# A bracketleftalt2 3 2 i 3 i 2 i 0 1 i 3 i 1 i 0

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A=bracketleft.alt232+i3i2-i01+i-3i1-i0bracketright.alt2THEOREM 8.10The Eigenvalues of a Hermitian MatrixIfAis a Hermitian matrix, then its eigenvalues are real numbers.A2.REMARKNote that this theorem impliesthat the eigenvalues of arealsymmetric matrixare real, asstated in Theorem 7.7.
8.5Unitary and Hermitian Matrices433To find the eigenvectors of a complex matrix, use a procedure similar to that usedfor a real matrix. For instance, in Example 5, to find eigenvectors corresponding to theeigenvalueλ1= -1, substituteλ= -1 into the equationbracketleft.alt2λ-3-2-i-3i-2+iλ-1-i3i-1+iλbracketright.alt2bracketleft.alt2v1v2v3bracketright.alt2=bracketleft.alt2000bracketright.alt2to obtainbracketleft.alt2-4-2-i-3i-2+i-1-1-i3i-1+i-1bracketright.alt2bracketleft.alt2v1v2v3bracketright.alt2=bracketleft.alt2000bracketright.alt2.Solve this equation to verify thatv1=bracketleft.alt2-11+2i1bracketright.alt2is an eigenvector. In a similar manner, verify thatv2=bracketleft.alt21-21i6-9i13bracketright.alt2andv3=bracketleft.alt21+3i-2-i5bracketright.alt2are eigenvectors corresponding toλ2=6 andλ3= -2, respectively.LINEARALGEBRAAPPLIEDQuantum mechanics had its start in the early twentiethcentury as scientists began to study subatomic particlesand light. Collecting data on energy levels of atoms, andthe rates of transition between levels, they found thatatoms could be induced to more excited states by theabsorption of light. German physicist Werner Heisenberg(1901–1976) laid a mathematical foundation for quantummechanics using matrices. Studying the dispersion of light,he used vectors to represent energy levels of states andHermitian matrices to represent “observables” such asmomentum, position, spin, and energy. A measurementyields precisely one real value and leaves the system inprecisely one of a set of mutually exclusive (orthogonal)states. So, the eigenvalues are the possible values thatcan result from a measurement of an observable, andthe eigenvectors are the corresponding states of thesystem after the measurement. Let matrixAbe a diagonalHermitian matrix that represents an observable. Thenconsider a physical system whose state is representedby the column vectoru. To measure the value of theobservableAin the system of stateu, find the productu*Au=[u1u2u3]bracketleft.alt2a11000a22000a33bracketright.alt2bracketleft.alt2u1u2u3bracketright.alt2=[(u1u1)a11+(u2u2)a22+(u3u3)a33].

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Complex number