Normal matrices are matrices that include Hermitian matrices and enjoy several of the same properties as Hermitian matrices. 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). Physical transformations as unitary operators 50 Chapter 2. The following simple Proposition is indispensable. 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!!!! Since the matrix A+AT is symmetric the study of quadratic forms is reduced to the symmetric case. 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). 239 Example 9.0.2. Suppose v;w 2 V. Then jjv +wjj2 = jjvjj2 +2ℜ(v;w)+jjwjj2: 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. Henceforth V is a Hermitian inner product space. Since A is Hermitian, we have A H = A = T. The diagonal elements of a Hermitian matrix are real. A matrix A is called Hermitianif A∗ = A. 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. Real Hermitian is the same as symmetric. 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). ! 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. I The sum of two Hermitian matrices is a Hermitian matrix. 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. That is, if a matrix is unitarily diagonalizable, then I The matrix must be symmetric if it has only real values. Meaning An = Hermitian. Proposition 0.1. Example 5: A Hermitian matrix. 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). Indeed, while we proved that Hermitian matrices are unitarily diagonalizable, we did not establish any converse. I The matrix cannot be symmetric if it has complex values. So for a real matrix A∗ = AT. A = j: 1-2j,-1-2j: 0 = … Next we need to setup some technical lemmas for the proof of the main theorem. I The product of two Hermitian matrices is a Hermitian matrix i AB = BA. I All Eigenvalues of a Hermitian matrix are real. (b) Show that the … 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. 2 Note that we could have put the overline representing scalar complex conjugation in the lower left instead of the upper right. Example 9.0.3. 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].