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Econ 306 Lecture 1 Matrix Algebra Lecture Outline 1. DEFINITION 2. ADDITION AND SUBTRACTION 3. MULTIPLICATION 4. THE TRACE OF A MATRIX 5. THE DETERMINANT OF A SQUARE MATRIX 6. THE RANK OF A MATRIX 7. INVERSE MATRICIES AND GENERASLISED INVERSES 8. IDEMPOTENT MATRICES 9. KRONECKER PRODUCT a11 a 22 . A= . am1 a12 . . a1n a22 . . . . . . . . . . . . . . amn = A[aij] To transpose a matrix: a11 a12 A a 21 a 22 a13 a 23 a11 A AT a12 a13 a 21 a 22 a 23 If A = AT the matrix is SYMMETRIC A special symmetric matrix is the IDENTITY MATRIX 1 0 . . 0 0 1 . . . I . . . . . . . . . . 0 0 . . 1 1 0 I2 0 1 The identity matrix is also an example of a DIAGONAL MATRIX diag (a1 , a 2 ,..., a n ) For A = B both matrices must be of the same size and aij = bij Equality is TRANSITIVE C=A+B cij=aij+bij 6 8 A 2 4 0 3 B 9 7 6 5 C 7 11 Matrices must be CONFORMABLE Matrix addition and subtraction are COMMUTATIVE A B B A and ASSOCIATIVE ( A B) C A ( B C ) also ( A B) A B T T T SCALAR multiplication B = kA bij = kaij 1 2 A 3 4 k=2 2 4 B 6 8 Two forms of multiplication C = AB C = BA A premultiplies B A postmultiplies B For multiplication to be possible the number of columns in A must equal the number of rows in B A = (m x p) B = (p x n) p cij aik bkj k 1 C = AB = (m x n) 6 8 A 2 4 3 8 1 D 9 2 5 F = AD (6 8) (8 2) (6 1) (8 5) (6 3) (8 9) (2 3) (4 9) (2 8) (4 2) (2 1) (4 5) 90 32 46 30 24 18 In general AB BA except when Both matrices are diagonal Scalar multiplication When one matrix is an identity matrix AN INNER PRODUCT is the product of a row vector and a column vector J = (T x 1) with all elements 1 JTJ = T Two vectors are ORTHOGONAL IF THEIR INNER PRODUCT =0 2 a 5 5 b 2 aTb = 0 The inner product is commutative aTb = bTa √of the inner product of a vector with itself is the EUCLIDEAN VECTOR NORM For multiplication the DISTRIBUTIVE law holds A(B + C) = AB + AC As does the associative law A(BC) = (AB)C The CANCELLATION law does not hold If AB = 0, then it does not follow that A or B or both are 0 1 2 2 4 A B 1 2 3 6 The TRANSPOSE OF A PRODUCT (AB)T=BTAT If A = AT and B = BT then (AB)T = BTAT = BA does not necessarily = AB If A is symmetric but B is not note that BTAB is symmetric since (BTAB)T = BTATB = BTAB If A = I then BTB is symmetric The TRACE of a matrix tr(A) = a11 + a22 + …. + ann tr(AT) = tr(A) tr(kA) = ktr(A) tr(In) = n If A and B are square tr(A + B) = tr(A) + tr(B) tr(AB) = tr(BA) tr(ABC) = tr(BCA) = tr(CAB) The DETERMINANT of a matrix det A or │A│ a11 a12 A a a 22 21 │A│= a11 - a12 a11 A a 21 a31 a12 a 22 a32 a13 a 23 a33 A a11 a22 a23 a32 a33 a12 a21 a23 a31 a33 a13 a21 a22 a31 a32 a11( a22a33-a23a32) - a12(a21a33-a23a31) + a13(a21a32-a22a31) ( a22a33-a23a32) is called the minor of a11 and is usually denoted│Aij│ - in this case │A11│ 3 5 4 A 6 9 7 2 8 1 9 7 6 7 6 9 A 3 5 4 8 1 2 1 2 8 = 3(9 – 56) - 5(6 – 14) + 4(48 – 18) = -141 + 40 + 120 = 19 The COFACTOR of the elements of aij denoted by cij is cij = (-1)i+j│Aij│ n n j 1 i A aijcij aijcij PROPERTIES OF DETERMINANTS 1. 