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In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of + being , for real numbers and ).
3 Ιαν 2024 · A matrix \(A = \left[ a_{ij} \right]\) is called a complex matrix if every entry \(a_{ij}\) is a complex number. The notion of conjugation for complex numbers extends to matrices as follows: Define the conjugate of \(A = \left[ a_{ij} \right]\) to be the matrix
Example 8.7.2. In C4, find a unit vector u that is a positive real multiple of z = (1 − i, i, 2, 3 + 4i). Solution. kzk = √2 + 1 + 4 + 25 = √32 = 4√2, so take u = 1 4√2 z. Transposition of complex matrices is defined just as in the real case, and the following notion is fun-damental.
In this post we explain what a conjugate transpose matrix is and how to calculate the complex conjugate transpose of a matrix. Also, we show you an example of the conjugate transpose of a matrix and all the properties of this type of matrix.
28 Οκτ 2024 · The conjugate transpose of an m×n matrix A is the n×m matrix defined by A^ (H)=A^_^ (T), (1) where A^ (T) denotes the transpose of the matrix A and A^_ denotes the conjugate matrix. In all common spaces (i.e., separable Hilbert spaces), the conjugate and transpose operations commute, so A^ (H)=A^_^ (T)=A^ (T)^_.
The conjugate transpose of a matrix is the matrix defined by where denotes transposition and the over-line denotes complex conjugation. Remember that the complex conjugate of a matrix is obtained by taking the complex conjugate of each of its entries (see the lecture on complex matrices).
The transpose of the matrix representation of a complex number corresponds to complex conjugation. The inverse of the matrix representation of a complex number corresponds to the reciprocal of the complex number.
