Unitary Matrix Properties Proof.Sure, I can provide proofs for five properties of unitary matrices. A unitary matrix is a square matrix that satisfies the condition U†U = I, where U† represents the conjugate transpose (also known as the Hermitian transpose) of U, and I is the identity matrix. Here are five important properties of unitary matrices along with their proofs:
Unitary Matrix Properties Proof
Property 1: Inverse of a Unitary Matrix is Unitary Statement: If U is a unitary matrix, then its inverse U^-1 is also unitary.
Proof: Starting with U†U = I, we want to show that (U^-1)†U^-1 = I. We know that (AB)† = B†A† for matrices A and B, and (U^-1)† = (U†)^-1. Therefore,
(U^-1)†U^-1 = (U†)^-1U†U^-1 = (U†)^-1I = (U†)^-1 = U^-1
Thus, (U^-1)†U^-1 = I, showing that the inverse of a unitary matrix is also unitary.
Property 2: Eigenvalues of Unitary Matrices have Modulus 1 Statement: The eigenvalues of a unitary matrix have a modulus (magnitude) of 1.
Proof: Let λ be an eigenvalue of the unitary matrix U, and let v be the corresponding eigenvector. By definition of eigenvalue and eigenvector, we have Uv = λv. Taking the conjugate transpose of both sides:
(Uv)† = (λv)†
v†U† = λ*v†
Since U†U = I, we can substitute it in:
v†I = λ*v†
v† = λ*v†
Now, multiply both sides by v:
v†v = λ*v†v
Since v†v is the norm (magnitude) of vector v, and we know it’s a real number, we can conclude that:
|λ| * ||v||^2 = |λ| = |λ| * 1
This implies that |λ| = 1, so the eigenvalues of U have modulus 1.
Property 3: Unitary Matrices Preserve Inner Product Statement: If U is a unitary matrix, then for any vectors x and y, the inner product (x, y) is preserved under the transformation: (Ux, Uy) = (x, y).
Proof: Using the properties of the conjugate transpose, we have:
(Ux, Uy) = (Ux)†Uy = x†U†Uy
Since U†U = I:
x†U†Uy = x†Iy = x†y = (x, y)
Hence, the inner product is preserved.
Property 4: Unitary Matrices Preserve Norm Statement: If U is a unitary matrix, then for any vector x, the norm of the transformed vector is the same as the norm of the original vector: ||Ux|| = ||x||.
Proof: Using the properties of the conjugate transpose, we have:
||Ux||^2 = (Ux)†Ux = x†U†Ux
Since U†U = I:
x†U†Ux = x†Ix = x†x = ||x||^2
Taking the square root of both sides gives:
||Ux|| = ||x||
Property 5: Unitary Matrices Preserve Orthogonality Statement: If U is a unitary matrix and x and y are orthogonal vectors, then Ux and Uy are also orthogonal.
Proof: Let x and y be orthogonal vectors, meaning their inner product is 0: (x, y) = 0. Using Property 3 (Unitary Matrices Preserve Inner Product), we know that (Ux, Uy) = (x, y) = 0, which shows that Ux and Uy are also orthogonal.
These are five important properties of unitary matrices along with their proofs. Unitary matrices have many other interesting properties as well, and these properties are fundamental in various areas of mathematics and applications.