2. 3. │AT│=│A│ a11 ka12 a11 a12 ka a21 ka22 a 21 a 22 If A is (n x n) then │kA│ = kn│A│ 4. If a square matrix has two equal rows or columns its determinant is zero 5. If any row (or column) is the multiple of any other row (or column) then its determinant is zero 6. The value of a determinant is unchanged if a multiple of one row (or column) is added to another row (or column) 7. If A is a diagonal matrix of order n then its determinant is a11a22 …. ann 8. If A is a triangular matrix of order n then its determinant is a11a22 …. ann 9. If B is the matrix obtained from a square matrix A by interchanging any two rows (or columns) then det B = -det A 10. If A and B are square matrices of the same order then │AB│ = │A││B│ 11. If A1, A2 , …. , As are square matrices then │diag(A1, A2 , …. , As )│ = │A1││A2│ … │As│ 12. In general │A + B│does not equal│A│ + │B│ The RANK of a matrix is equal to the highest order nonzero determinant that can be formed from its submatrices 4 5 2 14 3 9 6 21 A 8 10 7 28 1 2 9 5 det A = 0 4 5 2 3 9 6 63 8 10 7 Rank of A = 3 The rank of a matrix can also be measured by the maximum number of linearily independent columns of A This also equals the maximum number of linearily independent rows 4 5 2 14 3 9 6 21 1 2 0 (1) 0 8 10 7 28 1 2 9 5 c1a1 + c2a2 + c3a3 + c4a4 = 0 A FULL COLUMN RANK matrix has the same number of linearily independent columns (rows) equal to the number of columns A FULL ROW RANK matrix has the same number of linearily independent rows (columns) equal to the number of rows If A does not have full row and column rank it is SINGULAR If A does have full row and column rank it is NONSINGULAR rank (In) = n rank (kA) = rank (A) rank (AT) = rank (A) If A is (m x n) then rank (A) is ≤ min {m, n} rank AB ≤ min{rank (A), rank (B)} INVERSES If A and B are matrices of order n such that AB = BA = In then B is called the inverse of A A has an inverse iff it is of full column and row rank – nonsingular A-1 = CT / │A│ CT is the transpose of the matrix of co-factors 1 2 A 3 4 │A│= 4 – 6 = -2 c11 = 4 c12 = -3 c21 = -2 c22 = 1 1 4 3 A 2 2 1 T 1 1 2 A 3 / 2 1 / 2 1 PROPERTIES OF INVERSES 1. I-1 = I 2. (A-1)-1 = A 3. AB = I BA = I 4. A non-singular A-1 non-singular 5. A and B non-singular (AB)-1 = B-1A-1 LEFT INVERSE of a (m x n) matrix A is the (n x m) matrix B such that BA = In RIGHT INVERSE of a (m x n) matrix A is the (n x m) matrix C such that AC = Im (T x k) design matrix X which has rank k < T has an infinite number of left inverses including (XTX)-1XT IDEMPOTENT MATRICES AA = A 0.4 0.8 A 0.3 0.6 Consider M = [I – X(XTX)-1XT] KRONECKER PRODUCT A B 1 3 A 2 0 2 2 0 B 1 0 3 2 1 1 A B 2 2 1 2 0 2 3 0 3 1 2 0 2 0 0 3 1 2 1 4 2 6 0 0 0 2 0 4 0 0 3 0 6 6 3 0 0 0 9 0 0 2 0 0 3 2 0 0 3