(tB), A( B) ( A)B (AB); (A + B)C AC + BC if A and B are m n and C is n p; 4 D(A + B) DA + DB if A and B are m n and D is p m We prove the associative law only: First observe that (AB)C and A(BC) are both of size m q Let A [a ij, B [b jk, C [c kl Then p p n ((AB)C) il (AB) ik c kl a ij b jk c kl k p k j n a ij b jk c kl k j ;
5 6 CHAPTER MATRICES Similarly (A(BC)) il n j k p a ij b jk c kl However the double summations are equal For sums of the form n p p n and j k d jk k j represent the sum of the np elements of the rectangular array [d jk, by rows and by columns, respectively Consequently ((AB)C) il (A(BC)) il for i m, l q Hence (AB)C A(BC) d jk The system of m linear equations in n unknowns a x + a x + + a n x n b a x + a x + + a n x n b a m x + a m x + + a mn x n b m is equivalent to a single matrix equation a a a n x a a a n x a m a m a mn x n that is AX B, where A [a ij is the coefficient matrix of the system, x b x b X is the vector of unknowns and B is the vector of x n constants Another useful matrix equation equivalent to the above system of linear equations is x a a a m + x a a a m + + x n a n a n a mn b b b m b m, b b b m
6 LINEAR TRANSFORMATIONS 7 EXAMPLE The system is equivalent to the matrix equation and to the equation x [ [ [ + y x + y + z x y + z 0 x y z [ + z Linear transformations [ 0 [ 0 An n dimensional column vector is an n matrix over F The collection of all n dimensional column vectors is denoted by F n Every matrix is associated with an important type of function called a linear transformation DEFINITION (Linear transformation) We can associate with A M m n (F), the function T A : F n F m, defined by T A (X) AX for all X F n More explicitly, using components, the above function takes the form y y a x + a x + + a n x n a x + a x + + a n x n y m a m x + a m x + + a mn x n, where y, y,,y m are the components of the column vector T A (X) The function just defined has the property that T A (sx + ty ) st A (X) + tt A (Y ) () for all s, t F and all n dimensional column vectors X, Y For T A (sx + ty ) A(sX + ty ) s(ax) + t(ay ) st A (X) + tt A (Y )
7 8 CHAPTER MATRICES REMARK It is easy to prove that if T : F n F m is a function satisfying equation, then T T A, where A is the m n matrix whose columns are T(E ),, T(E n ), respectively, where E,,E n are the n dimensional unit vectors defined by E 0 0,, E n One well known example of a linear transformation arises from rotating the (x, y) plane in dimensional Euclidean space, anticlockwise through θ radians Here a point (x, y) will be transformed into the point (x, y ), where x xcos θ y sinθ y xsinθ + y cos θ In dimensional Euclidean space, the equations 0 0 x xcos θ y sinθ, y xsinθ + y cos θ, z z; x x, y y cos φ z sinφ, z y sinφ + z cos φ; x xcos ψ + z sinψ, y y, z xsinψ + z cos ψ; correspond to rotations about the positive z, x and y axes, anticlockwise through θ, φ, ψ radians, respectively The product of two matrices is related to the product of the corresponding linear transformations: If A is m n and B is n p, then the function T A T B : F p F m, obtained by first performing T B, then T A is in fact equal to the linear transformation T AB For if X F p, we have T A T B (X) A(BX) (AB)X T AB (X) The following example is useful for producing rotations in dimensional animated design (See [7, pages 97 ) EXAMPLE The linear transformation resulting from successively rotating dimensional space about the positive z, x, y axes, anticlockwise through θ, φ, ψ radians respectively, is equal to T ABC, where
