Hadamard Matrices and Orthonormal Bases Calvin Hotchkiss CBMS Boot Camp Iowa State University June 2, 2018 Hadamard Matrices and ONB Iowa State Let v T be the transpose of a vector (or matrix) v, v its (pointwise) conjugate, and v ∗ its conjugate transpose. Definition 1 (Inner product) T T Let x = (x1, x2,..., xn) and y = (y1, y2,..., yn) be column n ∗ vectors in C . Then hx, yi = y x. Two vectors x and y are called orthogonal if hx, yi = 0. The norm of a vector x, jjxjj = phx, xi. Preliminaries n T We start out today in C := f(x1, x2,... xn) : xj 2 Cg. This is a complete vector space with an inner product, what we call a Hilbert Space. Hadamard Matrices and ONB Iowa State Definition 1 (Inner product) T T Let x = (x1, x2,..., xn) and y = (y1, y2,..., yn) be column n ∗ vectors in C . Then hx, yi = y x. Two vectors x and y are called orthogonal if hx, yi = 0. The norm of a vector x, jjxjj = phx, xi. Preliminaries n T We start out today in C := f(x1, x2,... xn) : xj 2 Cg. This is a complete vector space with an inner product, what we call a Hilbert Space. Let v T be the transpose of a vector (or matrix) v, v its (pointwise) conjugate, and v ∗ its conjugate transpose. Hadamard Matrices and ONB Iowa State Two vectors x and y are called orthogonal if hx, yi = 0. The norm of a vector x, jjxjj = phx, xi. Preliminaries n T We start out today in C := f(x1, x2,... xn) : xj 2 Cg. This is a complete vector space with an inner product, what we call a Hilbert Space. Let v T be the transpose of a vector (or matrix) v, v its (pointwise) conjugate, and v ∗ its conjugate transpose. Definition 1 (Inner product) T T Let x = (x1, x2,..., xn) and y = (y1, y2,..., yn) be column n ∗ vectors in C . Then hx, yi = y x. Hadamard Matrices and ONB Iowa State The norm of a vector x, jjxjj = phx, xi. Preliminaries n T We start out today in C := f(x1, x2,... xn) : xj 2 Cg. This is a complete vector space with an inner product, what we call a Hilbert Space. Let v T be the transpose of a vector (or matrix) v, v its (pointwise) conjugate, and v ∗ its conjugate transpose. Definition 1 (Inner product) T T Let x = (x1, x2,..., xn) and y = (y1, y2,..., yn) be column n ∗ vectors in C . Then hx, yi = y x. Two vectors x and y are called orthogonal if hx, yi = 0. Hadamard Matrices and ONB Iowa State Preliminaries n T We start out today in C := f(x1, x2,... xn) : xj 2 Cg. This is a complete vector space with an inner product, what we call a Hilbert Space. Let v T be the transpose of a vector (or matrix) v, v its (pointwise) conjugate, and v ∗ its conjugate transpose. Definition 1 (Inner product) T T Let x = (x1, x2,..., xn) and y = (y1, y2,..., yn) be column n ∗ vectors in C . Then hx, yi = y x. Two vectors x and y are called orthogonal if hx, yi = 0. The norm of a vector x, jjxjj = phx, xi. Hadamard Matrices and ONB Iowa State n Any basis B = fb1, b2,..., bng for C has the property that, for n any v 2 C , there exist unique a1, a2,..., an in C such that: v = a1b1 + a2b2 + ··· + anbn. An orthonormal basis is particularly convenient, since it has the property that aj = hv, bj i for all j, so that n v = hv, bj i bj . j=1 X Orthonormal Bases A set of mutually orthogonal vectors is called orthonormal if they n all have norm 1. If they are additionally a basis for C , they are called an orthonormal basis. Hadamard Matrices and ONB Iowa State An orthonormal basis is particularly convenient, since it has the property that aj = hv, bj i for all j, so that n v = hv, bj i bj . j=1 X Orthonormal Bases A set of mutually orthogonal vectors is called orthonormal if they n all have norm 1. If they are additionally a basis for C , they are called an orthonormal basis. n Any basis B = fb1, b2,..., bng for C has the property that, for n any v 2 C , there exist unique a1, a2,..., an in C such that: v = a1b1 + a2b2 + ··· + anbn. Hadamard Matrices and ONB Iowa State Orthonormal Bases A set of mutually orthogonal vectors is called orthonormal if they n all have norm 1. If they are additionally a basis for C , they are called an orthonormal basis. n Any basis B = fb1, b2,..., bng for C has the property that, for n any v 2 C , there exist unique a1, a2,..., an in C such that: v = a1b1 + a2b2 + ··· + anbn. An orthonormal basis is particularly convenient, since it has the property that aj = hv, bj i for all j, so that n v = hv, bj i bj . j=1 X Hadamard Matrices and ONB Iowa State u1 Let u1 = v1, then e1 = . jju1jj u2 Let u2 = v2 - hv2, e1i e1. Then e2 = . jju2jj u3 Let u3 = v3 - hv3, e1i e1 - hv3, e2i e2. Then e3 = . Continue jju3jj until: uk uk = vk - hvk , e1i e1 - ··· - hvk , ek-1i ek-1. Then ek = . jjuk jj The Gram{Schmidt Process Given a set of linearly independent vectors S = fv1,..., vk g, the Gram{Schmidt process generates an orthonormal set 0 S = fe1,..., ek g that spans the same subspace as S. In the process it will generate a set fu1,..., uk g that is orthogonal but not necessarily orthonormal. Hadamard Matrices and ONB Iowa State u2 Let u2 = v2 - hv2, e1i e1. Then e2 = . jju2jj u3 Let u3 = v3 - hv3, e1i e1 - hv3, e2i e2. Then e3 = . Continue jju3jj until: uk uk = vk - hvk , e1i e1 - ··· - hvk , ek-1i ek-1. Then ek = . jjuk jj The Gram{Schmidt Process Given a set of linearly independent vectors S = fv1,..., vk g, the Gram{Schmidt process generates an orthonormal set 0 S = fe1,..., ek g that spans the same subspace as S. In the process it will generate a set fu1,..., uk g that is orthogonal but not necessarily orthonormal. u1 Let u1 = v1, then e1 = . jju1jj Hadamard Matrices and ONB Iowa State u3 Let u3 = v3 - hv3, e1i e1 - hv3, e2i e2. Then e3 = . Continue jju3jj until: uk uk = vk - hvk , e1i e1 - ··· - hvk , ek-1i ek-1. Then ek = . jjuk jj The Gram{Schmidt Process Given a set of linearly independent vectors S = fv1,..., vk g, the Gram{Schmidt process generates an orthonormal set 0 S = fe1,..., ek g that spans the same subspace as S. In the process it will generate a set fu1,..., uk g that is orthogonal but not necessarily orthonormal. u1 Let u1 = v1, then e1 = . jju1jj u2 Let u2 = v2 - hv2, e1i e1. Then e2 = . jju2jj Hadamard Matrices and ONB Iowa State The Gram{Schmidt Process Given a set of linearly independent vectors S = fv1,..., vk g, the Gram{Schmidt process generates an orthonormal set 0 S = fe1,..., ek g that spans the same subspace as S. In the process it will generate a set fu1,..., uk g that is orthogonal but not necessarily orthonormal. u1 Let u1 = v1, then e1 = . jju1jj u2 Let u2 = v2 - hv2, e1i e1. Then e2 = . jju2jj u3 Let u3 = v3 - hv3, e1i e1 - hv3, e2i e2. Then e3 = . Continue jju3jj until: uk uk = vk - hvk , e1i e1 - ··· - hvk , ek-1i ek-1. Then ek = . jjuk jj Hadamard Matrices and ONB Iowa State The columns (and also the rows) of U form an orthonormal basis n for C . n If B = fb1, b2,..., bng is an orthonormal basis for C , then UB = fUb1, Ub2,..., Ubng is also an orthornomal basis for B. n If x is a column vector in C , written in terms of the standard basis, then Ux is the same vector written in terms of the ONB formed by the columns of U. Unitary matrices ∗ ∗ ∗ An n × n matrix U is called unitary if U U = UU = In, where U represents the conjugate transpose of the matrix U. Hadamard Matrices and ONB Iowa State n If B = fb1, b2,..., bng is an orthonormal basis for C , then UB = fUb1, Ub2,..., Ubng is also an orthornomal basis for B. n If x is a column vector in C , written in terms of the standard basis, then Ux is the same vector written in terms of the ONB formed by the columns of U. Unitary matrices ∗ ∗ ∗ An n × n matrix U is called unitary if U U = UU = In, where U represents the conjugate transpose of the matrix U. The columns (and also the rows) of U form an orthonormal basis n for C . Hadamard Matrices and ONB Iowa State Unitary matrices ∗ ∗ ∗ An n × n matrix U is called unitary if U U = UU = In, where U represents the conjugate transpose of the matrix U. The columns (and also the rows) of U form an orthonormal basis n for C . n If B = fb1, b2,..., bng is an orthonormal basis for C , then UB = fUb1, Ub2,..., Ubng is also an orthornomal basis for B. n If x is a column vector in C , written in terms of the standard basis, then Ux is the same vector written in terms of the ONB formed by the columns of U. Hadamard Matrices and ONB Iowa State This means that the columns (and also the rows) of H form an n p orthogonal basis for C , with each vector having norm n.