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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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What is the definition of an idempotent matrix?
A square matrix A is idempotent if A^2 = A. It's really simple
Is the row space of matrix an equivalent to the column space of matrix AT which is the transpose of matrix A?
Since the columns of AT equal the rows of A by definition, they also span the same space, so yes, they are equivalent.
What is the definition of an anti-symmetric matrix?
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.
What is the definition of a Hermitian matrix?
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.
What is an orthogonal matrix?
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.