8 LINEAR TRANSFORMATIONS (x, y) 9 l (x, y ) θ Figure : Reflection in a line C A cos θ sinθ 0 sinθ cos θ cos ψ 0 sinψ 0 0 sinψ 0 cos ψ, B The matrix ABC is quite complicated: A(BC) cos ψ 0 sin ψ 0 0 sinψ 0 cos ψ cos φ sinφ 0 sinφ cos φ cos θ sinθ 0 cos φsinθ cos φcos θ sinφ sin φsinθ sin φcos θ cos φ cos ψ cos θ+sin ψ sin φ sin θ cos ψ sin θ+sin ψ sin φcos θ sin ψ cos φ cos φsin θ cos φcos θ sin φ sin ψ cos θ+cos ψ sin φsin θ sin ψ sin θ+cos ψ sin φcos θ cos ψ cos φ EXAMPLE Another example from geometry is reflection of the plane in a line l inclined at an angle θ to the positive x axis We reduce the problem to the simpler case θ 0, where the equations of transformation are x x, y y First rotate the plane clockwise through θ radians, thereby taking l into the x axis; next reflect the plane in the x axis; then rotate the plane anticlockwise through θ radians, thereby restoring l to its original position
9 0 (x, y) CHAPTER MATRICES l (x, y ) θ Figure : Projection on a line In terms of matrices, we get transformation equations [ [ [ cos θ sinθ 0 cos ( θ) sin( θ) y sinθ cos θ 0 sin( θ) cos ( θ) [ [ [ cos θ sinθ cos θ sinθ x sinθ cos θ sinθ cos θ y [ [ cos θ sinθ x sinθ cosθ y [ x [ x y The more general transformation [ [ x cos θ sinθ a sinθ cos θ y [ x y [ u + v, a > 0, represents a rotation, followed by a scaling and then by a translation Such transformations are important in computer graphics See [, 4 EXAMPLE Our last example of a geometrical linear transformation arises from projecting the plane onto a line l through the origin, inclined at angle θ to the positive x axis Again we reduce that problem to the simpler case where l is the x axis and the equations of transformation are x x, y 0 In terms of matrices, we get transformation equations [ x y [ cos θ sinθ sinθ cos θ [ [ cos ( θ) sin( θ) sin ( θ) cos ( θ) [ x y
10 RECURRENCE RELATIONS [ [ [ cos θ 0 cos θ sinθ x sinθ 0 sinθ cos θ y [ cos [ θ cos θ sin θ x sin θ cos θ sin θ y Recurrence relations DEFINITION (The identity matrix) The n n matrix I n [δ ij, defined by δ ij if i j, δ ij 0 if i j, is called the n n identity matrix of order n In other words, the columns of the identity matrix of order n are the unit vectors E,,E n, respectively For example, I [ 0 0 THEOREM If A is m n, then I m A A AI n DEFINITION (k th power of a matrix) If A is an n n matrix, we define A k recursively as follows: A 0 I n and A k+ A k A for k 0 For example A A 0 A I n A A and hence A A A AA The usual index laws hold provided AB BA: A m A n A m+n, (A m ) n A mn ; (AB) n A n B n ; A m B n B n A m ; 4 (A + B) A + AB + B ; 5 (A + B) n n i0 ( n i ) A i B n i ; 6 (A + B)(A B) A B We now state a basic property of the natural numbers AXIOM (MATHEMATICAL INDUCTION) If P n denotes a mathematical statement for each n, satisfying (i) P is true,
11 CHAPTER MATRICES (ii) the truth of P n implies that of P n+ for each n, then P n is true for all n [ EXAMPLE Let A [ A n + 6n 4n 9n 6n Prove that if n Solution We use the principle of mathematical induction Take P n to be the statement [ A n + 6n 4n 9n 6n Then P asserts that A [ [ which is true Now let n and assume that P n is true We have to deduce that [ [ A n+ + 6(n + ) 4(n + ) 7 + 6n 4n + 4 9(n + ) 6(n + ) 9n 9 5 6n Now, A n+ A [ n A [ + 6n 4n 7 4 9n 6n 9 5 [ ( + 6n)7 + (4n)( 9) ( + 6n)4 + (4n)( 5) ( 9n)7 + ( 6n)( 9) ( 9n)4 + ( 6n)( 5) [ 7 + 6n 4n + 4, 9n 9 5 6n and the induction goes through The last example has an application to the solution of a system of recurrence relations:
12 4 PROBLEMS EXAMPLE The following system of recurrence relations holds for all n 0: x n+ 7x n + 4y n y n+ 9x n 5y n Solve the system for x n and y n in terms of x 0 and y 0 Solution Combine the above equations into a single matrix equation [ [ [ xn+ 7 4 xn, y n+ 9 5 y n [ [ 7 4 xn or X n+ AX n, where A and X 9 5 n y n We see that X AX 0 X AX A(AX 0 ) A X 0 X n A n X 0 (The truth of the equation X n A n X 0 for n, strictly speaking follows by mathematical induction; however for simple cases such as the above, it is customary to omit the strict proof and supply instead a few lines of motivation for the inductive statement) Hence the previous example gives [ [ [ xn + 6n 4n x0 X y n n 9n 6n y 0 [ ( + 6n)x0 + (4n)y 0, ( 9n)x 0 + ( 6n)y 0 and hence x n (+6n)x 0 +4ny 0 and y n ( 9n)x 0 +( 6n)y 0, for n 4 PROBLEMS Let A, B, C, D be matrices defined by A 0, B 5 0 4,
