Notes on Determinant


 Ira Joseph
 1 years ago
 Views:
Transcription
1 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 actually calculating the solution. This note is the preliminary to the calculation of eigenvalues and eigenvectors. We want to determine whether a system of homogeneous linear equations Ax = 0, where A is an n n matrix, has nontrivial solution or not. It is obvious that x = 0 is a solution. Any solution with at least one component nonzero is said to be nontrivial. Corollary 10 at the end of this note gives a necessary and sufficient condition for the existence of nontrivial solution. It is analogous to the discriminant of a quadratic equation ax 2 + bx + c = 0. We can determine whether the roots are distinct or repeated from the discriminant b 2 4ac, without actually calculating the roots. The objectives of this notes are: 1. Define determinant properly. 2. Derive some basic properties of determinant determinant The determinant of 2 2 matrix is defined as [ ] a b det := ad bc. (1) c d (The symbol := means true by definition.) If A is a 2 2 matrix with nonzero determinant, we can solve Ax = b by x = 1 [ ] d b b. det A c a The solution is uniquely determined by the above equation. If det A = 0, then either Ax = b has no solution or infinitely many solutions. To emphasize the dependency on the rows of the matrix, we let u and v be the first and second rows of a 2 2 matrix, respectively, and write the determinant function as det(u, v). With this notation, we can readily check from (1) that the determinant function satisfies the following properties: 1. det(u, v) = det(v, u). 2. det(u 1 + u 2, v) = det(u 1, v) + det(u 2, v). 3. det(u, v 1 + v 2 ) = det(u, v 1 ) + det(u, v 2 ). 4. det(cu, v) = c det(u, v), for any constant c. 5. det(u, cv) = c det(u, v), for any constant c. 1
2 6. det([1 0], [0 1]) = 1. Here, u, u 1, u 2, v, v 1 and v 2 are row vectors of length two, and c is a real constant which may equal zero. We want to generalize the notion of determinant to general n n matrices, so that we can efficiently determine whether an n n linear system has unique solution. 2 How to define n n determinant? There are several approaches in defining determinant beyond the 2 2 case. We shall give two of them below. The first one is recursive, and the second one is not. In the following, the matrix A = [a ij ] is an n n matrix. The (i, j)entry of A is denoted by a ij. [ ] a b Expansion on the first row. For 2 2 matrices, we define the determinant as ad bc. c d Recursively, suppose that we know how to calculate (n 1) (n 1) determinant for some integer n 3. In order to simplify notations, for positive integers between 1 and n, we let A ij be the determinant of the submatrix obtained by removing the ith row and the jth column. The value A ij is called the minor corresponding to the (i, j)entry of A. In terms of the minors, the determinant of the n n matrix A can be defined by det(a) := a 11 A 11 a 12 A 12 + a 13 A 13 + ( 1) n+1 a 1n A 1n. (2) This definition is usually called the Laplace expansion on the first row. Example: det = = (21) 2(12) + 3( 15) = Definition in terms of permutations There are n! ways to write 1 to n in some order. Each of them is called a permutation of 1, 2,..., n. For example, (2, 5, 1, 4, 3) and (1, 4, 5, 2, 3) are two permutations of 1 to 5. For a sequence of distinct numbers, (a 1, a 2,..., a n ), we define the corresponding inversion number as the number of pairs which are not in the natural order; we count the number of pairs a i and a j such that a i > a j and i < j. Mathematically, we let inv(a 1, a 2,..., a n ) := #{(i, j) : a i > a j and i < j}. The symbol #S stands for the number of elements in a set S. In other words, the inversion number of permutation (p 1, p 2,..., p n ) counts the number of integers r and s such that r is to the left of s and r is larger than s. A permutation is in ascending order if and only if the inversion number is equal to zero. Example: The inversion number of (2, 5, 1, 4, 3) is 5, because we can find five pairs (2, 1), (5, 1), (5, 3), (4, 3) and (5, 4). The inversion number of (1, 4, 5, 2, 3) is 4, because of (4, 2), (5, 2), (4, 3) and (5, 3). Inversion number is defined for any sequence of distinct numbers, not necessarily a permutation of 1 to n. For example, the inversion number of (1, 16, 4, 9) is 2. For an n n matrix A, we can define the determinant of A by det A := ( 1) inv(p 1,p 2,...,p n ) a 1,p1 a 2,p2 a 3,p3 a 4,p4 a n,pn, (3) with the summation over all permutations of 1 to n. There are n! terms, and each term consists of a sign ( 1) inv(p1,p2,...,pn) and the product of n entries in A. This formula is sometime called the Leibniz formula. We note that the determinant only needs the information whether the inversion number is even or odd; if the inversion number is even, then the corresponding term has positive sign, otherwise the sign is negative. 2
