Chapter 5. Determinants. Po-Ning Chen, Professor. Department of Electrical and Computer Engineering. National Chiao Tung University
|
|
- Grace Dalton
- 7 years ago
- Views:
Transcription
1 Chapter 5 Determinants Po-Ning Chen, Professor Department of Electrical and Computer Engineering National Chiao Tung University Hsin Chu, Taiwan 30010, R.O.C.
2 5.1 The properties of determinants 5-1 Recall for Gauss-Jordan method and square A, Ax = b has a unique solution if all pivots are non-zero. Note: A set of numbers are non-zeros if, and only if, their product is non-zero! Ax = b has a unique solution if, and only if, the product of all pivots is non-zero. Hence, the product of all pivots is an important decisive factor for a linear equation Ax = b (or for a matrix A). Definition (Determinant): The determinant of a square matrix A, denoted by det(a), is equal to the product of all pivots, multiplying by ( 1) m, where m is the number of row exchanges preformed during the derivation of these pivots. Note: det(a) is sometimes denoted by A (This notation is identical to take the absolute value ); however, it needs to be pointed out that the latter notation may be a little misleading because it is possible that det(a) = A < 0.
3 5.1 The properties of determinants 5-2 Alternative calculations of det(a): Pivotformula: det(a) = product of all pivots (if no row exchanges are performed in the process of producing pivots) Leibniz formula: det(a) = n sgn(σ) σ P n i=1 a i,σ(i) where P n is the set of all permutation of (1, 2,...,n), and 1 if σ can be recovered to (1, 2,...,n)byevennumber sgn(σ) = of pair-wise switching 1 otherwise Example. det(a 3 3 ) = ( 1) 0 a 1,1 a 2,2 a 3,3 + ( 1) 1 a 1,1 a 2,3 a 3,2 + ( 1) 1 a 1,2 a 2,1 a 3,3 + ( 1) 2 a 1,2 a 2,3 a 3,1 + ( 1) 2 a 1,3 a 2,1 a 3,2 + ( 1) 1 a 1,3 a 2,2 a 3,1
4 5.1 The properties of determinants 5-3 Leibniz formula (continued): Each product in the Leibniz formula takes exactly one entry from each row (analogously, one entry from each column). So, there are n! products in the Leibniz formula. (There are n selections from the 1st row, (n 1) selections from the 2nd row, (n 2) selections from the 3rd row,...,etc.) Co-factor formula: n det(a) = a l,j C l,j where the co-factor C l,j is equal to ( 1) l+j multiplying the determinant of the matrix that results from A by removing the l-th row and the j-th column. j=1 The proofs of equivalences of these formulas will be given later. We will base on the Pivot formula (by its definition) to show ten fundamental properties of determinant.
5 5.1 Properties of determinant 5-4 Ten properties listed in the textbook (first only proved for row operations; their validity for column operations will be given in Property 10) 1. Unit determinant for identity matrix. (Hint: By Pivot formula or by definition.) 2. Sign reversal by row/column switch. (Hint: By Pivot formula.) By Pivot formula, a row switch matrix à of A simply resultes in different permutation matrix P to satisfy PA = P à = LU. The property follows since P gives one more row exchange during the LU decomposition.
6 5.1 Properties of determinant Linearity with respect to any row/column. (Hint: By Pivot formula.) ([ ]) ([ ]) t a1,1 t a Multiplication: det 1,2 a1,1 a = t det 1,2 a 2,1 a 2,2 a 2,1 a 2,2 Addition: ([ ]) ([ ]) ([ ]) a1,1 + a det 1,1 a 1,2 + a 1,2 a1,1 a = det 1,2 a +det 1,1 a 1,2 a 2,1 a 2,2 a 2,1 a 2,2 a 2,1 a 2,2 Note: det(t A) =t n det(a) t det(a) fort 1. As an additional note, [ ] [ ] [ ] a1,1 a t 1,2 t a1,1 t a = 1,2 t a1,1 t a 1,2. a 2,1 a 2,2 t a 2,1 t a 2,2 a 2,1 a 2,2 Recall that when the rows of A are edges of a box, the absolute value of det(a) gives the volume of the box! So, extending each edge by t-fold will increase the volume by t n -fold.
