Matrices PDF Lecture Notes

Introduction Some Special Matrices Complex Matrices MATRICES Dr. Gabriel Obed Fosu Department of Mathematics Kwame Nkrumah University of Science and Technology March 9, 2022 Dr. Gabriel Obed Fosu...

Introduction Some Special Matrices Complex Matrices MATRICES Dr. Gabriel Obed Fosu Department of Mathematics Kwame Nkrumah University of Science and Technology March 9, 2022 Dr. Gabriel Obed Fosu 1/35  Introduction Some Special Matrices Complex Matrices Outline 1 Introduction Matrix Arithmetic and Properties 2 Some Special Matrices Nonsingular Matrices Symmetric Matrices Orthogonal and Orthonormal Matrix 3 Complex Matrices Hermitian Matrices Dr. Gabriel Obed Fosu 2/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Outline of Presentation 1 Introduction Matrix Arithmetic and Properties 2 Some Special Matrices Nonsingular Matrices Symmetric Matrices Orthogonal and Orthonormal Matrix 3 Complex Matrices Hermitian Matrices Dr. Gabriel Obed Fosu 3/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition 1 A matrix is a rectangular array of numbers, symbols, or anything with m rows and n columns which is used to represent a mathematical object or a property of such an object. The symbol Rm×n denotes the collection of all m × n matrices whose entries are real numbers. Matrices represent linear maps. Dr. Gabriel Obed Fosu 4/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition 1 A matrix is a rectangular array of numbers, symbols, or anything with m rows and n columns which is used to represent a mathematical object or a property of such an object. The symbol Rm×n denotes the collection of all m × n matrices whose entries are real numbers. Matrices represent linear maps. 2 Matrices will usually be denoted by capital letters, and the notation A = [aij ] specifies that the matrix is composed of entries aij located in the ith row and jth column of A. Example of 2 × 3 matrix   1 2 −9 A= 3 −3 2 Dr. Gabriel Obed Fosu 4/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition 1 A matrix is a rectangular array of numbers, symbols, or anything with m rows and n columns which is used to represent a mathematical object or a property of such an object. The symbol Rm×n denotes the collection of all m × n matrices whose entries are real numbers. Matrices represent linear maps. 2 Matrices will usually be denoted by capital letters, and the notation A = [aij ] specifies that the matrix is composed of entries aij located in the ith row and jth column of A. Example of 2 × 3 matrix   1 2 −9 A= 3 −3 2 3 A vector is a matrix with either one row or one column. Column vector (3 × 1):   1 x = −3 7 Row vector (1 × 4):   y = 6 Dr. Gabriel Obed Fosu 1 4/350 −12  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Square and Zero Matrices Definition (Square matrix) Matrices of size (n, n) are called square matrices or n-square matrices of order n. Examples of 2 × 2 and 3 × 3 matrices are respectively given as     1 2 3 1 2 4 , 5 6 3 4 7 8 8 Dr. Gabriel Obed Fosu 5/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Square and Zero Matrices Definition (Square matrix) Matrices of size (n, n) are called square matrices or n-square matrices of order n. Examples of 2 × 2 and 3 × 3 matrices are respectively given as     1 2 3 1 2 4 , 5 6 3 4 7 8 8 Definition (The zero matrix) Each m × n matrix, all of whose elements are zero, is called the zero matrix (of size m × n) and is denoted by the symbol 0.   