3 Example: For n = 4, we check that inv(1, 2, 3, 4) = 0, inv(2, 1, 3, 4) = 1, inv(3, 1, 2, 4) = 2, inv(4, 1, 2, 3) = 3, inv(1, 2, 4, 3) = 1, inv(2, 1, 4, 3) = 2, inv(3, 1, 4, 2) = 3, inv(4, 1, 3, 2) = 4, inv(1, 3, 2, 4) = 1, inv(2, 3, 1, 4) = 2, inv(3, 2, 1, 4) = 3, inv(4, 2, 1, 3) = 4, inv(1, 3, 4, 2) = 2, inv(2, 3, 4, 1) = 3, inv(3, 2, 4, 1) = 4, inv(4, 2, 3, 1) = 5, inv(1, 4, 2, 3) = 2, inv(2, 4, 1, 3) = 3, inv(3, 4, 1, 2) = 4, inv(4, 3, 1, 2) = 5, inv(1, 4, 3, 2) = 3, inv(2, 4, 3, 1) = 4, inv(3, 4, 2, 1) = 5, inv(4, 3, 2, 1) = 6. Using the inversion number of the 24 permutations of 1,2,3,4, we can write the determinant of a 4 4 matrix a 11 a 12 a 13 a 14 A = a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 as det A := a 11 a 22 a 33 a 44 a 12 a 21 a 33 a 44 + a 13 a 21 a 32 a 44 a 14 a 21 a 32 a 43 a 11 a 22 a 34 a 43 + a 12 a 21 a 34 a 43 a 13 a 21 a 34 a 42 + a 14 a 21 a 33 a 42 a 11 a 23 a 32 a 44 + a 12 a 23 a 31 a 44 a 13 a 22 a 31 a 44 + a 14 a 22 a 31 a 43 + a 11 a 23 a 34 a 42 a 12 a 23 a 34 a 41 + a 13 a 22 a 34 a 41 a 14 a 22 a 33 a 41 + a 11 a 24 a 32 a 43 a 12 a 24 a 31 a 43 + a 13 a 24 a 31 a 42 a 14 a 23 a 31 a 42 a 11 a 24 a 33 a 42 + a 12 a 24 a 33 a 41 a 13 a 24 a 32 a 41 + a 14 a 23 a 32 a 41. We note that in each term, the first subscripts are in ascending order, while the second subscripts form a permutation of 1, 2, 3 and 4. There are 24 terms. Half of them have positive sign and half of them have negative sign. Proposition 1. The two definitions given above are equivalent. Proof We want to show that, for any n n matrix, the calculations in the two definitions of determinant give the same result. We proceed by mathematical induction. [ ] a11 a According to the first definition, the determinant 2 2 matrix 12 is a a 21 a 11 a 22 a 12 a 21. It is in 22 accordance with the second definition, because the inversion number of the permutation (1, 2) is 0 and the inversion number of (2, 1) is 1, [ ] a11 a det 12 = ( 1) inv(1,2) a a 21 a 11 a 22 + ( 1) inv(2,1) a 12 a 21 = a 11 a 22 a 12 a Let n be an integer larger than or equal to 3. Suppose that the two definitions of determinant agree with each other for matrices of size (n 1) (n 1). We will use the fact that if (p 1, p 2,..., p n ) is a permutation of 1 to n, then there are exactly p 1 1 pairs of number (p 1, p k ) with p 1 > p k. Therefore, inv(p 1, p 2,..., p n ) = (p 1 1) + inv(p 2, p 3,..., p n ) (4) if (p 1, p 2,..., p n ) is a permutation of 1 to n. In (4), inv(p 2, p 3,..., p n ) is the inversion number of (p 2, p 3,..., p n ), which is a permutation of {1, 2,..., p 1 1, p 1 + 1,..., n}. From the definition in (3), we can group the terms according to p 1, and get det A = ( 1) inv(p 1,p 2,...,p n ) a 1,p1 a 2,p2 a 3,p3 a 4,p4 a n,pn, = n ( 1) p1 1 a 1,p1 p 1=1 (p 2,p 3,...,p n ) ( 1) inv(p2,p3,...,pn) a 2,p2 a 3,p3 a 4,p4 a n,pn. 3
4 By the induction hypothesis, the summation ( 1) inv(p2,p3,...,pn) a 2,p2 a 3,p3 a 4,p4 a n,pn (p 2,p 3,...,p n ) is precisely the minor corresponding to the (1, p 1 )entry of A. Therefore, det A = which coincides with the first definition in (2). n ( 1) p1 1 A 1,p1, p 1=1 At this point, a computer scientist would say: the computational complexity involved in each of the two definitions given above are so high that they are not suitable for calculating determinant of large size. Indeed, the Leibniz formula has n! terms, and n! increases exponentially with n. If we apply the recursive formula in the straightforward way, the computational complexity is also in the order of n!. For example the calculation of a matrix using the first recursive definition directly requires more than three million multiplications. A more efficient method of calculating the determinant will be given in the next section. 3 Axioms of determinant It turns out that the properties of determinant are more important then how to calculate the determinant. We shall identify several basic and desirable properties of determinant, called axioms, and verify that the two equivalent ways of calculating determinant in the last section satisfy these axioms. We write the desired function on matrices as a function which whose inputs are n row vectors of length n, and the output is a real number, det(v 1, v 2,..., v n ). As in the 2 2 case, we want to construct a function which (at least) satisfies the following properties. (Axiom 1) If any two row vectors in det(v 1, v 2,..., v n ) are exchanged, then the value of the determinant is multiplied by 1. (Axiom 2) det(u + u, v 2,..., v n ) = det(u, v 2,..., v n ) + det(u, v 2,..., v n ) for any two row vectors u and u of length n. (Axiom 3) det(cv 1, v 2,..., v n ) = c det(v 1, v 2,..., v n ) for any constant c. (Axiom 4) For i = 1, 2,..., n, let e i be the standard basis of R n, i.e., Then det(e 1, e 2,..., e n ) = 1. e i := (0, 0,..., 0, 1, 0, 0,..., 0). } {{ } i These four properties are called the axioms of determinant. We first derive some immediate properties from the four axioms. Proposition 2. If det is a function which satisfies Axioms 1 to 4, then 1. If any two row vectors are identical, then det(v 1, v 2,..., v n ) is equal to 0. 4