7 5.1 Properties of determinant Two equivalent rows/columns make zero determinant. Can be proved by Property 2 since det(a) = det(a) implies det(a) =0. 5. Subtracting one row (column) by a multiple of the other row (column) makes no change on determinant. This property can be proved by Properties 3 and 4. ([ ]) ([ ]) ([ ]) a1,1 + t a det 2,1 a 1,2 + t a 2,2 a1,1 a = det 2,2 a2,1 a +t det 2,2 a 2,1 a 2,2 a 1,2 a 2,2 a 2,1 a 2,2 }{{} =0 So the Guass-Jordan method (or more specifically, forward elimination) will not change the determinant. 6. Zero rows/columns give zero determinant. This property can be proved by Property 3. ([ ]) ([ ]) 0 a1,1 0 a det 1,2 a1,1 a =0 det 1,2 a 2,1 a 2,2 a 2,1 a 2,2
8 5.1 Properties of determinant Determinant of a triangular matrix is the product of diagonal terms. This property holds by Pivot formula or by definition. Also, from Property 5, we know that the forward elimination can be used to determine the determinant as n det(a) =det(u) = u i,i. Note: If there are odd number of row exchanges performed during forward elimination, then det(a) = det(u). i=1
9 5.1 Properties of determinant det(a) 0 if, and only if, A is invertible. This property follows from det(a) = det(u), i.e., from the discussion in Property 7. Equivalently, we can state that det(a) = 0 if, and only if, A is singular (or non-invertible).
10 5.1 Properties of determinant det(ab) =det(a) det(b) Proof: We first show that det(eb)=det(e) det(b) for any elementary matrix E, where an elementary matrix is one that performs 1) row-pair combination (i.e., new row j =oldrowj c j,i row i), 2) multiplying one row (i.e., new row j = c j,j old row j), and 3) row switching. As a result, the E that makes valid det(eb) =det(e) det(b) canbe obtained from the identity matrix I by 1) replacing (j, i)th element by c j,i ; (Property 5) 2) replacing (j, j)th element by c j,j ; (Property 3: Multiplication with t = c j,j ) 3) switching the respective two rows. (Property 2) Now suppose one of det(a) anddet(b) is zero. So, by Property 8, one of A and B is not invertible, so is AB; thus,det(ab) =det(a)det(b) =0.
11 5.1 Properties of determinant 5-10 Proof (cont): Now suppose both det(a) anddet(b) are not zero. Then, by Gauss- Jordan method (or rref), we can respectively decompose A and B into A = E 1 E 2 E k and B = E 1E 2 E k. (Exercise: What is rref(a) ifdet(a) 0?) As a result, det(ab) = det(e 1 E 2 E k E 1E 2 E k ) = det(e 1 )det(e 2 ) det(e k )det(e 1)det(E 2) det(e k ) = det(e 1 E 2 E k ) det(e 1 E 2 E k ) = det(a) det(b). Note: The formal proof above is exactly what has stated in the textbook (that gives the idea behind the proof in an informal way).
12 5.1 Properties of determinant det(a) =det(a T ) This property can be proved by A = E 1 E 2 E k and det(e j )=det(ej T ) for every j.
13 5.1 A powerful tool: Elementary matrix 5-12 The elementary matrix E can be obtained from the identity matrix I by 1) replacing (j, i)th element by c j,i ; In such case, det(e) = 1. (Proved by Property 5) 2) replacing (j, j)th element by c j,j ; In such case, det(e) =c j,j. (Proved by Property 3: Multiplication with t = c j,j ) 3) switching the respective two rows. In such case, det(e) = 1. (Proved by Property 2) So, det(a) =det(e 1 )det(e 2 ) det(e k )=(+1) #1 ( 1) #3 n j=1 c j,j
14 5.1 Eigenvalues 5-13 Problems 22 and 23: (Problem 22, Section 5.1) From ad bc, find the determinant of A and A 1 and A λi: [ ] 2 1 A = and A 1 = 1 [ ] [ ] λ 1 and A λi = λ Which two numbers λ lead to det(a λi) = 0? Write down the matrix A λi for each of those numbers λ it should not be invertible. [ ] 4 1 (Problem 23, Section 5.1) From A = find A 2 and A 1 and A λi and 2 3 their determinants. Which two numbers λ lead to det(a λi) =0? The eigenvalues of a matrix A is the solutions λ of det(a λi) =0. This is important because for these λ s, there exists a non-zero vector v such that Av = λv. Equivalently, there exists a non-zero vector satisfying (A λi)v =0. Vector v is usually referred to as the eigenvector associated with eigenvalue λ.
15 5.1 Co-factor 5-14 Problem 30 (also, Problem 27 in Section 5.2): (Problem 30, Section 5.1) (Calculus question) Show that the partial derivatives of ln(deta) givea 1! [ ] f/ a f/ c f(a, b, c, d) =ln(ad bc) leads to. f/ b f/ d (Problem 27, Section 5.2) (A calculus question) Show that the derivative of deta with respect to a 1,1 is the cofactor C 1,1. The other entries are fixed we are only changing a 1,1. Let s look at the clue of the answers of these two problems first. We will come back to give a formal proof about cofactor matrix can be used to provide the inverse of A later in Section 5.2.
16 5.1 Co-factor 5-15 Can the Co-factor matrix be used to provide the inverse of a matrix A? Direct answer: Yes. ln det(a) ln det(a) a C 1,1 C 2,1 C 1,1 a 2,1 n,1 A 1 1 ln det(a) ln det(a) C = 1,2 C 2,2 C n,2 det(a) = a 1,2 a 2, C 1,n C 2,n C n,n ln det(a) a 2,n ln det(a) a 1,n ln det(a) a n,1 ln det(a) a n,2 ln det(a) a n,n where the first equality (blue-colored part) is directly given without proof, and the second equality follows from the co-factor formula below. Co-factor formula: det(a) = n a i,j C i,j where the co-factor C i,j is equal to ( 1) i+j multiplying the determinant of the matrix that results from A by removing the i-th row and the j-th column. j=1
17 5.1 Co-factor 5-16 Problem 11 in Section 5.2: (Problem 11, Section 5.2) Find all cofactors and put them into cofactor matrices C, D. FindAC AC T and detb. [ ] a b A = B = c d Puzzle asked in the solution: detb = 21 and detd =( 21) 2. Why? See the next page. C 1,1 C 1,n C =..... is called the cofactor matrix of A. C n,1 C n,n C T is called the adjugate matrix of A, denoted by adj(a). So, A 1 1 = det(a) CT. Hence in Problem 11, AC T = det(a) I but AC = det(a) A(A 1 ) T. The solution given on page 535 of the textbook for C should be C T.
18 5.2 Inverse and co-factors 5-17 Lemma: Suppose A is invertible. Then, AC T = det(a)i. Proof: By following the co-factor formula, the diagonal terms of AC T are all det(a). When inner-producting the jth row (a j,1,a j,2,...,a j,n ) with the co-factors of ith row (C i,1,c i,2,...,c i,n ), where i j, we can replace the ith row by the j-th row to form a new matrix A,andthen,det(A )=0sincetworows are equal. Then the co-factors of i-th row of A are exactly the same as the co-factors of i-th row of A,andthei-th row of A is (a j,1,a j,2,...,a j,n ). Hence, det(a ) = a i,1c i,1 + a i,2c i,2 + + a i,nc i,n = a j,1 C i,1 + a j,2 C i,2 + + a j,n C i,n =0. For example, when n = 3, the lemma says a 1,1 a 1,2 a 1,3 C 1,1 C 2,1 C 3,1 det(a) 0 0 a 2,1 a 2,2 a 2,3 C 1,2 C 2,2 C 3,2 = 0 det(a) 0 a 3,1 a 3,2 a 3,3 C 1,3 C 2,3 C 3,3 0 0 det(a) This answers the puzzle in Problem 11, Section 5.2.
19 5.2 Pivots and submatrices 5-18 An interesting property based on the pivot formula: Define A 1 = [ a 1,1 ], Ak = a 1,1 a 1,k....., A n = a k,1 a k,k a 1,1 a 1,k a 1,n a k,1 a k,k a n,1 a n,n Suppose the LU decomposition of A n gives pivots from forward elimination without row exchanges: d 1 u 1,2 u 1,3 u 1,n l 2, d 2 u 2,3 d 2,n A n = l 3,1 l 3, d 3 d 3,n l n,1 l n,2 l n, d n Then, det(a k )=d 1 d 2 d k. Idea behind the proof: Pivot d k only depends on those elements in A k.
20 det 5.2 Leibniz formula revisited 5-19 In Section 5.1, we define and prove ten properties of determinants based on the Pivot formula. We now show that the Pivot formula is equivalent to the Leibniz formula by these properties. det(a) = n sgn(σ) a i,σ(i). σ P n By Property 3, a 1,1 a 1,2 a 1,3 det a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 a 1, a 1, a 1,3 = det a 2,1 a 2,2 a 2,3 +det a 2,1 a 2,2 a 2,3 + det a 2,1 a 2,2 a 2,3 } a 3,1 a 3,2 {{ a 3,3 } a 3,1 a 3,2 a 3,3 a 3,1 a 3,2 a 3,3 ([ ]) a 1,1 0 0 a1,1 0 0 a 2, det 0 a 2,2 0 +det a 1, a 2,3 a 3,1 a 3,2 a 3,3 a 3,1 a 3,2 a 3,3 a 3,1 a 3,2 a 3,3 }{{} ([ ]) a 1,1 0 0 a1,1 0 0 det 0 0 a 2,3 +det +det a 3, a 2,3 0 a 3,2 0 i=1 a 1, a 2,3 0 0 a 3,3
