Physics Lournal

Conjugate Transpose

The conjugate transpose of a matrix combines the concept of a mathematical conjugate, and a transpose.

A conjugate is formed by changing the sign between two terms in a binomial, for instance:

$x + y \rightarrow x - y$ .

A transpose, or transposition of a matrix, flips a matrix over it's diagonal, from the top left, to the bottom right.

The conjugate transpose, also known as the Hermitian transpose, is the application of both of these concepts to matrices, that have complex (numeral) entries:

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted $\dagger$ , which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with:

$\sigma_{y} = \begin{pmatrix} 0 & -i\ i & 0 \end{pmatrix} \implies \sigma_{y}^{\dagger} = \begin{pmatrix} 0 & -(i)\ -(-i) & 0 \end{pmatrix}\ = \begin{pmatrix} 0 & -i\ i & 0 \end{pmatrix} = \sigma_{y}$

Referenced in

Qiskit: Intro to Linear Algebra

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted $\dagger$ , which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with:

**__Conjugate Transpose__**

Conjugate Transpose

The conjugate transpose of a matrix combines the concept of a mathematical conjugate, and a transpose.

A conjugate is formed by changing the sign between two terms in a binomial, for instance:

x+y→x−yx + y \rightarrow x - yx+y→x−y.

A transpose, or transposition of a matrix, flips a matrix over it's diagonal, from the top left, to the bottom right.

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted †\dagger†, which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with:

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted †\dagger†, which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with:

Referenced in

Qiskit: Intro to Linear Algebra

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted †\dagger†, which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with:

$x + y \rightarrow x - y$ .

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted $\dagger$ , which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with:

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted $\dagger$ , which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with:

A Hermitian matrix is a matrix is that is equivalent to its Conjugate Transpose denoted $\dagger$ , which means if you flip the signs of the imaginary components of the matrix, and then reflect these components over the top left diagonal, the resulting matrix will be the one you started with: