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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.

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Q: What is the definition of a skew-Hermitian matrix?

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A square matrix A is idempotent if A^2 = A. It's really simple

Since the columns of AT equal the rows of A by definition, they also span the same space, so yes, they are equivalent.

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.

The Definition of an Anti-Symmetric Matrix:If a square matrix, A, is equal to its negative transpose, -A', then A is an anti-symmetric matrix.Notes:1. All diagonal elements of A must be zero.2. The cross elements of A must have the same magnitude, but opposite sign.

A matrix A is orthogonal if itstranspose is equal to it inverse. So AT is the transpose of A and A-1 is the inverse. We have AT=A-1 So we have : AAT= I, the identity matrix Since it is MUCH easier to find a transpose than an inverse, these matrices are easy to compute with. Furthermore, rotation matrices are orthogonal. The inverse of an orthogonal matrix is also orthogonal which can be easily proved directly from the definition.

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A square matrix A is idempotent if A^2 = A. It's really simple

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