When is a matrix unitary

When the vectors are arrays of real numbers, the inner product is the usual dot product between two vectors: where denotes the transpose of.

Definition A complex matrix is said to be unitary if and only if it is invertible and its inverse is equal to its conjugate transpose, that is,. Remember that is the inverse of a matrix if and only if it satisfies where is the identity matrix. As a consequence, the following two propositions hold. Proposition is a unitary matrix if and only if.

Example Define the complex matrix The conjugate transpose of is The matrix product between and is Then, is unitary. Proposition A matrix is unitary if and only if its columns form an orthonormal set. Note that the -th entry of the identity matrix is Moreover, by the very definition of matrix product, the -th entry of the product is the product between the -th row of denoted by and the -th column of denoted by : In turn, by the definition of conjugate transpose, the -th row of is equal to the conjugate transpose of the -th column of.

Therefore, we have that Having established these facts, let us prove the "if" part of the proposition. Suppose that the columns of form an orthonormal set. Then, which implies for any and. As a consequence, which means that is unitary. Let us now prove the "only if" part. Suppose that is unitary. Then, which implies As a consequence, the columns of are orthonormal. Example Consider again the matrix and denote its two columns by The two columns have unit norm because and They are orthogonal because.

Proposition A matrix is unitary if and only if its transpose is unitary. We already know that is unitary if and only We can transpose both sides of the equation and obtain the equivalent condition where we have used the fact that the order of conjugation and transposition does not matter.

The latter condition is satisfied if and only if is unitary, which proves the proposition. Proposition A matrix is unitary if and only if its rows form an orthonormal set. The rows of are the columns of , which is unitary 1 if and only if it has orthonormal columns; 2 if and only if is unitary. Don't have an account? Signup here. Unitary Matrix. A Unitary Matrix is a form of a complex square matrix in which its conjugate transpose is also its inverse. This means that a matrix is flipped over its diagonal row and the conjugate of its inverse is calculated.

If the resulting output, called the conjugate transpose is equal to the inverse of the initial matrix, then it is unitary. Unitary matrices have a few properties specific to their form. For example,. Unitary matrices leave the length of a complex vector unchanged. For real matrices , unitary is the same as orthogonal.

In fact, there are some similarities between orthogonal matrices and unitary matrices. The rows of a unitary matrix are a unitary basis. That is, each row has length one, and their Hermitian inner product is zero. Similarly, the columns are also a unitary basis. In fact, given any unitary basis, the matrix whose rows are that basis is a unitary matrix. It is automatically the case that the columns are another unitary basis.

In particular, a unitary matrix is always invertible, and. Note that transpose is a much simpler computation than inverse. A similarity transformation of a Hermitian matrix with a unitary matrix gives.

Unitary matrices are normal matrices. If is a unitary matrix, then the permanent. The unitary matrices are precisely those matrices which preserve the Hermitian inner product.