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Chapter 2 Matrices and Linear Transformations

Linear Algebra. Chapter 2 Matrices and Linear Transformations. 大葉大學 資訊工程系 黃鈴玲. 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices. a ij : the element of matrix A in row i and column j. Definition

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Chapter 2 Matrices and Linear Transformations

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  1. Linear Algebra Chapter 2Matrices and Linear Transformations 大葉大學 資訊工程系 黃鈴玲

  2. 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices • aij: the element of matrix A in row i and column j. Definition Two matrices are equal if they are of the same size and if their corresponding elements are equal. ( 代表 for every, for all) Thus A = B if aij = bij  i, j.

  3. Addition of Matrices Definition Let A and B be matrices of the same size. TheirsumA + B is the matrix obtained by adding together the corresponding elements of A and B. The matrix A + B will be of the same size as A and B. If A and B are not of the same size, they cannot be added, and we say that the sum does not exist.

  4. Determine A + B and A + C, if the sum exist. Example Solution (自行練習) (2) A andC are not of the same size, A + C does not exist.

  5. Example Observe that A and 3A are both 2  3 matrices. Scalar Multiplication of matrices Definition Let A be a matrix and c be a scalar. The scalar multiple of A by c, denoted cA, is the matrix obtained by multiplying every element of A by c. The matrix cA will be the same size as A.

  6. Negation and Subtraction Definition The matrix (-1)C is written –C and is called the negative of C. Example We now define subtraction in terms of addition and scalar multiplication. Let A – B = A + (–1)B Example

  7. Matrix Multiplication Examples

  8. Example 1 Sol. 無法相乘AB does not exist. Note.In general, ABBA. 如此題之BA存在

  9. Definition Let the number of columns in a matrix Abe the same as the number of rows in a matrix B. The product AB then exists. Let A: mn matrix, B: nk matrix, The product matrix C=ABhas elements C is a mk matrix. If the number of columns in A does not equal the number of row B, we say that the product does not exist.

  10. Example 3 Determine c23. Let C = AB, Example 2 Sol. BA does not exist. 隨堂作業：5(f) Note.In general, ABBA. 隨堂作業：11(d)

  11. A m r Example If A is a 5  6 matrix and B is an 6 7 matrix. Because A has six columns and B has six rows. Thus AB exits. And AB will be a 5 7 matrix. Size of a Product Matrix If A is an m r matrix and B is an r n matrix, then AB will be an m n matrix. B = AB r n m n 隨堂作業：8(f)

  12. Special Matrices Definition A zero matrix is a matrix in which all the elements are zeros. A diagonal matrix is a square matrix in which all the elements not on the main diagonal are zeros. An identity matrix is a diagonal matrix in which every diagonal element is 1.

  13. Example 4 Theorem 2.1 Let A be m n matrix and Omn be the zero m n matrix. Let B be an n n square matrix. On and In be the zero and identity n n matrices. Then A + Omn = Omn + A = A BOn = OnB = On BIn = InB = B

  14. Matrix multiplication in terms of columns (a) A: mn, B: nrLet the columns of B be the matrices B1, B2, …, Br. Write B=[B1B2 … Br]. Thus AB=A[B1B2 … Br]=[AB1AB2 … ABr]. Example   and

  15. Matrix multiplication in terms of columns (b) A: mn, B: n1, where A=[A1A2 … An] and . We get . Example  

  16. Partitioning of Matrices A matrix can be subdivided into a number of submatrices. Example where Example 

  17. Example 5 Let Consider the following partition of A. Under this partition A is interpreted as a 22 matrix. For the product ABto be exist, B must be partitioned into a matrix having two rows. Let  隨堂作業：21(a)

  18. Exercise 17 Let A be a matrix whose third row is all zeros. Let B be any matrix such that the product AB exists. Prove that the third row of AB is all zeros. Homework • Exercise 2.1:5, 8, 11, 17, 21 Solution

  19. 2.2 Algebraic Properties of Matrix Operations Theorem 2.2 -1 Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed. Properties of Matrix Addition and scalar Multiplication 1. A + B = B + A Commutative property of addition 2. A + (B + C) = (A + B) + C Associative property of addition 3. A + O = O + A = A (whereO is the appropriate zero matrix) 4. c(A + B) = cA + cB Distributive property of addition 5. (a + b)C = aC + bC Distributive property of addition 6. (ab)C = a(bC)

  20. Theorem 2.2 -2 Let A, B, and C be matrices and a, b, and c be scalars. Assume that the size of the matrices are such that the operations can be performed. Properties of Matrix Multiplication 1. A(BC) = (AB)C Associative property of multiplication 2. A(B + C) = AB + AC Distributive property of multiplication 3. (A + B)C = AC + BCDistributive property of multiplication 4. AIn = InA = A (whereIn is the appropriate zero matrix) 5. c(AB) = (cA)B = A(cB) Note: AB BA in general. Multiplication of matrices is not commutative.