21 5.2 Leibniz formula revisited 5-20 = a 1, a 1, a 1,3 = det 0 a 2,2 0 + det 0 0 a 2,3 + det a 2, a 3,3 a 3, a 3,2 0 a 1, a 1, a 1,3 + det 0 0 a 2,3 + det a 2, det 0 a 2,2 0 0 a 3, a 3,3 a 3, = a 1,1 a 2,2 a 3,3 det a 1,2 a 2,3 a 3,1 det a 1,3 a 2,1 a 3,2 det a 1,1 a 2,3 a 3,2 det a 1,2 a 2,1 a 3,3 det a 1,3 a 2,2 a 3,1 det = a 1,1 a 2,2 a 3,3 + a 1,2 a 2,3 a 3,1 + a 1,3 a 2,1 a 3,2 a 1,1 a 2,3 a 3,2 a 1,2 a 2,1 a 3,3 a 1,3 a 2,2 a 3,1
22 5.2 Leibniz formula revisited 5-21 The usefulness of the Leibniz formula can be seen in the below example. (So, it is called the big formula in the textbook.) Example. Find the determinant of A = Solution. The Leibniz formula needs to choose exactly one element from each row. The elements chosen from the remaining rows cannot be located at the same columns as the ones that have been chosen (because the Leibniz formula also needs to choose exactly one element from each column). So, the only nonzero elements at the 1st and 4th rows need to be chosen (otherwise the product will be zero); so, det(a) =det 0 1 =sgn(2, 1, 4, 3) a 1,2a 2,1 a 3,4 a 4,3 =
23 5.2 Co-factor formula revisited 5-22 The co-factor formula can be proved by the Leibniz formula. Co-factor formula: det(a) = n a l,j C l,j where the co-factor C l,j is equal to ( 1) l+j multiplying the determinant of the matrix that results from A by removing the l-th row and the j-th column. det(a 3 3 ) = ( 1) 0 a 1,1 a 2,2 a 3,3 + ( 1) 1 a 1,1 a 2,3 a 3,2 + ( 1) 1 a 1,2 a 2,1 a 3,3 j=1 +( 1) 2 a 1,2 a 2,3 a 3,1 + ( 1) 2 a 1,3 a 2,1 a 3,2 + ( 1) 1 a 1,3 a 2,2 a 3,1 = sgn(1, 2, 3)a 1,1 a 2,2 a 3,3 + sgn(1, 3, 2)a 1,1 a 2,3 a 3,2 + sgn(2, 1, 3)a 1,2 a 2,1 a 3,3 +sgn(2, 3, 1)a 1,2 a 2,3 a 3,1 + sgn(3, 1, 2)a 1,3 a 2,1 a 3,2 + sgn(3, 2, 1)a 1,3 a 2,2 a 3,1 = a 2,1 (sgn(2, 1, 3)a 1,2 a 3,3 + sgn(3, 1, 2)a 1,3 a 3,2 ) +a 2,2 (sgn(1, 2, 3)a 1,1 a 3,3 + sgn(3, 2, 1)a 1,3 a 3,1 ) +a 2,3 (sgn(1, 3, 2)a 1,1 a 3,2 + sgn(2, 3, 1)a 1,2 a 3,1 ) = a 2,1 ( 1) 2+1 (sgn(2, 3)a 1,2 a 3,3 + sgn(3, 2)a 1,3 a 3,2 ) +a 2,2 ( 1) 2+2 (sgn(1, 3)a 1,1 a 3,3 + sgn(3, 1)a 1,3 a 3,1 ) +a 2,3 ( 1) 2+3 (sgn(1, 2)a 1,1 a 3,2 + sgn(2, 1)a 1,2 a 3,1 ) = a 2,1 C 2,1 + a 2,2 C 2,2 + a 2,3 C 2,3
24 5.2 Co-factor formula revisited 5-23 det(a) = σ P n sgn(σ(1),σ(2),,σ(n)) = = n j=1 a l,j σ P n :σ(l)=j n a l,j ( 1) l+j j=1 n i=1 a i,σ(i) n sgn(σ(1),,σ(l),,σ(n)) a }{{} i,σ(i) =j i=1,i l sgn(σ(1),,σ(l 1),σ(l +1),,σ(n)) σ P n :σ(l)=j n i=1,i l a i,σ(i) } {{ } C l,j Example. Find the determinant of A = 0 a 1,2 0 0 a 2,1 0 a 2,3 0 0 a 3,2 0 a 3, a 4,3 0
25 5.2 Co-factor formula revisited 5-24 Solution. 0 a 1,2 0 0 a det(a) = det 2,1 0 a 2,3 0 0 a 3,2 0 a 3,4 0 0 a 4,3 0 a 2,1 a 2,3 0 a 2,1 a 2,3 0 = a 1,2 ( 1) 1+2 det 0 0 a 3,4 = a 1,2 det 0 0 a 3,4 0 a 4,3 0 0 a 4,3 0 ([ ]) a2,1 a = a 1,2 a 3,4 det 2,3 = a 0 a 1,2 a 3,4 a 2,1 a 4,3 4,3 Correction: Problem 14 : The matrices in Problem 15 (should be Problem 13). Problem 18 : Go back to B n to Problem 14 (should be Problem 17).
