### Need help from an expert?

The world’s top online tutoring provider trusted by students, parents, and schools globally.

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`

.

Study and Practice for Free

Trusted by 100,000+ Students Worldwide

Achieve Top Grades in your Exams with our Free Resources.

Practice Questions, Study Notes, and Past Exam Papers for all Subjects!

The world’s top online tutoring provider trusted by students, parents, and schools globally.

Loading...

Loading...