5 2. Suppose that the ith row vector can be decomposed as the sum u + u. Then 3. For any constant c, we have det(v 1,..., v i 1, u + u,..., v n ) = det(v 1,..., v i 1, u,..., v n ) + det(v 1,..., v i 1, u,..., v n ). det(v 1,..., v i 1, cv i,..., v n ) = c det(v 1,..., v i 1, v i,..., v n ). 4. For any two distinct row indices i j, adding any constant multiple of the r j to r i does not change the value of determinant. 5. If one of the input row vector is an allzero vector, then the determinant is equal to zero. Proof (1) Suppose that the ith row and the jth row are identical. Let the determinant function value be x. Since the ith and the jth row are the same, exchanging them gives the same determinant value x. However, by Axiom 1, if the ith and the jth row are exchanged, the value of the resulting determinant is x. Hence, x = x. This is possible only if x = 0. (2) We substitute v i by u + u, det(v 1,..., v i 1, u + u,..., v n ) = det(u + u,..., v i 1, v 1,..., v n ) = det(u,..., v i 1, v 1,..., v n ) det(u,..., v i 1, v 1,..., v n ) = det(v 1,..., v i 1, u,..., v n ) + det(v 1,..., v i 1, u,..., v n ). The first and third equalities follow from Axiom 1, while the second equality follows from Axiom 2. (3) The proof of part 3 is similar to part 2, and is omitted. (4) For any constant c, det(v 1,..., v i + cv j,..., v j..., v n ) = det(v 1,..., v i,..., v j..., v n ) + det(v 1,..., cv j,..., v j..., v n ) = det(v 1,..., v i,..., v j..., v n ) + c det(v 1,..., v j,..., v j..., v n ) = det(v 1,..., v i,..., v j..., v n ). The first and second equalities follows from Axiom 2 and 3, respectively. The last equality follows from the first part of the proposition. (5) Suppose that the components of v i are all zero. Let c be any real number which is not equal to 1. We have cv i = v i = [ ], and by part (3) of the proposition, we get Therefore det(v 1,..., v i 1, v i,..., v n ) = det(v 1,..., v i 1, cv i,..., v n ) This implies that det(v 1,..., v i 1, v i,..., v n ) = 0. = c det(v 1,..., v i 1, v i,..., v n ). (1 c) det(v 1,..., v i 1, v i,..., v n ) = 0. We now verify that the second definition of determinant satisfies the four axioms. Proposition 3. The function defined in (3) satisfies Axioms 1 to 4. 5
6 Proof In order to avoid potential confusion, we let v i = [a i1 a i2... a in ] be row vector of length n, for i = 1, 2,..., n, and let f(v 1, v 2,..., v n ) be a function defined as f(v 1, v 2,..., v n ) := ( 1) inv(p 1,p 2,...,p n ) a 1,p1 a 2,p2 a n,pn, (5) where the summation is over all possible permutations (p 1,..., p n ) of 1 to n. (Verification of Axiom 4) Let v i be the ith basis vector e i. Then { 1 if i = j, a ij = 0 otherwise. Hence, all terms in (5) are zero except the term corresponding to (p 1, p 2,..., p n ) = (1, 2,..., n). This gives f(e 1, e 2,..., e n ) = inv(1, 2,..., n)a 11 a 22 a nn = = 1. (Verification of Axiom 3) Suppose that the ith row v i is multiplied by a constant c. We have f(v 1, v 2,..., cv i,..., v n ) = ( 1) inv(p 1,p 2,...,p n ) a 1,p1 a 2,p2 (ca i,pi ) a n,pn, = c ( 1) inv(p1,p2,...,pn) a 1,p1 a 2,p2 a i,pi a n,pn, = cf(v 1, v 2,..., cv i,..., v n ). (Verification of Axiom 2) The verification of Axiom 2 is similar to the verification of Axiom 3, and is omitted. (Verification of Axiom 1) For Axiom 1, we need the property that if we swap two numbers in (p 1, p 2,..., p n ), then the parity of the inversion number changes, i.e., if the inversion number is even before the swapping, then the inversion after the swapping is odd, and vice versa. Suppose that we interchange row i and row j, with i j. To express the idea algebraically, we let v i = [a j1 a j2... a jn ] v j = [a i1 a i2... a in ] p i = p j, and p j = p i. The determinant of the matrix after exchanging rows i and j is f(v 1, v 2,..., v i,..., v j,..., v n ) = ( 1) inv(p 1,p 2,...,p n ) a 1,p1 a 2,p2 a j,pi a i,pj a n,pn = = (p 1,..., p i,..., p j,...,p n) ( 1) inv(p1,p2,...,pn) a 1,p1 a 2,p2 a i,pj a j,pi a n,pn ( 1) ( 1) inv(p 1,..., p i,..., p j,...,p n ) = f(v 1, v 2,..., v i,..., v j,..., v n ). a 1,p1 a 2,p2 a i, pi a j, pj a n,pn 6
7 This completes the proof of the proposition. 4 Calculation of determinant by row reductions We have verified that we can construct a function which satisfies the four axioms of determinant. The next question we want to ask is Can we construct two different functions, both of which satisfy the four axioms of determinant? The answer is no, because the function value is uniquely determined by the four axioms. Given an n n matrix A, we can calculate the determinant of A by rowreducing it to an upper triangular or a lower triangular matrix, and use the property that the determinant of an upper triangular or a lower triangular matrix is equal to the product of the diagonal entries. The effect of elementary row operations on the determinant is summarized in the following table. Elementary row operation Effect on determinant Exchange two rows Multiply by 1 Multiply a row by a nonzero constant c Multiply by c Add a constant multiple of a row to another row No change Example: Find the determinant of A = We first exchange row 1 and row 3, in order to make the topleft corner equal to 1, det(a) = det The value is multiplied by 1, because of Axiom 1. After subtracting two times row 1 from row 3, and subtracting two times row 1 from row 4, we get det(a) = det The value does not change after these two row operations. Now, we add four times row 2 to row 3, and 2 times row 2 to row 4, det(a) = det The matrix on the righthand side is upper triangular. The entries above the diagonal becomes zero after a few more row reductions of the third type, det(a) = det
