50 Chapter 2. A square matrix is called Hermitian if it is self-adjoint. The following simple Proposition is indispensable. 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. 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!!!!! " By the spectral theorem for Hermitian matrices (which, for sake of completeness, we prove below), one can diagonalise using a sequence . Henceforth V is a Hermitian inner product space. Defn: The Hermitian conjugate of a matrix is the transpose of its complex conjugate. 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. 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). 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. Suppose v;w 2 V. Then jjv +wjj2 = jjvjj2 +2ℜ(v;w)+jjwjj2: The matrix element Amn is defined by ... and A is said to be a Hermitian Operator. For example, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} And eigenvalues are 1 and -1. The diagonal entries of Λ are the eigen-values of A, and columns of U are eigenvectors of A. ProofofTheorem2. Some complications arise, due to complex conjugation. It is true that: Every eigenvalue of a Hermitian matrix is real. But does this mean that : if all of the eigenvalues of a matrix is real, then the matrix is Hermitian? 2 Moreover, for every Her-mitian matrix A, there exists a unitary matrix U such that AU = UΛ, where Λ is a real diagonal matrix. This is equivalent to the condition a_(ij)=a^__(ji), (2) where z^_ denotes the complex conjugate. Proposition 0.1. Example: Find the eigenvalues and eigenvectors of the real symmetric (special case of Hermitian) matrix below. ... Any real nonsymmetric matrix is not Hermitian. of real eigenvalues, together with an orthonormal basis of eigenvectors . 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]. Example 9.0.3. Thus all Hermitian matrices are diagonalizable. 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