13 4 CHAPTER MATRICES C 4, D [ 4 0 Which of the following matrices are defined? Compute those matrices which are defined A + B, A + C, AB, BA, CD, DC, D [Answers: A + C, BA, CD, D ; 0 5 4, , 4 0 4, [ Let A then [ 0 0 Show that if B is a such that AB I, a b B a b a + b for suitable numbers a and b Use the associative law to show that (BA) B B If A [ a b c d, prove that A (a + d)a + (ad bc)i 0 [ 4 4 If A, use the fact A 0 4A I and mathematical induction, to prove that A n (n ) A + n I if n 5 A sequence of numbers x, x,,x n, satisfies the recurrence relation x n+ ax n +bx n for n, where a and b are constants Prove that [ xn+ x n [ xn A x n,
14 4 PROBLEMS 5 [ [ [ a b xn+ x where A and hence express in terms of 0 x n x 0 If a 4 and b, use the previous question to find a formula for x n in terms of x and x 0 [Answer: [ a a 6 Let A 0 x n n x + n x 0 (a) Prove that [ A n (n + )a n na n+ na n ( n)a n if n (b) A sequence x 0, x,,x n, satisfies x n+ ax n a x n for n Use part (a) and the previous question to prove that x n na n x + ( n)a n x 0 for n [ a b 7 Let A and suppose that λ c d and λ are the roots of the quadratic polynomial x (a+d)x+ad bc (λ and λ may be equal) Let k n be defined by k 0 0, k and for n k n n i λ n i λ i Prove that k n+ (λ + λ )k n λ λ k n, if n Also prove that { (λ n k n λ n )/(λ λ ) if λ λ, nλ n if λ λ Use mathematical induction to prove that if n, A n k n A λ λ k n I, [Hint: Use the equation A (a + d)a (ad bc)i
15 6 CHAPTER MATRICES 8 Use Question 7 to prove that if A if n A n n [ [ + ( )n, then [ 9 The Fibonacci numbers are defined by the equations F 0 0, F and F n+ F n + F n if n Prove that if n 0 F n 5 (( + ) n ( 5 ) n ) 5 0 Let r > be an integer Let a and b be arbitrary positive integers Sequences x n and y n of positive integers are defined in terms of a and b by the recurrence relations x n+ x n + ry n y n+ x n + y n, for n 0, where x 0 a and y 0 b Use Question 7 to prove that x n y n r as n 5 Non singular matrices DEFINITION 5 (Non singular matrix) A matrix A M n n (F) is called non singular or invertible if there exists a matrix B M n n (F) such that AB I n BA Any matrix B with the above property is called an inverse of A If A does not have an inverse, A is called singular THEOREM 5 (Inverses are unique) If A has inverses B and C, then B C
16 5 NON SINGULAR MATRICES 7 Proof Let B and C be inverses of A Then AB I n BA and AC I n CA Then B(AC) BI n B and (BA)C I n C C Hence because B(AC) (BA)C, we deduce that B C REMARK 5 If A has an inverse, it is denoted by A So AA I n A A Also if A is non singular, it follows that A is also non singular and (A ) A THEOREM 5 If A and B are non singular matrices of the same size, then so is AB Moreover Proof Similarly (AB) B A (AB)(B A ) A(BB )A AI n A AA I n (B A )(AB) I n REMARK 5 The above result generalizes to a product of m non singular matrices: If A,,A m are non singular n n matrices, then the product A A m is also non singular Moreover (A A m ) A m A (Thus the inverse of the product equals the product of the inverses in the reverse order) EXAMPLE 5 If A and B are n n matrices satisfying A B (AB) I n, prove that AB BA Solution Assume A B (AB) I n Then A, B, AB are non singular and A A, B B, (AB) AB But (AB) B A and hence AB BA [ [ a b EXAMPLE 5 A is singular For suppose B 4 8 c d is an inverse of A Then the equation AB I gives [ [ [ a b c d 0
17 8 CHAPTER MATRICES and equating the corresponding elements of column of both sides gives the system which is clearly inconsistent THEOREM 5 Let A non singular Also a + c 4a + 8c 0 [ a b c d A [ and ad bc 0 Then A is d b c a REMARK 5 The expression ad bc is called the determinant of A and is denoted by the symbols deta or a b c d Proof Verify that the matrix B [ AB I BA d b c a satisfies the equation EXAMPLE 5 Let A Verify that A 5I, deduce that A is non singular and find A Solution After verifying that A 5I, we notice that ( ) ( ) A 5 A I 5 A A Hence A is non singular and A 5 A THEOREM 54 If the coefficient matrix A of a system of n equations in n unknowns is non singular, then the system AX B has the unique solution X A B Proof Assume that A exists