0 0 S= 0 0 Dr. Gabriel Obed Fosu 5/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Identity Matrix Definition (The identity matrix) The n × n matrix I = [δij ], defined by δij = 1 if i = j, and δij = 0 if i 6= j, is called the n × n identity matrix of order n. Dr. Gabriel Obed Fosu 6/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Identity Matrix Definition (The identity matrix) The n × n matrix I = [δij ], defined by δij = 1 if i = j, and δij = 0 if i 6= j, is called the n × n identity matrix of order n. Example of 3 × 3 identity matrix is   1 0 0 I = 0 1 0 0 0 1 When any n × n matrix A is multiplied by the identity matrix, either on the left or the right, the result is A. Thus, the identity matrix acts like 1 in the real number system. Dr. Gabriel Obed Fosu 6/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition (Equality of matrices) Matrices A and B are said to be equal if they have the same size and their corresponding elements are equal; i.e., A and B have dimension m × n, and A = [aij ], B = [bij ], with aij = bij for 1 ≤ i ≤ m, 1 ≤ j ≤ n. Dr. Gabriel Obed Fosu 7/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition (Equality of matrices) Matrices A and B are said to be equal if they have the same size and their corresponding elements are equal; i.e., A and B have dimension m × n, and A = [aij ], B = [bij ], with aij = bij for 1 ≤ i ≤ m, 1 ≤ j ≤ n. Definition (Addition of matrices) Let A = [aij ] and B = [bij ] be of the same size. Then A + B is the matrix obtained by adding corresponding elements of A and B; that is, A + B = [aij ] + [bij ] = [aij + bij ] Dr. Gabriel Obed Fosu 7/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition (Equality of matrices) Matrices A and B are said to be equal if they have the same size and their corresponding elements are equal; i.e., A and B have dimension m × n, and A = [aij ], B = [bij ], with aij = bij for 1 ≤ i ≤ m, 1 ≤ j ≤ n. Definition (Addition of matrices) Let A = [aij ] and B = [bij ] be of the same size. Then A + B is the matrix obtained by adding corresponding elements of A and B; that is, A + B = [aij ] + [bij ] = [aij + bij ] Definition (Scalar multiple of a matrix) Let A = [aij ] and t be a number (scalar). Then tA is the matrix obtained by multiplying all elements of A by t; that is, tA = t[aij ] = [taij ] Dr. Gabriel Obed Fosu 7/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition (Negative of a matrix) Let A = [aij ]. Then −A is the matrix obtained by replacing the elements of A by their negatives. Dr. Gabriel Obed Fosu 8/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Definition (Negative of a matrix) Let A = [aij ]. Then −A is the matrix obtained by replacing the elements of A by their negatives. Definition (Subtraction of matrices) Matrix subtraction is defined for two matrices A = [aij ] and B = [bij ] of the same size, in the usual way; that is, A − B = [aij ] − [bij ] Dr. Gabriel Obed Fosu 8/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Example If     1 2 5 6 A= ,B = 3 4 0 −1   6 8 A+B = 3 3 Dr. Gabriel Obed Fosu 9/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Example If     1 2 5 6 A= ,B = 3 4 0 −1   6 8 A+B = 3 3   −4 −4 A−B = 3 5 Dr. Gabriel Obed Fosu 9/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Example If     1 2 5 6 A= ,B = 3 4 0 −1   6 8 A+B = 3 3   −4 −4 A−B = 3 5   10 12 2B = 0 −2 Dr. Gabriel Obed Fosu 9/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Properties The matrix operations of addition, scalar multiplication, negation and subtraction satisfy the following laws of arithmetic. Let s and t be arbitrary scalars and A, B, C be matrices of the same size 1 (A + B) + C = A + (B + C) 2 A+B =B+A 3 0+A=A 4 A + (−A) = 0 5 (s + t)A = sA + tA, (s − t)A = sA − tA 6 t(A + B) = tA + tB, t(A − B) = tA − tB 7 s(tA) = (st)A 8 1A = A, 0A = 0, (−1)A = −A 9 tA = 0 =⇒ t = 0 or A = 0 Dr. Gabriel Obed Fosu 10/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Matrix Product Let A = [aij ] be a matrix of size m × p and B = [bjk ] be a matrix of size p × n (i.e., the number