26 5.2 Co-factor formula revisited 5-25 (Problem 13, Section 5.2) The n by n determinant C n has 1 s above and below the main diagonal: C 1 = 0 C 2 = C 3 = C 4 = (a) What are these determinants C 1, C 2, C 3, C 4?Hint:ReduceC 4 to C 2. (b) By cofactors find the relation between C n and C n 1 and C n 2.FindC 10. (Problem 14, Section 5.2) The matrices in Problem have 1 s just above and below the main diagonal. Going down the matrix, which order of columns (if any) gives all 1 s? Explain why that permutation is even for n =4, 8, 12,... and odd for = 2, 6, 10,...Then C n =0(oddn) C n =1(n =4, 8,...) C n = 1 (n =2, 6,...). Hint: Permute the red-color 1 s above. They give the only non-zero term in the Leibniz formula.
27 5.2 Co-factor formula revisited 5-26 (Problem 17, Section 5.2) The matrix B n is the 1, 2, 1 matrix A n except that b 1,1 =1insteadofa 1,1 = 2. Using cofactors of the last row of B 4 show that B 4 =2 B 3 B 2 =1. B 4 = B 3 = B 2 = [ 1 ] The recursion B n =2 B n 1 B n 2 is satisfied when every B n =1. This recursion is the same as for the A s in Example 6. The difference is in the starting values 1, 1, 1 for the determinants of sizes n =1, 2, 3. B n = A n 1 0 = (Problem 18, Section 5.2) Go back to B n in Problem It is the same as A n except for b 1,1 = 1. So use linearity in the first row, where [ ] equals [ ] minus [ ] : A n 1 A n Linearity gives B n = A n A n 1. Hint: B n =1and A n = n +1.
28 5.3 Crammer s rules, inverses, and volumns 5-27 Now we know how to compute A 1 by the co-factors when A is a square matrix and det(a) 0. C 1,1 C 2,1 C n,1 A 1 1 C = 1,2 C 2,2 C n,2 det(a) C 1,n C 2,n C n,n WecanthengobacktosolveAx = b by this inverse, i.e., x = A 1 b. Cramer however provides a different solutions based on determinants under the condition that det(a) 0.
29 5.3 Key idea of Cramer s rule 5-28 x 1 Let x = x 2 be the solution of Ax = b. x 3 Then we can easily prove that a 1,1 a 1,2 a 1,3 x a 2,1 a 2,2 a 2,3 x a 3,1 a 3,2 a 3,3 x a 1,1 a 1,2 a 1,3 x 1 a 1,1 a 1,2 a 1,3 0 a 1,1 a 1,2 a 1,3 0 = a 2,1 a 2,2 a 2,3 x 2, a 2,1 a 2,2 a 2,3 1, a 2,1 a 2,2 a 2,3 0 a 3,1 a 3,2 a 3,3 x 3 a 3,1 a 3,2 a 3,3 0 a 3,1 a 3,2 a 3,3 1 b 1 a 1,2 a 1,3 = b 2 a 2,2 a 2,3 b 3 a 3,2 a 3,3
30 5.3 Key idea of Cramer s rule 5-29 Then by the product rule, x b 1 a 1,2 a 1,3 det(a) det x = det(a) x 1 = det b 2 a 2,2 a 2,3 x b 3 a 3,2 a 3,3 We finally obtain Similarly, x 2 = det det x 1 = a 1,1 b 1 a 1,3 a 2,1 b 2 a 2,3 a 3,1 b 3 a 3,3 a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 det det b 1 a 1,2 a 1,3 b 2 a 2,2 a 2,3 b 3 a 3,2 a 3,3 a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 and x 3 = det det a 1,1 a 1,2 b 1 a 2,1 a 2,2 b 2 a 3,1 a 3,2 b 3 a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3
31 5.3 Areas, volumes and determinants 5-30 Recall that all properties of determinants are derived from the first three properties. 1. Unit determinant for identity matrix. (Hint: By Pivot formula or by definition.) 2. Sign reversal by row/column switch. (Hint: By Pivot formula.) 3. Linearity with respect to any row/column. (Hint: By Pivot formula.) ([ ]) ([ ]) ta1,1 ta Multiplication: det 1,2 a1,1 a = t det 1,2 a 2,1 a 2,2 a 2,1 a 2,2 Addition: ([ ]) ([ ]) ([ ]) a1,1 + a det 1,1 a 1,2 + a 1,2 a1,1 a = det 1,2 a +det 1,1 a 1,2 a 2,1 a 2,2 a 2,1 a 2,2 a 2,1 a 2,2 As it turns out, all functions that satisfy these three properties should have the same value.