  21. Proof of Thm 2.2 (A+B=B+A) Consider the (i,j)th elements of matrices A+B and B+A:  A+B=B+A Example 1

  22. Compute ABC. Example 2 Sol. ABC = D22 23 31 21 ABC = (AB)C = A(BC)

  23. Caution • In algebra we know that the following cancellation laws apply. • If ab = ac and a  0 then b = c. • If pq = 0 then p = 0 or q = 0. • However the corresponding results are not true for matrices. • AB = ACdoes not imply that B = C. • PQ = Odoes not imply that P = O or Q = O. Example

  24. Powers of Matrices Definition If A is a square matrix, then Theorem 2.3 If A is an n n square matrix and r and s are nonnegative integers, then 1. ArAs = Ar+s. 2. (Ar)s = Ars. 3. A0 = In (by definition)

  25. Example 4 Simplify the following matrix expression. 注意這兩項不能合併！ Solution Example 3 Solution

  26. Systems of Linear Equations A system of m linear equations in n variables as follows Let We can write the system of equations in the matrix form AX = B 隨堂作業：32(c)

  27. Solutions to Systems of Linear Equations Consider a homogeneous system of linear equations AX=0. Let X1 and X2 be solutions. ThenAX1=0 and AX2=0  A(X1 + X2) = AX1 + AX2 = 0  X1 + X2 is also a solution If c is a scalar, A(cX1) = cAX1 = 0  cX1 is also a solution Note.The set of solutions to a homogeneous system of linear equationsis closed under addition and under scalar multiplication. It is a subspace.

  28. Example 5 Consider the following homogeneous system of linear equations. It can be shown that there are many solutions,x1=2r, x2=3r, x3 = r. The solutions are vectors in R3 of the form (2r,3r, r) or r(2, 3, 1). The solutions form a subspace of R3 of dimension 1. Note. x1=0, …, xn = 0, is a solution to every homogeneous system. The set of solutions to every homogeneous system passes through the origin. Figure 2.4 隨堂作業：36(a)

  29. Solutions to Nonhomogeneous systems Exercise 41 Show that a set of solutions to a system of nonhomogeneous linear equations is not closed under addition or under scalar multiplication, and that it is thus not a subspace of Rn. Proof Let AX=Y be a nonhomogeneous system of linear equations, so Y0. Let X1 and X2 be two solutions. ThenAX1=Y and AX2=Y  A(X1 + X2) = AX1 + AX2 = 2Y X1 + X2 is not a solution. (可省略) If c is a scalar, A(cX1) = cAX1 = cY cX1 is not a solution. The set of solutions is not a subspace of Rn.

  30. Example Consider the following nonhomogeneous system of linear equations. The general solution of this system is (2r+4, 3r+2, r). (2r+4, 3r+2, r) = r(2, 3, 1)+(4, 2, 0) Note that r(2, 3, 1) is the general solution of the corresponding homogeneous system. The vector (4, 2, 0) is the specific solution to the nonhomogeneous system corresponding to r=0. Figure 2.5 隨堂作業：40

  31. Idempotent and Nilpotent Matrices(Exercise 24~30) Definition • A square matrix A is said to be idempotent (等冪) if A2=A. • A square matrix A is said to nilpotent (冪零) if there is a positive integer p such that Ap=0. The least integer p such that Ap=0 is called the degree of nilpotency of the matrix. Example 隨堂作業：28, 29(b)

  32. Exercise 21 A, B: diagonalmatrix of the same size, c: scalar Prove that A+B and cA are diagonal matrices. Homework • Exercise 2.2:6, 13, 21, 25, 27, 28, 29, 32, 36, 40, 41

  33. Exercise 25 Exercise 27 Determine b, c, and d such that is idempotent. Prove that if A and B are idempotent and AB=BA, then AB is idempotent. Homework

  34. Example 1 2.3 Symmetric Matrices Definition Thetranspose (轉置) of a matrix A, denoted At, is the matrix whose columns are the rows of the given matrix A.

  35. Theorem 2.4 Properties of Transpose Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed. 1. (A + B)t = At + Bt Transpose of a sum 2. (cA)t = cAt Transpose of a scalar multiple 3. (AB)t = BtAt Transpose of a product 4. (At)t = A

  36. Theorem 2.4 Properties of Transpose Proof for 3. (AB)t= BtAt

  37. match match Symmetric Matrix Definition A symmetric matrix is a matrix that is equal to its transpose. Example 隨堂作業：2(c)

  38. Example 3 Let A and B be symmetric matrices of the same size. Let C be a linear combination of A and B. Prove that the product C is symmetric. Proof Let C = aA+bB, where a and b are scalars. Ct = (aA+bB)t = (aA)t + (bB)t by Thm 2.4 (1) = aAt + bBtby Thm 2.4 (2) =aA+ bB since A and B are symmetric = C Thus C is symmetric.

  39. If and only if • Let p and q be statements.Suppose that p implies q (if p then q), written p q,and that also q  p, we say that“pif and only ifq” (若且唯若) (通常簡寫為 iff ) (也說成: p is necessary and sufficient for q)

  40. Example 4 Let A and B be symmetric matrices of the same size. Prove that the product AB is symmetric if and only if AB = BA. Proof *We have to show (a) AB is symmetric AB = BA, and the converse, (b) AB is symmetric AB = BA. () Let AB be symmetric, then AB= (AB)tby definition of symmetric matrix = BtAt by Thm 2.4 (3) = BA since A and B are symmetric () Let AB = BA, then (AB)t = (BA)t = AtBt by Thm 2.4 (3) = AB since A and B are symmetric

  41. Example 3 相關題目 Let A be a symmetric matrix. Prove that A2 is symmetric. Proof

  42. Example 5 Determine the trace of the matrix Solution We get Trace of a matrix Definition Let A be a square matrix. The trace of A, denoted tr(A) is the sum of the diagonal elements of A. Thus if A is an n n matrix. tr(A) = a11 + a22 + … + ann 隨堂作業：15(b)

  43. Theorem 2.5 Properties of Trace Let A and B be matrices and c be a scalar. Assume that the sizes of the matrices are such that the operations can be performed. 1. tr(A + B) = tr(A) + tr(B) 2. tr(AB) = tr(BA) 3. tr(cA) = c tr (A) 4. tr(At) = tr(A) Proof of (1) Since the diagonal element of A + B are (a11+b11), (a22+b22), …, (ann+bnn), we get tr(A + B) = (a11 + b11) + (a22 + b22) + …+ (ann + bnn) = (a11 + a22 + … + ann) + (b11 + b22 + … + bnn) = tr(A) + tr(B).

  44. Example of (2) tr(AB)=tr(BA)

  45. The conjugate (共軛複數) of a complex number z = a + bi isdefined and written z = a- bi. Matrices with Complex Elements The element of a matrix may be complex numbers. A complex number is of the form z = a + bi Where a and b are real numbers and a is called the real part and b the imaginary part (虛部) of z.

  46. Compute A + B, 2A, and AB. Example 7 Solution

  47. Definition (i) The conjugate of a matrix A is denoted A and is obtained by taking the conjugate of each element of the matrix. (ii) The conjugate transpose of A is written and defined by A*=At. (iii) A square matrix C is said to be hermitian if C=C*. Example (i), (ii) Example (iii) 隨堂作業：23

  48. Exercise 11 A matrix A is said to be antisymmetric if A = -At. (a) give an example of an antisymmetric matrix.(b) Prove that an antisymmetric matrix is a square matrix having diagonal elements zero.(c) Prove that the sum of two antisymmetric matrices of the same size is an antisymmetric matrix. Homework • Exercise 2.3:2, 5, 11, 15, 23

  49. Example 1 Prove that the matrix has inverse Proof 2.4 The Inverse of a Matrix Definition Let A be an n n matrix. If a matrix B can be found such that AB = BA = In, then A is said to be invertible(可逆) and B is called the inverse(反矩陣) of A. If such a matrix B does not exist, then A has no inverse. (denote B = A-1, andA-k=(A-1)k ) Thus AB = BA = I2, proving that the matrix A has inverse B.

  50. Thm2.2 乘法結合律 Theorem 2.7 If a matrix has an inverse, that inverse is unique. Proof Let B and C be inverses of A. Thus AB = BA = In, and AC = CA = In. Multiply both sides of the equation AB = In by C. C(AB) = CIn (CA)B = C InB = C B = C Thus an invertible matrix has only one inverse.

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