of columns of A equals the number of rows of B). Then the product AB is an m × n matrix. That is, if     a b e f A= ,B = c d g h then   ae + bg af + bh AB = ce + dg cf + dh Dr. Gabriel Obed Fosu 11/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Matrix Product Let A = [aij ] be a matrix of size m × p and B = [bjk ] be a matrix of size p × n (i.e., the number of columns of A equals the number of rows of B). Then the product AB is an m × n matrix. That is, if     a b e f A= ,B = c d g h then   ae + bg af + bh AB = ce + dg cf + dh Example      1 2 5 6 19 22 = 3 4 7 8 43 50     1   3 4 = 11 2 Matrix multiplication is not commutative AB 6= BA Dr. Gabriel Obed Fosu 11/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Trace Definition (Trace) If A is an n × n matrix, the trace of A, written trace(A), is the sum of the main diagonal elements; that is, Xn trace(A) = a11 + a22 + · · · + ann = aii i=1 Dr. Gabriel Obed Fosu 12/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Trace Definition (Trace) If A is an n × n matrix, the trace of A, written trace(A), is the sum of the main diagonal elements; that is, Xn trace(A) = a11 + a22 + · · · + ann = aii i=1 Example If     1 2 5 6 A= ,B = 3 4 0 −1 then trace(A) = 1 + 4 = 5 trace(B) = 5 + (−1) = 4 Dr. Gabriel Obed Fosu 12/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Properties of Trace trace(A + B) = trace(A) + trace(B) trace(cA) = c · trace(A), where c is a scalar. trace(AB) = trace(BA) Dr. Gabriel Obed Fosu 13/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Power of a Matrix Definition (kth power of a matrix) If A is an n × n matrix, we define Ak as follows: A0 = I and Ak = A × A × A · · · A × A; A occurs k times for k ≥ 1. Example A4 = A × A × A × A Dr. Gabriel Obed Fosu 14/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices The transpose of a matrix Definition (The transpose of a matrix) Let A be an m×n matrix. Then AT , the transpose of A, is the matrix obtained by interchanging the rows and columns of A. In other words if A = [aij ], then (AT )ij = aji. Dr. Gabriel Obed Fosu 15/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices The transpose of a matrix Definition (The transpose of a matrix) Let A be an m×n matrix. Then AT , the transpose of A, is the matrix obtained by interchanging the rows and columns of A. In other words if A = [aij ], then (AT )ij = aji. If   1 2 3 A= 0 −1 15 then   1 0 AT = 2 −1 3 15 Dr. Gabriel Obed Fosu 15/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Properties of Transpose 1 (AT )T = A Dr. Gabriel Obed Fosu 16/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Properties of Transpose 1 (AT )T = A 2 (A ± B)T = AT ± B T if A and B are m × n Dr. Gabriel Obed Fosu 16/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Properties of Transpose 1 (AT )T = A 2 (A ± B)T = AT ± B T if A and B are m × n 3 (sA)T = sAT if s is a scalar Dr. Gabriel Obed Fosu 16/35  Introduction Some Special Matrices Matrix Arithmetic and Properties Complex Matrices Properties of Transpose 1 (AT )T = A 2 (A ± B)T = AT ± B T if A and B are m × n 3 (sA)T = sAT if s is a scalar 4 (AB)T = B T AT if A is m × k and B is k × n Dr. Gabriel Obed Fosu 16/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Outline of Presentation 1 Introduction Matrix Arithmetic and Properties 2 Some Special Matrices Nonsingular Matrices Symmetric Matrices Orthogonal and Orthonormal Matrix 3 Complex Matrices Hermitian Matrices Dr. Gabriel Obed Fosu 17/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Diagonal Matrix) The aii , 1 ≤ i ≤ n, entries of a square matrix are called the diagonal elements. If the nondiagonal elements are all zero, then the matrix is called a diagonal matrix. It is denoted by A = diag(a11 , a22 ,... , ann ). Some examples are     14 0 0 10 0  0 −9 0 , 0 −13 0 0 3 Dr. Gabriel Obed Fosu 18/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Bidiagonal Matrix) A bidiagonal matrix is a matrix with nonzero entries along the main diagonal and either the diagonal above or the diagonal below. Dr. Gabriel Obed Fosu 19/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Bidiagonal Matrix) A bidiagonal matrix is a matrix with nonzero entries along the main diagonal and either the diagonal above or the diagonal below. The matrix B1 is an upper bidiagonal matrix.   