8 We can pull out the two factor of 5 from the determinant by Axiom 3, and obtain det(a) = 5 5 det = The last equality follows from Axiom 4. Example: Find the determinant of B = by row reduction. After subtracting row 1 and row 2 from row 3, we obtain As the last row is allzero, the determinant is zero by part (5) of Proposition 2, det B = det = We note that the calculations in these two examples only depend on the properties of determinant. In general, we have the following Proposition 4. There is one and only one function which satisfies the four axioms of determinant. From now on, any function which satisfies the four axioms is called the determinant function. 5 Using determinant to test linear independence Given a set of n vectors of length n, we can test whether they are linearly independent by putting these n vectors together to form an n n matrix, and calculating the determinant. If the determinant is nonzero, then the n vectors are linearly independent, otherwise they are linearly dependent. Proposition 5. Let u 1, u 2,..., u n be column vectors of length n, and let M be the n n matrix whose columns are precisely u 1, u 2,..., u n. We have det M = 0 if and only if u 1, u 2,..., u n are linearly dependent. Proof Let c 1, c 2,..., c n be real numbers such that c 1 u 1 + c 2 u c n u n = 0. The real numbers c 1, c 2,..., c n are solution to c 1 0 c 2.. M = 0.. (6) 0 c n 8
9 We want to test whether we can find a nontrivial solution to (6). We rowreduce the matrix M to reduced row echelon form (RREF). Recall that whether the determinant of M is zero or not is invariant under elementary row operations. Therefore, the determinant of M is nonzero if and only if the determinant of the RREF is nonzero. We count the number of pivots in the RREF and consider two cases: (i) there are exactly n pivots in the RREF, (ii) there are strictly less than n pivots in the RREF. We we call that in RREF, each row contains at most one pivot, and each column contains at most one pivot. Case 1, there are exactly n pivots: Since each pivot occupies one row and one column, there is exactly one pivot in each row and each column. The RREF of M is the n n identity matrix, which has determinant equal to 1 by Axiom 4. Hence det(m) 0 in this case. Since M can be rowreduced to the identity matrix, the only choice for c 1 to c n is c 1 = c 2 = = c n = 0. Therefore u 1, u 2,..., u n are linearly independent. Case 2, there are strictly less than n pivots: The last row of the RREF of M is an allzero row. The determinant of the RREF is zero by part (5) of Proposition 2. By part (5) of Proposition 2, det(m) = 0 in this case. Since the last row is zero, the number of variables, n, is strictly larger than the number of essentially different equations. We can find (infinitely many) nontrivial solution to the system of linear equations in (6). This implies that u 1, u 2,..., u n are linearly dependent. Example: We want to test whether the following four vectors are linearly independent, , 1 2, 2 2, Concatenate the four vectors and form the matrix We have already calculated that the determinant of this matrix is 25, which is nonzero. Therefore, these four vectors are linearly independent. 6 Computing determinant by column expansion We used several times the property that if a matrix contains an allzero row, then the determinant is zero. Similar thing are true for columns. Proposition 6. If a matrix contains an allzero column, then the determinant of this matrix is zero. Proof Let the columns of the matrix be denoted by u 1, u 2,..., u n. Suppose that the jth column, u j, is the zero vector. Then, columns vectors u 1, u 2,..., u n are linearly dependent, because the linear combination c 1 u 1 + c 2 u c j u j c n u n is equal to the zero vector by setting c j = 1 and the rest of the coefficients to be zero. From Proposition 5, we conclude that the determinant of the matrix is zero. 9
10 We can now show that the determinant can be calculated by Laplace expansion on the first column. Consider the 4 4 case as an example: a e The first row can be decomposed as a sum of two vectors [a ] = [a 0 0 0] + []. By the second axiom of determinant, we get a a e = e + e By Laplace expansion on the first row, the first determinant on the right hand side can be simplified to a 3 3 determinant, a e = a v w x + e Repeat the same procedure to the second row of e, we obtain a e = a v w x + e e = a v w x + 0 = a v w x e v w x + 0. We have used the property that the determinant is multiplied by 1 after exchanging two rows. Apply the 10
11 same steps to rows 3 and 4, we get a v w x e v w x + q v w x = a v w x e v w x + q v w x u v w x = a v w x e v w x + q v w x u. In the last step, we have used the property that if there is a zero column vector in a matrix, then the determinant is zero. In general, we have the following Proposition 7. Let A be an n n matrix and A ij be the determinant of the submatrix obtained by deleting row i and column j. The determinant of A can be computed by expansion on the first column, Example: det A = a 11 A 11 a 21 A 21 + a 31 A ( 1) n+1 a n1 A n1. det = = 2(8) + (13) = Duality between rows and columns Proposition 8. Let T denote the transpose operator. We have for square matrix A. Proof det(a T ) = det(a) We can proceed by mathematical induction. The statement is true for 2 2 matrix, [ ] [ ] a b a c det = ad bc = det. c d b d Suppose that det(b T ) = det(b) for all (n 1) (n 1) matrices B. Let A be an n n matrix, and let A ij be the minor corresponding to the (i, j)entry. We let a ij = a ji be the (i, j)entry of matrix A T, and A ij be the minor corresponding to the (i, j)entry of AT. By the induction hypothesis, we have A ij = A ji. We calculate the determinant of A by expanding on the first column, det A T = a 11A 11 a 21A 21 + a 31A ( 1) n+1 a n1a n1 = a 11 A 11 a 12 A 12 + a 13 A ( 1) n+1 a 1n A 1n. The last line is precisely the Laplace expansion of det(a) on the first row. Therefore det A T = det A. Using Proposition 8, every row property of determinant can be translated to a column property. For example, if we exchange two columns of a matrix, the determinant is multiplied by 1. If we add a scalar multiple of a column to another column, the determinant does not change. Combining Propositions 5 and 8, we obtain the following important theorem 11
12 Theorem 9. For an n n matrix A, the followings are equivalent: 1. the columns of A are linearly independent. 2. det(a) det(a T ) the rows of A are linearly independent. ( Equivalence means that if one statement is true, then all the other statements are also true, but if one statement is false, then all the other statements are also false.) Proof The equivalence between (1) and (2) is precisely the content of Proposition 5. The equivalence between (2) and (3) follows directly from Proposition 8. The equivalence between (3) and (4) follows from the dual of Proposition 5, by replacing all columns by rows, and by A by A T. The following corollary is an immediate consequence of Theorem 9. Corollary 10. Let A be an n n matrix. The system of linear equation has a nontrivial solution if and only if det(a) = 0. Example Consider the system of linear equations Ax = 0 x + y + z = 0 sx 2y z = 0 x y + 2z = 0 where s is a parameter. We want to find the value of s so that we can find a nontrivial solution to the above linear system. By Corollary 10, we solve det s 2 1 = for the parameter s. The above determinant is equal to det s 2 1 = det s [ 2 1 = 1 det 2 1 = 4 3s. ] s det [ 1 ] The first equality follows by subtracting row 1 from row 3, and the second by expansion on the first column. By setting the determinant to zero, we see that the system of linear equations has nontrivial solution if and only if s = 4/3. Indeed, x = 2427, y = 809, z = 1618 is a nontrivial solution when s = 4/3. When s 4/3, the only solution is x = y = z = 0. 12
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 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 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 information2.5 Elementary Row Operations and the Determinant
2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)
More information1 Determinants. Definition 1
Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described
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 information2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors
2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the
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 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 informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional 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 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
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 information1 Gaussian Elimination
Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 GaussJordan reduction and the Reduced
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 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 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 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 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 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 wellknown properties Volumes of parallelepipeds are
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
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 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 informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
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 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 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 315: Linear Algebra Solutions to Midterm Exam I
Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =
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 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 informationMATH10212 Linear Algebra B Homework 7
MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments
More informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationCofactor Expansion: Cramer s Rule
Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating
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 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,
MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call
More informationSolution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.
Solutions to Math 30 Takehome prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)
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 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 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 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 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 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 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 information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
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 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 informationLecture Notes: Matrix Inverse. 1 Inverse Definition
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationSection 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =
Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and
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 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 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 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 information1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither.
Math Exam  Practice Problem Solutions. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. (a) 5 (c) Since each row has a leading that
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 informationMatrix Calculations: Inverse and Basis Transformation
Matrix Calculations: Inverse and asis Transformation A. Kissinger (and H. Geuvers) Institute for Computing and Information ciences Intelligent ystems Version: spring 25 A. Kissinger (and H. Geuvers) Version:
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 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 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 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 informationA matrix over a field F is a rectangular array of elements from F. The symbol
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
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 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 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 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 informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
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 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 informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More informationDeterminants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number
LECTURE 13 Determinants 1. Calculating the Area of a Parallelogram Definition 13.1. Let A be a matrix. [ a c b d ] The determinant of A is the number det A) = ad bc Now consider the parallelogram formed
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
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 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 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 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 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 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 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 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 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 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 informationDeterminants. Chapter Properties of the Determinant
Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation
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 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 informationNON 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 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 informationSergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014
Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of
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 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 ndimensional column
More informationRow Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for inclass presentation
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 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 informationInverses. Stephen Boyd. EE103 Stanford University. October 27, 2015
Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudoinverse Left and right inverses 2 Left inverses a number
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 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 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 informationDirect Methods for Solving Linear Systems. Linear Systems of Equations
Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationImages and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232
Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been
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 informationPractice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.
Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More information