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