5 1 0 0 0 10 −1 0   0 0  (1) 9 2 0 0 0 −2 Dr. Gabriel Obed Fosu 19/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Bidiagonal Matrix) A bidiagonal matrix is a matrix with nonzero entries along the main diagonal and either the diagonal above or the diagonal below. The matrix B1 is an upper bidiagonal matrix. The matrix B2 is a lower bidiagonal matrix.     5 1 0 0 5 0 0 0 0 10 −1 0  2 10 0 0  0 0  (1)   (2) 9 2 0 9 9 0 0 0 0 −2 0 0 −1 −2 Dr. Gabriel Obed Fosu 19/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Tridiagonal Matrix) A tridiagonal matrix has only nonzero entries along the main diagonal and the diagonals above and below. Dr. Gabriel Obed Fosu 20/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Tridiagonal Matrix) A tridiagonal matrix has only nonzero entries along the main diagonal and the diagonals above and below. T is a 5 × 5 tridiagonal matrix.   2 1 0 0 0 3 4 5 0 0 (3)   0 A= −1 −3 2 0 0 0 1 2 10 0 0 0 −6 7 Dr. Gabriel Obed Fosu 20/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Nonsingular matrix Definition (Nonsingular matrix) An n × n matrix A is called nonsingular or invertible if there exists an n × n matrix B such that AB = BA = I The matrix B is the inverse of A. If A does not have an inverse, A is called singular. Dr. Gabriel Obed Fosu 21/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Nonsingular matrix Definition (Nonsingular matrix) An n × n matrix A is called nonsingular or invertible if there exists an n × n matrix B such that AB = BA = I The matrix B is the inverse of A. If A does not have an inverse, A is called singular. Properties: Let denote the inverse of A by A−1 , then 1 (A−1 )A = I = A(A−1 ) Dr. Gabriel Obed Fosu 21/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Nonsingular matrix Definition (Nonsingular matrix) An n × n matrix A is called nonsingular or invertible if there exists an n × n matrix B such that AB = BA = I The matrix B is the inverse of A. If A does not have an inverse, A is called singular. Properties: Let denote the inverse of A by A−1 , then 1 (A−1 )A = I = A(A−1 ) 2 (A−1 )−1 = A Dr. Gabriel Obed Fosu 21/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Nonsingular matrix Definition (Nonsingular matrix) An n × n matrix A is called nonsingular or invertible if there exists an n × n matrix B such that AB = BA = I The matrix B is the inverse of A. If A does not have an inverse, A is called singular. Properties: Let denote the inverse of A by A−1 , then 1 (A−1 )A = I = A(A−1 ) 2 (A−1 )−1 = A 3 (AB)−1 = B −1 A−1 Dr. Gabriel Obed Fosu 21/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Nonsingular matrix Definition (Nonsingular matrix) An n × n matrix A is called nonsingular or invertible if there exists an n × n matrix B such that AB = BA = I The matrix B is the inverse of A. If A does not have an inverse, A is called singular. Properties: Let denote the inverse of A by A−1 , then 1 (A−1 )A = I = A(A−1 ) 2 (A−1 )−1 = A 3 (AB)−1 = B −1 A−1 4 if A is nonsingular, then AT is also nonsingular and (AT )−1 = (A−1 )T Dr. Gabriel Obed Fosu 21/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Nonsingular matrix Definition (Nonsingular matrix) An n × n matrix A is called