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Define the Skew-Hermitian matrix.

A skew-Hermitian matrix is a square matrix whose conjugate transpose is equal to its negative.

A skew-Hermitian matrix is a complex square matrix A such that A^H = -A, where A^H is the conjugate transpose of A. In other words, the entries of A satisfy the relation a_ij = -conj(a_ji) for all i and j, where conj denotes the complex conjugate, which is a fundamental concept in types of numbers.

Skew-Hermitian matrices have some interesting properties. For example, the eigenvalues of a skew-Hermitian matrix are purely imaginary or zero. To see why, suppose λ is an eigenvalue of A with eigenvector x. Then we have Ax = λx, and taking the conjugate transpose of both sides gives x^H A^H = conj(λ) x^H. Substituting A^H = -A and rearranging, we get x^H A x = -conj(λ) x^H x. Since x^H x is a positive real number, we must have λ = -conj(λ), which implies that λ is purely imaginary or zero. To understand more about complex numbers in this context, consider exploring the trigonometric form of complex numbers.

Another important property of skew-Hermitian matrices is that they are diagonalizable by a unitary matrix. That is, there exists a unitary matrix U such that U^H A U = D, where D is a diagonal matrix. To see why, note that the eigenvalues of A are purely imaginary or zero, so we can write A = PDP^-1, where P is invertible and D is diagonal with the eigenvalues of A on the diagonal. Then we can take U = P (or U = P^H, depending on the convention) to obtain the desired diagonalization. Understanding the diagonalization process is also vital for grasping concepts like differentiation of exponential and logarithmic functions.

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