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a). The diagonal entries of Λ are the eigen-values of A, and columns of U are eigenvectors of A. ProofofTheorem2. This is equivalent to the condition a_(ij)=a^__(ji), (2) where z^_ denotes the complex conjugate. The matrix element Amn is defined by ... and A is said to be a Hermitian Operator. ... Any real nonsymmetric matrix is not Hermitian. 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]. Basics of Hermitian Geometry 11.1 Sesquilinear Forms, Hermitian Forms, Hermitian Spaces, Pre-Hilbert Spaces In this chapter, we generalize the basic results of Eu-clidean geometry presented in Chapter 9 to vector spaces over the complex numbers. Therefore, a Hermitian matrix A=(a_(ij)) is defined as one for which A=A^(H), (1) where A^(H) denotes the conjugate transpose. Some complications arise, due to complex conjugation. 2 239 Example 9.0.2. 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!!!!! " It is true that: Every eigenvalue of a Hermitian matrix is real. For example, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} And eigenvalues are 1 and -1. The eigenvalue for the 1x1 is 3 = 3 and the normalized eigenvector is (c 11 ) =(1). Notice that this is a block diagonal matrix, consisting of a 2x2 and a 1x1. 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. of real eigenvalues, together with an orthonormal basis of eigenvectors . Since the matrix A+AT is symmetric the study of quadratic forms is reduced to the symmetric case. Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. Thus all Hermitian matrices are diagonalizable. 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). Henceforth V is a Hermitian inner product space. A square matrix is called Hermitian if it is self-adjoint. Suppose v;w 2 V. Then jjv +wjj2 = jjvjj2 +2ℜ(v;w)+jjwjj2: By the spectral theorem for Hermitian matrices (which, for sake of completeness, we prove below), one can diagonalise using a sequence . 50 Chapter 2. But does this mean that : if all of the eigenvalues of a matrix is real, then the matrix is Hermitian? So, for example, if M= 0 @ 1 i 0 2 1 i 1 + i 1 A; then its Hermitian conjugate Myis My= 1 0 1 + i i 2 1 i : In terms of matrix elements, [My] ij = ([M] ji): Note that for any matrix (Ay)y= A: Thus, the conjugate of the conjugate is the matrix … 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). The following simple Proposition is indispensable. Proposition 0.1. Example: Find the eigenvalues and eigenvectors of the real symmetric (special case of Hermitian) matrix below. Example 9.0.3. Let be a Hermitian matrix. Moreover, for every Her-mitian matrix A, there exists a unitary matrix U such that AU = UΛ, where Λ is a real diagonal matrix. 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