32 5.3 Areas, volumes and determinants 5-31 Example. Area of a parallelogram. Solution. It is a function of two vectors v and w. 1. Unit area for perpendicular unit vectors. f(v, w) =1 if v = w =1andv w =0. 2. Area remained unchanged by vector switch. f(v, w) =f(w, v). 3. Linearity with respect to any vector. Multiplication: f(t v, w) =t f(v, w). Addition: f(v + v, w) =f(v, w)+f(v, w). (See the next page.) As it turns out, for two-dimensional vectors v and w, ([ ]) f(v, w) = v det T. w T
33 5.3 Areas, volumes and determinants 5-32 (x 3,y 3 ) w v v (x 1,y 1 ) (x 2,y 2 ) Three points (x 1,y 1 ), (x 2,y 2 )and(x 3,y 3 ) also decide a parallelogram. Then the area of the parallelogram is ([ ]) f(v, w) = det x2 x 1 y 2 y 1 = x 3 x 1 y 3 y 1 x 1 y 1 1 det x 2 y 2 1 x 3 y 3 1.
34 5.3 Areas, volumes and determinants 5-33 Similarly, for three-dimensional vectors u, v and w, u T f(u, v, w) = det v T w T is the volume of the box spanned by the three vectors. Again, the volume of the box decided by four vertices (x 1,y 1,z 1 ), (x 2,y 2,z 2 ), (x 3,y 3,z 3 )and(x 4,y 4,z 4 ) can be written as x 2 x 1 y 2 y 1 z 2 z 1 x 1 y 1 z 1 1 f(u, v, w) = det x 3 x 1 y 3 y 1 z 3 z 1 x 4 x 1 y 4 x 1 z 4 z 1 = x det 2 y 2 z 2 1. x 3 y 3 z 3 1 x 4 y 4 z 4 1 Note that the determinant formula does not in any case simplify the calculation of the area and volume. It is just interesting to have some alternative expression so that we can take advantage of it in some special case.
35 5.3 Areas, volumes and determinants 5-34 How to find the area of the parallelogram spanned by two (n-dimensional) vectors v and w, wheren>2? It might have no determinant formula. But we know we can apply, e.g., Property 5, to yield that f(v, w) =f(v, w av) such that v (w av) = 0 (i.e., a = v w/ v 2 ). w av w w bv v (So all three parallelograms, i.e., the black one, the blue one and the red one, have thesamearea!) Then, f(v, w) =f(v, w av) = v w av = v w v w v v 2.
36 5.3 Jocobian: An application of area computation 5-35 Example. Application of determinant computation. In calculus, we can transform to and from the 2-dimensional (x, y)-axis to the polar (r, θ)-axis with { x(r, θ) =r cos(θ) y(r, θ) =r sin(θ) Linear approximation gives { x(r + r, θ + θ) x(r, θ)+ x(r,θ) r y(r + r, θ + θ) y(r, θ)+ y(r,θ) r r + x(r,θ) θ θ r + y(r,θ) θ θ This approximation is accurate when r and θ get smaller. So [ ] x y = [ ] x(r + r, θ + θ) x(r, θ) y(r + r, θ + θ) y(r, θ) [ x(r,θ) r y(r,θ) r x(r,θ) θ y(r,θ) θ ] } {{ } J(r,θ) [ ] r θ = J(r, θ) [ ] r θ
37 5.3 Jocobian: An application of area computation 5-36 Now for the area of the (very small) parallelogram spanned by (x 1,y 1 ), (x 2,y 2 ) and (x 3,y 3 ), we have [ ] [ ] x2 x 1 x 3 x 1 r2 r J(r, θ) 1 r 3 r 1 y 2 y 1 y 3 y 1 θ 2 θ 1 θ 3 θ 1 This gives the important result that for infinitesimal x and y (equivalently, r and θ), ([ da x,y = det x2 x 1 x 3 x 1 y 2 y 1 y 3 y 1]) ([ = det(j(r, θ)) det r2 r 1 r 3 r 1 θ 2 θ 1 θ 3 θ 1]) = det(j(r, θ)) da r,θ.
38 5.3 Cross product 5-37 For two vectors v and w, wehavedefinedtheinner product and the outer product respectively as inner product: v w = w T v and outer product :v w = vw T for which the outputs are respectively a scaler and an n n matrix. Now we introduce a third product: cross produdct for which the output is a vector. i j k v w det v 1 v 2 v 3 w 1 w 2 w 3 where ([ ]) ([ ]) ([ ]) v2 v det 3 v1 v i det 3 v1 v j + det 2 k, w 2 w 3 w 1 w 3 w 1 w i = 0, j = 1, and k =
39 5.3 Cross product 5-38 You have noticed that the cross product for 3-dimensional vectors is defined based on co-factors. This is no coincidence as the idea of the cross product is to output a vector that is orthogonal to both v and w. v (v w) ([ ]) ([ ]) ([ ]) v2 v = det 3 v1 v v i det 3 v1 v v j + det 2 v k w 2 w 3 w 1 w 3 w 1 w 2 ([ ]) ([ ]) ([ ]) v2 v = det 3 v1 v v w 2 w 1 det 3 v1 v v 3 w 1 w 2 + det 2 v 3 w 1 w 3 2 v 1 v 2 v 3 = det v 1 v 2 v 3 =0 w 1 w 2 w 3 An extension definition for the cross product of higher dimensional vectors is based on similar idea (to output a proper vector that is orthogonal to the original two), but will be more complicated (in choosing what we mean by proper. )
40 5.3 Cross product and area 5-39 proper = the length of the vector should be the area spanned by the original two. Example. Area of a parallelogram spanned by two 3-dimensional vectors v and w. 1. Unit area for perpendicular unit vectors. v w =1 if v = w =1andv w =0. Proof: v w 2 = (v 2 w 3 + v 3 w 2 ) 2 +(v 1 w 3 + v 3 w 1 ) 2 +(v 1 w 2 + v 2 w 1 ) 2 = (v1 2 + v2 2 + v3)(w w2 2 + w3) 2 (v 1 w 1 + v 2 w 2 + v 3 w 3 ) 2 = v 2 w 2 (v w) 2 =1. 2. Area remained unchanged by vector switch. v w = w v. 3. Linearity with respect to any vector. Multiplication: (t v) w = t (v w). Addition: (v + v ) w = v w + v w.
41 5.3 Cross product and area 5-40 As it turns out, for three-dimensional vectors v and w, Hence (as also can be known from area(v, w) = v w. v w 2 = v 2 w 2 (v w) 2 = v 2 w 2 v 2 w 2 cos 2 (θ) ) we have v w = v w sin(θ). 5.3 Cross product and right-hand rule Right-hand rule for cross product: v w points to the thumb when right hand fingers curl from v to w.
42 5.3 Triple product 5-41 Now we know inner product, outer product, cross product; let s introduce another product of vectors in the literature. Definition (Triple product): The triple product of three vectors is defined as (u v) w For 3-dimensional vectors, (u v) w = u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 gives the volume of the box spanned by u, v and w.
43 5.3 A potential way to derive the nullspace 5-42 Lemma: Suppose A is not invertible. Then, AC T = all-zero matrix where C is the cofactor matrix. Proof: The off-diagonal terms are zero by the same proof to show AC T = det(a)i for invertible A. The proof is completed by noting that the diagonal terms are equal to det(a) =0.
44 5.3 A potential way to derive the nullspace 5-43 This seems to give an alternative way to derive the null space of A, i.e, all solutions to satisfy Ax = 0. However, when A n n has rank no greater than (n 2), we have C i,j =0 for every i and j. So, this alternative method fails. This method works only when A n n has rank (n 1). Then, it is guaranteed that at least one column in C T is non-zero.
45 5.3 Remarks 5-44 Problem 40. Jacobi s formula. d det(a) =trace dx ( adj(a) da ) dx (Problem 40, Section 5.3) Suppose A is a 5 by 5 matrix. Its entries in row 1 multiply determinants (cofactors) in row 2 5 to give the determinant. Can you guess Jacobi formula for deta using 2 by 2 determinants from rows 1-2 times 3 by 3 determinants from rows 3 5? Test your formula on the 1, 2, 1 tridiagonal matrix that has determinant = 6. Example of Jacobi s formula. A = [ 2x 2 x x 1 ] = det(a) =x 2. adj(a) = transpose of cofactor matrix = [ ] d 4x 1 dx A = 1 0 adj(a) da dx = [ 1 x x 2x 2 ][ ] [ ] 4x 1 3x 1 = 1 0 2x 2 x d det(a) = 2x = 3x +( x) =trace dx [ ] 1 x x 2x 2 ( adj(a) da dx )
46 5.3 Remarks 5-45 The Jacobi s formula can be used to confirm the co-factor formula of the determinant. ( d det(a) = trace adj(a) da ) da i,j da i,j ( C 1,1 C 2,1 C n,1... ) C = trace 1,2 C 2,2 C n, C 1,n C 2,n C n,n }{{} 1in(i,j)th entry 0 C i,1 0 ( 0. 0 ) = trace 0 C i,j 0 = C i,j C i,n 0 }{{} jth column
47 5.3 Remarks 5-46 We can again apply the Jacobi s formula on C i,j.withk iand l j, d C i,j = C (i,k),(j,l) da k,l where C (i,k),(j,l) is ( 1) k +l multiplying the determinant M (i,k),(j,l) of (n 2) (n 2) submatrix of A with i, k rows and k, l columns removed, where { { k k k < i and l j l < j k 1 k>i j 1 l>j. So for k>iand l>j..... a det(a) = det i,j a i,l a k,j a k,l..... = + a i,j a k,l ( 1) i+j ( 1) (k 1)+(l 1) M (i,k),(j,l) + a i,l a k,j ( 1) i+l ( 1) k+(j 1) M (i,k),(j,l) + ([ ]) ai,j a = + det i,l ( 1) i+j+k+l M a k,j a (i,k),(j,l) + k,l n n ([ ]) ai,j a = det i,l ( 1) i+j+k+l M (i,k),(j,l), an extension of the cofactor formula. j=1 l=j+1 a k,j a k,l
48 5.3 Remarks 5-47 Problem 41. Cauchy-Binet formula. det (A m n B n m )= [m] det(a m [m] ) det(b [m] m ) where A m [m] is the submatrix of A with only m columns selected from A, and B [m] m is the submatrix of B with only m rows selected from B. The summation is carried out over all possible selections. There are ( n m) of them. (Problem 41, Section 5.3) The 2 by 2 matrix AB = (2 by 3)(3 by 2) has a Cauchy- Binet formula for detab: detab = sum of (2 by 2 determinants in A)(2 by 2 determinants in B) (a) Guess which 2 by 2 determinants to use from A and B. (b) Test your formula when the rows of A are 1, 2, 3and1, 4, 7withB = A T.
