MATH36001 Background Material 2015


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1 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 an m n matrix These elements can be taken from an arbitrary field F However for the purpose of this course, F will always be the set of all real or all complex numbers, denoted by R and C, respectively An m n matrix may also be written in terms of its elements as A = [a ij ], where a ij ( i m, j n) denotes the element in position (i, j) Note that i is the row number, and j is the column number R m n denotes the vector space of all real m n matrices, and C m n is the vector space of all complex m n matrices If m = n the matrix is square, and in the general case it is rectangular A submatrix of A is any matrix obtained by deleting rows and columns A block matrix A A q A = A p A pq is a partitioning of A into submatrices A ij whose dimensions must be consistent A row vector [ x x 2 x n ] is a n matrix A column vector y = y y 2 y m is an m matrix Note that F m F m R m and C m denote the vector space of real and complex column vectors with m elements, respectively 2 Zero and Identity Matrices The zero matrix O m n is the m n matrix all of whose elements are zero When m and n are clear from the context we may simply write O The n n identity matrix is I n = 0 0,
2 MATH3600: Background Material Page 2 or just I if the dimension is clear from the context The jth column of I is denoted by e j, and is called the jth unit vector, that is, 0 0 I = [ e e 2 e n ], e = 0, e 2 =,, e n = Householder s Notation Generally, we use capital letters A, B, C,, Λ for matrices, lower case letters a ij, b ij, c ij, δ ij, λ ij for matrix elements, lower case letters x, y, z, c, g, h for vectors, lower case Greek letters α, β, γ, θ, π for scalars 4 Basic Manipulations with Matrices Transposition: (F m n F n m ) C = A T c ij = a ji A T has rows and columns interchanged, so it is an n m matrix Conjugate transposition: (F m n F n m ) C = A c ij = a ji, where the bar denotes complex conjugate If A R m n then A = A T Properties of transposition: (A T ) T = A, (A ) = A, (αa) T = αa T, (αa) = αa, (A + B) T = A T + B T, (A + B) = A + B, (AB) T = B T A T, (AB) = B A Addition: (F m n F m n F m n ) C = A + B c ij = a ij + b ij Scalarmatrix multiplication: (F F m n F m n ) Properties of matrix addition: C = αa c ij = αa ij A + B = B + A matrix addition is commutative (A + B) + C = A + (B + C) matrix addition is associative α(a + B) = αa + αb, (α + β)a = αa + βa matrix addition and scalar multiplication are distributive
3 MATH3600: Background Material Page 3 Matrixmatrix multiplication: (F m r F r n F m n ) r C = AB c ij = a ik b kj Properties of matrix multiplication: A(BC) = (AB)C A(B + C) = AB + AC k= matrix multiplication is associative matrix multiplication is distributive Remember that AB BA in general, that is, matrix multiplication is not commutative Matrix powers: If A is a nonzero square matrix we define A 0 I, and for any positive integer, A k = k times {}}{ A A = A k A = AA k A square matrix is involutory if A 2 = I, idempotent if A 2 = A, and nilpotent if A k = 0 for some integer k > 0 Using powers of matrices, we can also define polynomials in matrices: if p(z) = c 0 + c z + + c k z k, then given A C n n, we define p(a) C n n by 5 Inner and Outer Products p(a) = c 0 I + c A + + c k A k The inner product of two vectors x, y C n is the scalar n x y = x i y i C i= The length of a vector x is given by x x Two nonzero vectors are orthogonal if their inner product x y is zero If, in addition, x x = y y =, the vectors are orthonormal The outer product of the vectors x C m and y C n is the m n matrix x y x y n xy = C m n x m y x m y n 6 Special Matrices A diagonal matrix D has all its offdiagonal elements zero It can be written D = diag(α i ), or equivalently α D = α 2 α n
4 MATH3600: Background Material Page 4 Here and below, a blank entry is understood to be zero An upper triangular matrix U has zero elements below the diagonal, that is u ij = 0 for i > j, U =, where is not necessarily zero Similarly, the matrix L is lower triangular if all elements above the main diagonal are zero The matrix A is block diagonal, block upper triangular, block lower triangular if it has the partitioned forms A A A 2 A p A A = A 22, A = A 22, A = A 2 A 22 A pp A pp A p A pp Here the matrices A ii are all square but do not necessarily have the same size A R n n is a symmetric matrix if A T = A; A C n n is a Hermitian matrix if A = A An n n Hermitian matrix A is positive definite if x Ax > 0 for all 0 x C n positive semidefinite if x Ax 0 for all x C n indefinite if (x Ax)(y Ay) < 0 for some x, y C n An orthogonal matrix Q R n n satisfies QQ T = I and Q T Q = I, so that if Q = [q,, q n ] then qi T q j = δ ij (Kronecker delta), where δ ij = if i = j, 0 otherwise The columns of Q are mutually orthogonal and of unit length A unitary matrix U C n n satisfies UU = U U = I The permutation matrix P ij is the identity matrix with its ith and jth rows interchanged 0 ith row P ij = 0 jth row P ij A swaps the ith and jth rows of A AP ij swaps the ith and jth columns of A Note that P ij is orthogonal and involutory (P 2 ij = I)