nonsingular or invertible if there exists an n × n matrix B such that AB = BA = I The matrix B is the inverse of A. If A does not have an inverse, A is called singular. Properties: Let denote the inverse of A by A−1 , then 1 (A−1 )A = I = A(A−1 ) 2 (A−1 )−1 = A 3 (AB)−1 = B −1 A−1 4 if A is nonsingular, then AT is also nonsingular and (AT )−1 = (A−1 )T Homogeneous system A linear system Ax = 0 is said to be homogeneous. If A is nonsingular, then x = A−1 (0) = 0, so the system has only 0 as its solution. It is said to have only the trivial solution. Dr. Gabriel Obed Fosu 21/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Symmetric and Skew Symmetric matrix Symmetric A matrix A is symmetric if AT = A. Another way of looking at this is that when the rows and columns are interchanged, the resulting matrix is A. Dr. Gabriel Obed Fosu 22/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Symmetric and Skew Symmetric matrix Symmetric A matrix A is symmetric if AT = A. Another way of looking at this is that when the rows and columns are interchanged, the resulting matrix is A. Skew symmetric A matrix A is skew symmetric if AT = −A. Dr. Gabriel Obed Fosu 22/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Symmetric and Skew Symmetric matrix Symmetric A matrix A is symmetric if AT = A. Another way of looking at this is that when the rows and columns are interchanged, the resulting matrix is A. Skew symmetric A matrix A is skew symmetric if AT = −A. Example     2 −3 4 0 2 1 The matrix A = −3 1 2 is symmetric and B = −2 0 −3 is skew symmetric. 4 2 3 −1 3 0 Dr. Gabriel Obed Fosu 22/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Symmetric Definite Matrices Definition (Symmetric Positive Definite Matrix)   x1  x2  A symmetric matrix A is positive definite if for every nonzero vector x = .    ..  xn xT Ax > 0 (4) The expression xT Ax is called the quadratic form associated with A. Note The sum of two positive definite matrices is positive definite. Dr. Gabriel Obed Fosu 23/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Symmetric Positive Semidefinite Matrix)   x1  x2  A is symmetric positive semidefinite if for every nonzero vector x = .    ..  xn xT Ax ≥ 0 (5) Dr. Gabriel Obed Fosu 24/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Example   1 0 The symmetric matrix A = is positive definite because for 0 1     x 0 x = 1 6= (6) x2 0 then    1 0 x1 xT Ax = [x1 x2 ] (7) 0 1 x2 = x21 + x22 (8) Since x21 + x22 > 0 then A is a symmetric positive definite matrix. Dr. Gabriel Obed Fosu 25/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Theorem 1 If A = (aij ) is positive definite, then aii > 0 for all i. 2 If A = (aij ) is positive definite, then the largest element in magnitude of all matrix entries must lie on the diagonal. Example   1 2 3 The matrix A = 4 0 1 cannot be positive definite because A has a diagonal element of 0 2 5 6 Example   1 −1 0 9 8 45 3 19 The matrix B =  0 15 16 35 cannot be positive definite because the largest element in  3 −55 2 22 magnitude (−55) is not on the diagonal of B. Dr. Gabriel Obed Fosu 26/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Theorem Suppose that a real symmetric tridiagonal matrix   b1 a1 a1 b2 a2   ... a2.. (9)   A= .   ... b   n−1 an−1  an−1 bn with diagonal entries all positive is strictly diagonally dominant, that is, bi > |ai−1 | + |ai |, 1≤i≤n Then A is positive definite. Dr. Gabriel Obed Fosu 27/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Symmetric Negative Definite Matrix)   x1  x2  A is symmetric negative definite if for every nonzero vector x = .    ..  