18 5 NON SINGULAR MATRICES 9 (Uniqueness) Assume that AX B Then (A A)X A B, (Existence) Let X A B Then I n X A B, X A B AX A(A B) (AA )B I n B B THEOREM 55 (Cramer s rule for equations in unknowns) The system ax + by e cx + dy f has a unique solution if a b c d 0, namely x, y, where e b f d and a e c f [ a b Proof Suppose 0 Then A has inverse c d [ A d b c a and we know that the system A [ x y [ e f has the unique solution [ [ x A e [ y f [ d b c a de bf ce + af [ e f [ [ / / Hence x /, y /
19 40 CHAPTER MATRICES COROLLARY 5 The homogeneous system ax + by 0 cx + dy 0 has only the trivial solution if a b c d 0 EXAMPLE 54 The system 7x + 8y 00 x 9y 0 has the unique solution x /, y /, where , , So x and y 0 79 THEOREM 56 Let A be a square matrix If A is non singular, the homogeneous system AX 0 has only the trivial solution Equivalently, if the homogenous system AX 0 has a non trivial solution, then A is singular Proof If A is non singular and AX 0, then X A 0 0 REMARK 54 If A,,A n denote the columns of A, then the equation AX x A + + x n A n holds Consequently theorem 56 tells us that if there exist x,,x n, not all zero, such that x A + + x n A n 0, that is, if the columns of A are linearly dependent, then A is singular An equivalent way of saying that the columns of A are linearly dependent is that one of the columns of A is expressible as a sum of certain scalar multiples of the remaining columns of A; that is one column is a linear combination of the remaining columns
20 5 NON SINGULAR MATRICES 4 EXAMPLE 55 A is singular For it can be verified that A has reduced row echelon form and consequently AX 0 has a non trivial solution x, y, z REMARK 55 More generally, if A is row equivalent to a matrix containing a zero row, then A is singular For then the homogeneous system AX 0 has a non trivial solution An important class of non singular matrices is that of the elementary row matrices DEFINITION 5 (Elementary row matrices) To each of the three types of elementary row operation, there corresponds an elementary row matrix, denoted by E ij, E i (t), E ij (t): E ij, (i j) is obtained from the identity matrix I n by interchanging rows i and j E i (t), (t 0) is obtained by multiplying the i th row of I n by t E ij (t), (i j) is obtained from I n by adding t times the j th row of I n to the i th row EXAMPLE 56 (n ) 0 0 E 0 0, E ( ) , E ( ) The elementary row matrices have the following distinguishing property: THEOREM 57 If a matrix A is pre multiplied by an elementary row matrix, the resulting matrix is the one obtained by performing the corresponding elementary row operation on A
21 4 CHAPTER MATRICES EXAMPLE 57 a b E c d e f a b c d e f a b e f c d COROLLARY 5 Elementary row matrices are non singular Indeed E ij E ij ; E i (t) E i (t ); (E ij (t)) E ij ( t) Proof Taking A I n in the above theorem, we deduce the following equations: E ij E ij I n E i (t)e i (t ) I n E i (t )E i (t) if t 0 E ij (t)e ij ( t) I n E ij ( t)e ij (t) EXAMPLE 58 Find the matrix A E (5)E ()E explicitly Also find A Solution A E (5)E () To find A, we have E (5) A (E (5)E ()E ) E (E ()) (E (5)) E E ( )E (5 ) E E ( ) E