5 MATH3600: Background Material Page 5 2 Basic Linear Algebra Definitions The rank of a matrix A, rank(a), is the maximum number of linearly independent rows or columns of A Recall that a set of vectors {v i } is linearly dependent if i α iv i = 0 for some scalar values α i not all zero, and otherwise linearly independent Two important subspaces associated with a matrix A C m n are range(a) = { y C m : y = Ax for some x C n}, null(a) = { x C n : Ax = 0 }, the null space of A If A = [a, a 2,, a n ] (ie, a j is the jth column of A), then range(a) = span{a, a 2,, a n }, rank(a) = dim(range(a)), the range of A where span(s) denotes the set of all linear combinations of vectors in the set S, and dim(v ) is the maximum number of linearly independent vectors in the vector space V For any A C m n, rank(a) + dim(null(a)) = n 3 Determinants If A = [α] C, then its determinant is given by det(a) = α The determinant of an n n matrix A can be defined in terms of order n determinants (expansion in cofactors): det(a) = = n a ij ( ) i+j det(a ij ) for any i, j= n a ij ( ) i+j det(a ij ) for any j i= Here A ij is an (n ) (n ) submatrix of A obtained by deleting the ith row and jth column The scalar ( ) i+j det A ij is called a cofactor of A Useful properties of the determinant include det(ab) = det(a) det(b), det(αa) = α n det(a) (α C) If A is block diagonal (or block triangular) with square diagonal blocks A, A 22,, A pp (of possibly different sizes) then det(a) = det(a ) det(a 22 ) det(a pp )
6 MATH3600: Background Material Page 6 4 Inverses If A, B C n n satisfy AB = I then B is the inverse of A, written B = A If A exists A is nonsingular; otherwise A is singular Also, (AB) = B A, (A ) T = (A T ) = A T Theorem For A C n n the following conditions are equivalent to A being nonsingular: null(a) = {0} (ie, there is no nonzero y C n such that Ay = 0 ) 2 rank(a) = n (ie, the rows or columns of A are linearly independent) 3 det(a) 0 4 None of A s eigenvalues is zero The adjugate adj of matrix is the transpose of the matrix of cofactors: adj(a) ij = ( ( ) i+j det A ij ) T, where A ij is the submatrix of A obtained by deleting the ith row and jth column A calculation using the cofactor expansion for the determinant shows that A adj(a) = det(a)i so that if A is nonsingular (det A 0) then A = adj(a) det(a) Exercises Let A = 2 [ ] 3 4, x =, and let b = Ax (i) Calculate b by using three vector inner products (ii) Calculate b by forming a linear combination of columns of A 2 Let A be the n n matrix with a j,j+ =, j =,, n and all other elements zero Represent A as a sum of vector outer products 3 (i) Show that A = is orthogonal and hence determine A (ii) Show that the product of two orthogonal matrices is an orthogonal matrix (iii) If Q is a real orthogonal matrix, what is det(q)? If U is a unitary matrix, what is det(u)?
7 MATH3600: Background Material Page 7 [ ] A B 4 Consider the block matrix with A C C D n n and D C m m nonsingular (i) Determine X C m m and Y C n n such that [ ] [ ] [ ] A B I O A B = C D CA I O X (block LU factorization) and that (ii) Prove that [ ] [ ] [ ] A B I BD Y O = C D O I C D (block UL factorization) det(a) det(d CA B) = det(d) det(a BD C), () det(i CB) = det(i BC), (2) det(i xy ) = y x, (3) where x, y C n 5 (i) Verify the Sherman Morrison formula: if A C n n is nonsingular and u, v C n are such that v A u, then (A + uv ) = A A uv A + v A u (4) (ii) Let α C When is I + αe i e T j exists nonsingular? Determine the inverse in those cases where it 6 Let x, x 2,, x n be nonzero orthogonal vectors Show that x,, x n are linearly independent Hint: assume the linearly dependence relation α i x i = 0 and show that orthogonality implies that all the α i must be zero 7 (i) Which is the only matrix that is both idempotent and involutory? (ii) Which is the only matrix that is both idempotent and nilpotent? (iii) Let x, y C n When is xy idempotent? When is it nilpotent? (iv) Prove that if A, B are idempotent and AB = BA then AB is also idempotent (v) Prove that an idempotent matrix is singular unless it is the identity matrix (vi) Prove that A is involutory if and only if (I A)(I + A) = O (vii) Let x C n and x x = Show that I 2xx is involutory The matrix I 2xx is called a Householder reflector (viii) Prove that if (I A) = k j=0 Aj for some integer k 0 then A is nilpotent 8 Let A, B be n n upper triangular matrices with diagonal elements a jj and b jj, respectively Show that (i) AB is triangular and the diagonal elements of AB are a ii b ii
8 MATH3600: Background Material Page 8 (ii) If a jj 0 for all j then A is nonsingular, A is triangular and the diagonal elements of A are /a jj 9 Let A C n n The trace of A is defined as the sum of its diagonal elements, ie, trace(a) = n i= a ii (i) Show that the trace is a linear function, ie, if A, B C n n and α, β C, then trace(αa + βb) = α trace(a) + β trace(b) (ii) Show that trace(ab) = trace(ba), even though in general AB BA (iii) Show that if S R n n is skewsymmetric, ie, S T = S, the trace(s) = 0 Prove the converse to this statement or provide a counterexample
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