xn xT Ax ≤ 0 (10) In this case, −A is positive definite. Definition (Symmetric Indefinite Matrix) A is symmetric indefinite if xT Ax assumes both positive and negative values. Alternatively, a matrix is symmetric indefinite if it has both positive and negative eigenvalues. Dr. Gabriel Obed Fosu 28/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Orthogonal Matrix) A matrix P is orthogonal if PTP = I (11) The inverse of P is its transpose. Alternatively, P is orthogonal if and only if the columns of P are orthogonal and have unit length. ( 0 if i = j hvi , vj i = (12) 1 if i 6= j Dr. Gabriel Obed Fosu 29/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Example   1 0 0 D = [e1 e2 e3 ] = 0 1 0 (13) 0 0 1 then he1 , e2 i = he1 , e3 i = he2 , e3 i = 0 (14) So the columns are orthogonal, and each has a unit length q √ |e1 | = |e2 | = |e3 | = a211 + a221 + a231 = 12 = 1 (15) Hence D is an orthogonal matrix. Dr. Gabriel Obed Fosu 30/35  Introduction Nonsingular Matrices Some Special Matrices Symmetric Matrices Complex Matrices Orthogonal and Orthonormal Matrix Definition (Orthonormal) 1 A set of orthogonal vectors, each with unit length, are said to be orthonormal. 2 D = [e1 e2 e3 ] is an orthogonal matrix, and each orthogonal vector e1 , e2 and e2 has a unit length. 3 Hence e1 , e2 , · · · , en are called the standard orthonormal basis. Dr. Gabriel Obed Fosu 31/35  Introduction Some Special Matrices Hermitian Matrices Complex Matrices Outline of Presentation 1 Introduction Matrix Arithmetic and Properties 2 Some Special Matrices Nonsingular Matrices Symmetric Matrices Orthogonal and Orthonormal Matrix 3 Complex Matrices Hermitian Matrices Dr. Gabriel Obed Fosu 32/35  Introduction Some Special Matrices Hermitian Matrices Complex Matrices If a = 1 + 3i then the complex conjugate is ā = 1 − 3i Dr. Gabriel Obed Fosu 33/35  Introduction Some Special Matrices Hermitian Matrices Complex Matrices If a = 1 + 3i then the complex conjugate is ā = 1 − 3i Definition (Complex conjugate matrix) The complex conjugate of an n × m complex matrix A = (zij ) is defined and denoted by Ā = (z̄ij )m×n. Dr. Gabriel Obed Fosu 33/35  Introduction Some Special Matrices Hermitian Matrices Complex Matrices If a = 1 + 3i then the complex conjugate is ā = 1 − 3i Definition (Complex conjugate matrix) The complex conjugate of an n × m complex matrix A = (zij ) is defined and denoted by Ā = (z̄ij )m×n. Definition (Hermitian and Skew Hermitian matrix) A complex n-square matrix A is said to be hermitian if ĀT = A or z̄ji = zij (16) and skew hermitian if ĀT = −A or z̄ji = −zij (17) Dr. Gabriel Obed Fosu 33/35  Introduction Some Special Matrices Hermitian Matrices Complex Matrices If a = 1 + 3i then the complex conjugate is ā = 1 − 3i Definition (Complex conjugate matrix) The complex conjugate of an n × m complex matrix A = (zij ) is defined and denoted by Ā = (z̄ij )m×n. Definition (Hermitian and Skew Hermitian matrix) A complex n-square matrix A is said to be hermitian if ĀT = A or z̄ji = zij (16) and skew hermitian if ĀT = −A or z̄ji = −zij (17) Example (M is Hermitian and N is skew Hermitian)     2 1−i 0 i 2+i 3 + 2i M = 1 + i −1 i  N =  −2 + i 3i −3i  0 −i 2 −3 + 2i −3i 0 Dr. Gabriel Obed Fosu 33/35  Introduction Some Special Matrices Hermitian Matrices Complex Matrices Exercises 1 Let A, B, C, D be matrices defined by       3 0 1 5 2 −3 −1   4 −1 A = −1 2 , B = −1 1 0 , C =  2 1 ,D =  2 0 1 1 −4 1 3 4 3 Which of the following matrices are defined? Compute those matrices which are defined. A + B, A + C, AB, BA, CD, DC, D2 , (C T )T 2 Rotate the line y = −x + 3 60°counterclockwise about the origin. Let σ1 = ( 01 10 ) , σ2 = 0i −i and σ1 = 10 −1 0 be the three Pauli matrices. Show that

3 0 xσ1 + yσ2 + 2σ3 is a hermitian matrix for any two real numbers x, y ∈ R. Dr. Gabriel Obed Fosu 34/35  Introduction Some Special Matrices Complex Matrices END OF LECTURE THANK YOU Dr. Gabriel Obed Fosu 35/35