Let . The conjugate transpose (also called adjoint) of is defined such that
- (Friedberg e6.2.10) Let such that the rows of form an orthonormal set. Then
The equivalent concept in the language transformations is the linear adjoint
(Friedberg 6.8) Let be a finite dimensional inner product space over and let be a linear transformation. Then, there exists a unique vector such that .
- Intuition: Let be an orthonormal basis . The vector
(Friedberg 6.9) Let be a finite dimensional inner product space and let be a linear operator on . Then there exists a unique linear function such that such that

This linear function is called the adjoint of .
(Friedberg 6.10) Let be a finite-dimensional inner product space and let be an orthonormal basis for . If is a linear operator on then
(Theorem 6.11) Let be a finite-dimensional inner product space, and let and be linear operator on . Then
- for any .
- (Friedberg e6.3.8) If is invertible then so is and in fact
(Theorem 6.11.1) Results analogous to (Theorem 6.11) can be stated for matrices. Let
- for any .
- If is invertible, then so is and in fact
- (Friedberg e6.3.16) If then the determinant satisfies
(Friedberg Lem6.12.2) Let . Then
(Friedberg Cor6.12.1) If such that . Then is invertible
(Friedberg 6.12) Let and . Then there exists such that and

Furthermore if and
(Friedberg e6.3.13) Let be a linear operator on a finite dimensional inner product space . The following statements are true
- For any ,

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Matrix Conjugate and Adjoint