22 5 NON SINGULAR MATRICES 4 REMARK 56 Recall that A and B are row equivalent if B is obtained from A by a sequence of elementary row operations If E,,E r are the respective corresponding elementary row matrices, then B E r ( (E (E A))) (E r E )A PA, where P E r E is non singular Conversely if B PA, where P is non singular, then A is row equivalent to B For as we shall now see, P is in fact a product of elementary row matrices THEOREM 58 Let A be non singular n n matrix Then (i) A is row equivalent to I n, (ii) A is a product of elementary row matrices Proof Assume that A is non singular and let B be the reduced row echelon form of A Then B has no zero rows, for otherwise the equation AX 0 would have a non trivial solution Consequently B I n It follows that there exist elementary row matrices E,,E r such that E r ( (E A)) B I n and hence A E Er, a product of elementary row matrices THEOREM 59 Let A be n n and suppose that A is row equivalent to I n Then A is non singular and A can be found by performing the same sequence of elementary row operations on I n as were used to convert A to I n Proof Suppose that E r E A I n In other words BA I n, where B E r E is non singular Then B (BA) B I n and so A B, which is non singular Also A ( B ) B Er (( (E I n )), which shows that A is obtained from I n by performing the same sequence of elementary row operations as were used to convert A to I n REMARK 57 It follows from theorem 59 that if A is singular, then A is row equivalent to a matrix whose last row is zero [ EXAMPLE 59 Show that A is non singular, find A and express A as a product of elementary row matrices
23 44 CHAPTER MATRICES Solution We form the partitioned matrix [A I which consists of A followed by I Then any sequence of elementary row operations which reduces A to I will reduce I to A Here [ 0 [A I 0 Hence R R R [ 0 0 R ( )R [ 0 0 R R R [ 0 0 Hence A is row equivalent to I and A is non singular Also [ A We also observe that E ( )E ( )E ( )A I A E ( )E ( )E ( ) A E ()E ( )E () The next result is the converse of Theorem 56 and is useful for proving the non singularity of certain types of matrices THEOREM 50 Let A be an n n matrix with the property that the homogeneous system AX 0 has only the trivial solution Then A is non singular Equivalently, if A is singular, then the homogeneous system AX 0 has a non trivial solution Proof If A is n n and the homogeneous system AX 0 has only the trivial solution, then it follows that the reduced row echelon form B of A cannot have zero rows and must therefore be I n Hence A is non singular COROLLARY 5 Suppose that A and B are n n and AB I n Then BA I n
24 5 NON SINGULAR MATRICES 45 Proof Let AB I n, where A and B are n n We first show that B is non singular Assume BX 0 Then A(BX) A0 0, so (AB)X 0, I n X 0 and hence X 0 Then from AB I n we deduce (AB)B I n B and hence A B The equation BB I n then gives BA I n Before we give the next example of the above criterion for nonsingularity, we introduce an important matrix operation DEFINITION 5 (The transpose of a matrix) Let A be an m n matrix Then A t, the transpose of A, is the matrix obtained by interchanging the rows and columns of A In other words if A [a ij, then ( A t) ji a ij Consequently A t is n m The transpose operation has the following properties: ( A t) t A; (A ± B) t A t ± B t if A and B are m n; (sa) t sa t if s is a scalar; 4 (AB) t B t A t if A is m n and B is n p; 5 If A is non singular, then A t is also non singular and ( A t ) ( A ) t ; 6 X t X x + + x n if X [x,,x n t is a column vector We prove only the fourth property First check that both (AB) t and B t A t have the same size (p m) Moreover, corresponding elements of both matrices are equal For if A [a ij and B [b jk, we have ( (AB) t ) ki (AB) ik n a ij b jk j n ( B t ) ( kj A t ) ji j ( B t A t) ki There are two important classes of matrices that can be defined concisely in terms of the transpose operation