49 5.3 Remarks 5-48 Example. A = Solution. [ ] and B = A T. [ ] 1 det(ab) = det = det ([ ]) ([ ]) det(a 2 [2] ) det(b [2] 2 ) = det det [2] ([ ]) ([ ]) det det + det = =24 ([ ]) =24 ([ ]) 2 3 det 4 7 ([ ])
MATRIX 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 informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
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 informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
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 three-space, we write a vector in terms
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 two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
More informationThe Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional 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 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 Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
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 informationLecture 4: Partitioned Matrices and Determinants
Lecture 4: Partitioned Matrices and Determinants 1 Elementary row operations Recall the elementary operations on the rows of a matrix, equivalent to premultiplying by an elementary matrix E: (1) multiplying
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
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 informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationUnit 18 Determinants
Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of
More informationSolution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
More informationEigenvalues and Eigenvectors
Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationNotes on Linear Algebra. Peter J. Cameron
Notes on Linear Algebra Peter J. Cameron ii Preface Linear algebra has two aspects. Abstractly, it is the study of vector spaces over fields, and their linear maps and bilinear forms. Concretely, it is
More informationUsing row reduction to calculate the inverse and the determinant of a square matrix
Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
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 informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
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 L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationMATH 551 - APPLIED MATRIX THEORY
MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationSection 6.1 - Inner Products and Norms
Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More information1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped
More informationDETERMINANTS TERRY A. LORING
DETERMINANTS TERRY A. LORING 1. Determinants: a Row Operation By-Product The determinant is best understood in terms of row operations, in my opinion. Most books start by defining the determinant via formulas
More informationLinear Algebra: Determinants, Inverses, Rank
D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the n-dimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationOperation Count; Numerical Linear Algebra
10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floating-point
More information3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationInner products on R n, and more
Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +
More informationMatrices 2. Solving Square Systems of Linear Equations; Inverse Matrices
Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,
More informationDETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH
DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More information26. Determinants I. 1. Prehistory
26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationChapter 7. Permutation Groups
Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral
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 information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationLecture Notes 2: Matrices as Systems of Linear Equations
2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably
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, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationMatrix algebra for beginners, Part I matrices, determinants, inverses
Matrix algebra for beginners, Part I matrices, determinants, inverses Jeremy Gunawardena Department of Systems Biology Harvard Medical School 200 Longwood Avenue, Cambridge, MA 02115, USA jeremy@hmsharvardedu
More information6. Cholesky factorization
6. Cholesky factorization EE103 (Fall 2011-12) triangular matrices forward and backward substitution the Cholesky factorization solving Ax = b with A positive definite inverse of a positive definite matrix
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationMatrices and Linear Algebra
Chapter 2 Matrices and Linear Algebra 2. Basics Definition 2... A matrix is an m n array of scalars from a given field F. The individual values in the matrix are called entries. Examples. 2 3 A = 2 4 2
More informationElementary Matrices and The LU Factorization
lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) on the identity (unit) matrix is called an elementary matrix. There are three
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationSection V.3: Dot Product
Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,
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 informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationOrthogonal Bases and the QR Algorithm
Orthogonal Bases and the QR Algorithm Orthogonal Bases by Peter J Olver University of Minnesota Throughout, we work in the Euclidean vector space V = R n, the space of column vectors with n real entries
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationMAT 242 Test 2 SOLUTIONS, FORM T
MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
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 informationA Primer on Index Notation
A Primer on John Crimaldi August 28, 2006 1. Index versus Index notation (a.k.a. Cartesian notation) is a powerful tool for manipulating multidimensional equations. However, there are times when the more
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 informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationLinear Algebra and TI 89
Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing
More informationLecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
More informationBrief Introduction to Vectors and Matrices
CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vector-valued
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
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 informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More information