A unitary operator U U does indeed satisfy U∗U = I U ∗ U = I, and therefore in particular is an isometry. However, unitary operators must also be surjective (by definition), and are therefore isometric and …

Mar 9, 2013 · I present a slight modification on the answer to a very similar question. I assume we are working over the complex numbers. Recall the eigenvalues of a skew-Hermitian matrix are …

Dec 15, 2025 · In other words, the well-known connection between unitary transformations and unitary matrices works with respect to any inner product (and its orthonormal bases).

So I'm trying to understand why the columns of a unitary matrix form an orthonormal basis. I know it has something to do with the inner product, but I don't fully understand that either (we learned all of this …

Proof of existence of square root of unitary and symmetric matrix Ask Question Asked 13 years, 7 months ago Modified 9 years, 4 months ago

Very good proof! However, an interesting thing is that you can perhaps stop at the third last step, because an equivalent condition of a unitary matrix is that its eigenvector lies on the unit circle, so …

are both unitary and Hermitian (for 0 ≤ θ ≤ 2π 0 ≤ θ ≤ 2 π). I call the latter type trivial, since its columns equal to plus/minus columns of the identity matrix. Do such matrices have any significance (in theory …

May 23, 2023 · Show that eigenvalues of a unitary matrix have absolute value 1. Proof below. Please verify, critique, or improve. Note: Many proofs are available; this question is to verify this proof. …

1 Let K ∈ {R,C} K ∈ {R, C} be either the field of real numbers R R or the field of complex numbers C C. Definition (Unitary matrix). A unitary matrix is a square matrix U ∈ Kn×n U ∈ K n × n such that U∗U =I …

Jun 21, 2020 · prove that an operator is unitary Ask Question Asked 5 years, 6 months ago Modified 5 years, 6 months ago