25 46 CHAPTER MATRICES DEFINITION 54 (Symmetric matrix) A matrix A is symmetric if A t A In other words A is square (n n say) and a ji a ij for all i n, j n Hence A is a general symmetric matrix [ a b b c DEFINITION 55 (Skew symmetric matrix) A matrix A is called skew symmetric if A t A In other words A is square (n n say) and a ji a ij for all i n, j n REMARK 58 Taking i j in the definition of skew symmetric matrix gives a ii a ii and so a ii 0 Hence [ A 0 b b 0 is a general skew symmetric matrix We can now state a second application of the above criterion for non singularity COROLLARY 54 Let B be an n n skew symmetric matrix Then A I n B is non singular Proof Let A I n B, where B t B By Theorem 50 it suffices to show that AX 0 implies X 0 We have (I n B)X 0, so X BX Hence X t X X t BX Taking transposes of both sides gives (X t BX) t (X t X) t X t B t (X t ) t X t (X t ) t X t ( B)X X t X X t BX X t X X t BX Hence X t X X t X and X t X 0 But if X [x,,x n t, then X t X x + + x n 0 and hence x 0,,x n 0
26 6 LEAST SQUARES SOLUTION OF EQUATIONS 47 6 Least squares solution of equations Suppose AX B represents a system of linear equations with real coefficients which may be inconsistent, because of the possibility of experimental errors in determining A or B For example, the system x y x + y 00 is inconsistent It can be proved that the associated system A t AX A t B is always consistent and that any solution of this system minimizes the sum r + + rm, where r,,r m (the residuals) are defined by r i a i x + + a in x n b i, for i,,m The equations represented by A t AX A t B are called the normal equations corresponding to the system AX B and any solution of the system of normal equations is called a least squares solution of the original system EXAMPLE 6 Find a least squares solution of the above inconsistent system 0 [ Solution Here A x 0, X, B y 00 Then A t A Also A t B [ 0 0 [ 0 0 So the normal equations are which have the unique solution [ [ x + y 400 x + y 500 x 00, y 600
27 48 CHAPTER MATRICES EXAMPLE 6 Points (x, y ),,(x n, y n ) are experimentally determined and should lie on a line y mx + c Find a least squares solution to the problem Solution The points have to satisfy mx + c y mx n + c y n, or Ax B, where A x x n, X [ m c, B y y n The normal equations are given by (A t A)X A t B Here [ x [ A t x x A n x + + x n x + + x n x + + x n n x n Also A t B [ x x n y y n [ x y + + x n y n y + + y n It is not difficult to prove that det (A t A) i<j n (x i x j ), which is positive unless x x n Hence if not all of x,,x n are equal, A t A is non singular and the normal equations have a unique solution This can be shown to be m i<j n (x i x j )(y i y j ), c i<j n REMARK 6 The matrix A t A is symmetric (x i y j x j y i )(x i x j )
28 7 PROBLEMS 49 7 PROBLEMS [ 4 Let A Prove that A is non singular, find A and express A as a product of elementary row matrices [ [Answer: A 4 A E ( )E ()E (4) is one such decomposition, A square matrix D [d ij is called diagonal if d ij 0 for i j (That is the off diagonal elements are zero) Prove that pre multiplication of a matrix A by a diagonal matrix D results in matrix DA whose rows are the rows of A multiplied by the respective diagonal elements of D State and prove a similar result for post multiplication by a diagonal matrix Let diag (a,,a n ) denote the diagonal matrix whose diagonal elements d ii are a,,a n, respectively Show that diag (a,,a n )diag (b,,b n ) diag (a b,,a n b n ) and deduce that if a a n 0, then diag (a,,a n ) is non singular and (diag (a,,a n )) diag (a,,a n ) Also prove that diag (a,,a n ) is singular if a i 0 for some i 0 0 Let A 6 Prove that A is non singular, find A and 7 9 express A as a product of elementary row matrices 7 [Answers: A 9, 0 0 A E E ()E E ()E ()E (4)E ( 9) is one such decomposition
29 50 CHAPTER MATRICES 4 Find the rational number k for which the matrix A is singular [Answer: k k 5 5 [ 5 Prove that A is singular and find a non singular matrix 4 P such that PA has last row zero [ 4 6 If A, verify that A A + I 0 and deduce that A (A I ) 7 Let A 0 0 (i) Verify that A A A + I (ii) Express A 4 in terms of A, A and I and hence calculate A 4 explicitly (iii) Use (i) to prove that A is non singular and find A explicitly [Answers: (ii) A 4 6A 8A + I (iii) A A A + I (i) Let B be an n n matrix such that B 0 If A I n B, prove that A is non singular and A I n + B + B Show that the system of linear equations AX b has the solution (ii) If B 0 r s 0 0 t (I B) explicitly X b + Bb + B b, verify that B 0 and use (i) to determine ;
