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Let A =[a ij] ∈M n.Consider the quadratic form on Cn or Rn deﬁned by Q(x)=xTAx = Σa ijx jx i = 1 2 Σ(a ij +a ji)x jx i = xT 1 2 (A+AT)x. in this basis, the matrix for U^ A(a) is diagonal with matrix elements e ia n. In summary, we can associate a Hermitian operator to any one-parameter family of unitary operators near the identity operator, and we can associate a family of unitary operators to any Hermitian operator by exponentiation. Hermitian Matrices We conclude this section with an observation that has important impli-cations for algorithms that approximate eigenvalues of very large Hermitian matrix A with those of the small matrix H = Q∗AQ for some subunitary matrix Q ∈ n×m for m n. (In engineering applications n = 106 is common, and n = 109 22 2). Thus, by Theorem 2, matrix transformation given by a symmetric/Hermitian matrix will be a self-adjoint operator on R n /C n , using the standard inner product. 239 Example 9.0.2. That is, if a matrix is unitarily diagonalizable, then I The matrix cannot be symmetric if it has complex values. Indeed, while we proved that Hermitian matrices are unitarily diagonalizable, we did not establish any converse. So for a real matrix A∗ = AT. I The sum of two Hermitian matrices is a Hermitian matrix. The Transformation matrix •The transformation matrix looks like this •The columns of U are the components of the old unit vectors in the new basis •If we specify at least one basis set in physical terms, then we can define other basis sets by specifying the elements of the transformation matrix!!!! For example, Cn with the standard Hermitian product (x,y) = x∗y = x 1y1 +...+x ny n. I recall that “Hermitian transpose” of A is denoted by A∗ and is obtained by transposing A and complex conjugating all entries. A matrix A is called Hermitianif A∗ = A. Real Hermitian is the same as symmetric. Next we need to setup some technical lemmas for the proof of the main theorem. 8.2 Hermitian Matrices 273 Proof If v is a unit eigenvector of A associated with an eigenvalue λ, then Av = λv and vhA = vhAh = (Av)h = (λv)h = λ∗vh Premultiplying both sides of the ﬁrst equality by vh, postmultiplying both sides of the second equality by v, and noting that vhv = kvk2 = 1, we get vhAv = λ = λ∗ Hence all eigenvalues of A are real. Normal matrices are matrices that include Hermitian matrices and enjoy several of the same properties as Hermitian matrices. I All Eigenvalues of a Hermitian matrix are real. I The matrix must be symmetric if it has only real values. y. Hermitian matrices have three key consequences for their eigenvalues/vectors: the eigenvalues λare real; the eigenvectors are orthogonal; 1 and the matrix is diagonalizable (in fact, the eigenvectors can be chosen in the form of an orthonormal basis). Since the matrix A+AT is symmetric the study of quadratic forms is reduced to the symmetric case. Hermitian, we’ll denote this matrix as H= a c c b , 1. where a and b are real and c is complex (real, imaginary or neither). The following simple Proposition is indispensable. ! Example 5: A Hermitian matrix. This Example is like Example One in that one can think of f 2 H as a an in nite-tuple with the continuous index x 2 [a;b]. 50 Chapter 2. Example 9.0.3. Meaning An = Hermitian. Proposition 0.1. A = 2: 1+j: 2-j, 1-j: 1: j: 2+j-j: 1 = 2: 1-j: 2+j (j 2 = -1) 1+j: 1-j: 2-j: j: 1: Now A T = => A is Hermitian (the ij-element is conjugate to the ji-element). Physical transformations as unitary operators Henceforth V is a Hermitian inner product space. I The product of two Hermitian matrices is a Hermitian matrix i AB = BA. A = j: 1-2j,-1-2j: 0 = … Since A is Hermitian, we have A H = A = T. The diagonal elements of a Hermitian matrix are real. 2 Note that we could have put the overline representing scalar complex conjugation in the lower left instead of the upper right. Suppose v;w 2 V. Then jjv +wjj2 = jjvjj2 +2ℜ(v;w)+jjwjj2: (b) Show that the …