Q: Can a Hermitian Matrix possess Complex Eigenvectors?

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Hermitian matrix (please note spelling): a square matrix with complex elements that is equal to its conjugate transpose.

This is the definition of eigenvectors and eigenvalues according to Wikipedia:Specifically, a non-zero column vector v is a (right) eigenvector of a matrix A if (and only if) there exists a number Î» such that Av = Î»v. The number Î» is called the eigenvalue corresponding to that vector. The set of all eigenvectors of a matrix, each paired with its corresponding eigenvalue, is called the eigensystemof that matrix

An eigenvector is a vector which, when transformed by a given matrix, is merely multiplied by a scalar constant; its direction isn't changed. An eigenvalue, in this context, is the factor by which the eigenvector is multiplied when transformed.

A complex number has an imaginary component and is of the form a + bi. (And i is the square root of -1 in this application.)A matrix is a table of numbers. For example, we might give the current (x,y,z) coordinates of a dozen asteroids using a 12 * 3 matrix.A complex matrix is a matrix of complex numbers.

It is the conjugate transpose of the matrix. Of course the conjugate parts only matters with complex entries. So here is a definition:A unitary matrix is a square matrix U whose entries are complex numbers and whose inverse is equal to its conjugate transpose U*. This means thatU*U = UU* = I. Where I is the identity matrix.

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Hermitian matrix defined:If a square matrix, A, is equal to its conjugate transpose, A†, then A is a Hermitian matrix.Notes:1. The main diagonal elements of a Hermitian matrix must be real.2. The cross elements of a Hermitian matrix are complex numbers having equal real part values, and equal-in-magnitude-but-opposite-in-sign imaginary parts.

Skew-Hermitian matrix defined:If the conjugate transpose, A†, of a square matrix, A, is equal to its negative, -A, then A is a skew-Hermitian matrix.Notes:1. The main diagonal elements of a skew-Hermitian matrix must be purely imaginary, including zero.2. The cross elements of a skew-Hermitian matrix are complex numbers having equal imaginary part values, and equal-in-magnitude-but-opposite-in-sign real parts.

Hermitian matrix (please note spelling): a square matrix with complex elements that is equal to its conjugate transpose.

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In quantum mechanics, the density matrix is a mathematical representation of the state of a quantum system that is used to describe mixtures of quantum states or states that have uncertainty. It provides a way to calculate the average values of observables and predict the outcomes of measurements on the system.

Absolutely not. They are rather quite different: hermitian matrices usually change the norm of vector while unitary ones do not (you can convince yourself by taking the spectral decomposition: eigenvalues of unitary operators are phase factors while an hermitian matrix has real numbers as eigenvalues so they modify the norm of vectors). So unitary matrices are good "maps" whiule hermitian ones are not. If you think about it a little bit you will be able to demonstrate the following: for every Hilbert space except C^2 a unitary matrix cannot be hermitian and vice versa. For the particular case H=C^2 this is not true (e.g. Pauli matrices are hermitian and unitary).

This is the definition of eigenvectors and eigenvalues according to Wikipedia:Specifically, a non-zero column vector v is a (right) eigenvector of a matrix A if (and only if) there exists a number Î» such that Av = Î»v. The number Î» is called the eigenvalue corresponding to that vector. The set of all eigenvectors of a matrix, each paired with its corresponding eigenvalue, is called the eigensystemof that matrix

An eigenvector is a vector which, when transformed by a given matrix, is merely multiplied by a scalar constant; its direction isn't changed. An eigenvalue, in this context, is the factor by which the eigenvector is multiplied when transformed.

A complex number has an imaginary component and is of the form a + bi. (And i is the square root of -1 in this application.)A matrix is a table of numbers. For example, we might give the current (x,y,z) coordinates of a dozen asteroids using a 12 * 3 matrix.A complex matrix is a matrix of complex numbers.

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It is the conjugate transpose of the matrix. Of course the conjugate parts only matters with complex entries. So here is a definition:A unitary matrix is a square matrix U whose entries are complex numbers and whose inverse is equal to its conjugate transpose U*. This means thatU*U = UU* = I. Where I is the identity matrix.

There are more people in the hierarchical structure then the matrix structure. The matrix structure is more complex than the hierarchical structure