30 7 PROBLEMS 5 [Answer: r s + rt 0 t Let A be n n (i) If A 0, prove that A is singular (ii) If A A and A I n, prove that A is singular 0 Use Question 7 to solve the system of equations x + y z a z b x + y + z c where a, b, c are given rationals Check your answer using the Gauss Jordan algorithm [Answer: x a b + c, y a + 4b c, z b Determine explicitly the following products of elementary row matrices (i) E E (ii) E (5)E (iii) E ()E ( ) (iv) (E (00)) (v) E (vi) (E (7)) (vii) (E (7)E ()) [Answers: (i) (iv) 4 / (v) 5 (ii) (vi) 5 (iii) (vii) Let A be the following product of 4 4 elementary row matrices: Find A and A explicitly [Answers: A A E ()E 4 E 4 () 7 5, A /
31 5 CHAPTER MATRICES Determine which of the following matrices over Z are non singular and find the inverse, where possible (a) (b) [Answer: (a) Determine which of the following matrices are non singular and find the inverse, where possible (a) (b) (c) (d) [Answers: (a) (e) (e) / 0 / / (b) 7 5 (f) / 0 0 / 5 5 (d) 4 / / /7 5 5 Let A be a non singular n n matrix Prove that A t is non singular and that (A t ) (A ) t 6 Prove that A [ a b c d has no inverse if ad bc 0 [Hint: Use the equation A (a + d)a + (ad bc)i 0 7 Prove that the real matrix A 5 is non singular by proving that A is row equivalent to I 4 a b a c b c 8 If P AP B, prove that P A n P B n for n
32 7 PROBLEMS 5 [ [ / /4 9 Let A, P / /4 [ 5/ 0 and deduce that 0 [ Let A [ a b c d A n ( ) 5 n [ 7 Verify that P AP 4 4 be a Markov matrix; that is a matrix whose elements [ b c are non negative and satisfy a+c b+d Also let P Prove that if A I then [ 0 (i) P is non singular and P AP 0 a + d (ii) A n [ [ b b 0 as n, if A b + c c c 0 If X [Answers: and Y , 4 [ ,, find XX t, X t X, Y Y t, Y t Y, Prove that the system of linear equations x + y 4 x + y 5 x + 5y , 6 is inconsistent and find a least squares solution of the system [Answer: x 6, y 7/6 The points (0, 0), (, 0), (, ), (, 4), (4, 8) are required to lie on a parabola y a + bx + cx Find a least squares solution for a, b, c Also prove that no parabola passes through these points [Answer: a 5, b, c
33 54 CHAPTER MATRICES 4 If A is a symmetric n n real matrix and B is n m, prove that B t AB is a symmetric m m matrix 5 If A is m n and B is n m, prove that AB is singular if m > n 6 Let A and B be n n If A or B is singular, prove that AB is also singular
NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that
NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationMATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More informationChapter 4: Binary Operations and Relations
c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationChapter 8. Matrices II: inverses. 8.1 What is an inverse?
Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationMatrix Inverse and Determinants
DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationDeterminants. Dr. Doreen De Leon Math 152, Fall 2015
Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationLecture Notes 1: Matrix Algebra Part B: Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationHomework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67
Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More information9 Matrices, determinants, inverse matrix, Cramer s Rule
AAC  Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:
More informationAlgebra and Linear Algebra
Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...
More informationMatrices: 2.3 The Inverse of Matrices
September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations.
More informationEC9A0: Presessional Advanced Mathematics Course
University of Warwick, EC9A0: Presessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Presessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond
More informationMATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationEIGENVALUES AND EIGENVECTORS
Chapter 6 EIGENVALUES AND EIGENVECTORS 61 Motivation We motivate the chapter on eigenvalues b discussing the equation ax + hx + b = c, where not all of a, h, b are zero The expression ax + hx + b is called
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationUndergraduate Matrix Theory. Linear Algebra
Undergraduate Matrix Theory and Linear Algebra a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn John S Alin Linfield College Colin L Starr Willamette University December 15, 2015 ii Contents 1 SYSTEMS OF LINEAR
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More information4.9 Markov matrices. DEFINITION 4.3 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if. (i) a ij 0 for 1 i, j n;
49 Markov matrices DEFINITION 43 A real n n matrix A = [a ij ] is called a Markov matrix, or row stochastic matrix if (i) a ij 0 for 1 i, j n; (ii) a ij = 1 for 1 i n Remark: (ii) is equivalent to AJ n
More informationChapter 1  Matrices & Determinants
Chapter 1  Matrices & Determinants Arthur Cayley (August 16, 1821  January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationUsing determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:
Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible
More informationMatrices, transposes, and inverses
Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrixvector multiplication: two views st perspective: A x is linear combination of columns of A 2 4
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationIntroduction to Matrix Algebra I
Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationMatrices and Systems of Linear Equations
Chapter Matrices and Systems of Linear Equations In Chapter we discuss how to solve a system of linear equations. If there are not too many equations or unknowns our task is not very difficult; what we
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
More informationL12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014
L12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.  Morpheus Primary concepts:
More informationMath 2331 Linear Algebra
2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More informationInterpolating Polynomials Handout March 7, 2012
Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More information( % . This matrix consists of $ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&
Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 08 4 01 1 0 11
More informationGRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics
GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic
More informationLecture 10: Invertible matrices. Finding the inverse of a matrix
Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices
More informationNotes on Linear Recurrence Sequences
Notes on Linear Recurrence Sequences April 8, 005 As far as preparing for the final exam, I only hold you responsible for knowing sections,,, 6 and 7 Definitions and Basic Examples An example of a linear
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationMatrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.
2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true
More informationQuaternions and isometries
6 Quaternions and isometries 6.1 Isometries of Euclidean space Our first objective is to understand isometries of R 3. Definition 6.1.1 A map f : R 3 R 3 is an isometry if it preserves distances; that
More informationEconomics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics
Economics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics Michael Best Spring 2015 Outline The Evaluation Problem Matrix Algebra Matrix Calculus OLS With Matrices The Evaluation Problem:
More informationSolutions to Review Problems
Chapter 1 Solutions to Review Problems Chapter 1 Exercise 42 Which of the following equations are not linear and why: (a x 2 1 + 3x 2 2x 3 = 5. (b x 1 + x 1 x 2 + 2x 3 = 1. (c x 1 + 2 x 2 + x 3 = 5. (a
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationDEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x
Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationElementary Row Operations and Matrix Multiplication
Contents 1 Elementary Row Operations and Matrix Multiplication 1.1 Theorem (Row Operations using Matrix Multiplication) 2 Inverses of Elementary Row Operation Matrices 2.1 Theorem (Inverses of Elementary
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More information= [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are
This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In
More informationMath 313 Lecture #10 2.2: The Inverse of a Matrix
Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationProblem Set 1 Solutions Math 109
Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twentyfour symmetries if reflections and products of reflections are allowed. Identify a symmetry which is
More information2.5 Gaussian Elimination
page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3
More information2Matrix Algebra. Matrix Addition, Scalar Multiplication, and Transposition
Matrix Algebra Section. Matrix Addition, Scalar Multiplication, and Transposition In the study of systems of linear equations in Chapter, we found it convenient to manipulate the augmented matrix of the
More informationMatrix Algebra and Applications
Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2  Matrices and Matrix Algebra Reading 1 Chapters
More informationSTRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2
STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of xaxis in anticlockwise direction, then the value of tan θ is called the slope
More informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 4. Gaussian Elimination In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor
More information5.3 Determinants and Cramer s Rule
290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given
More information8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
More informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More information0.1 Linear Transformations
.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called
More informationAPPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.
APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj
More information1 Quiz on Linear Equations with Answers. This is Theorem II.3.1. There are two statements of this theorem in the text.
1 Quiz on Linear Equations with Answers (a) State the defining equations for linear transformations. (i) L(u + v) = L(u) + L(v), vectors u and v. (ii) L(Av) = AL(v), vectors v and numbers A. or combine
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and
More informationPURE MATHEMATICS AM 27
AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationMath 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns
Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in
More informationCalculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants
Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwthaachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview
More information