The transpose of matrix X is defined as the matrix X^T such that the columns are interchanged with the rows and vice versa. The resulting matrix also changes its size if the matrix X is mxn then the transpose matrix X^T changes into nxm.
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The conjugate of matrix A is defined as matrix A such that A*=Ā^T. First the row elements of the matrix are changed with the elements of the columns and then the imaginary parts of every element is changed into negation.
Hermitian transpose and Hermitian conjugate are other names of conjugate transpose.
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