Properties of The Conjugate Transpose
- (A + B)* = A* + B* for any two matrices A and B of the same dimensions.
- (r A)* = r*A* for any complex number r and any matrix A. Here r* refers to the complex conjugate of r.
- (AB)* = B*A* for any m-by-n matrix A and any n-by-p matrix B. Note that the order of the factors is reversed.
- (A*)* = A for any matrix A.
- If A is a square matrix, then det(A*) = (det A)* and tr(A*) = (tr A)*
- A is invertible if and only if A* is invertible, and in that case we have (A*)−1 = (A−1)*.
- The eigenvalues of A* are the complex conjugates of the eigenvalues of A.
- for any m-by-n matrix A, any vector x in and any vector y in . Here denotes the standard complex inner product on and .
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