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Let $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ be either the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$.
$\begingroup$ 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 therefore, has magnitude 1. $\endgroup$
Unitary matrices are the complex versions, and they are the matrix representations of linear maps on complex vector spaces that preserve "complex distances". If you have a complex vector space then instead of using the scaler product like you would in a real vector space, you use the Hermitian product .
Decomposing a unitary matrix into block unitary matrices. 0. Exam question about JNF and matrix ...
Dot products of vectors within a matrix are inner products. You can think of them as transporting a"thing" to a new coordinate system $(x,y,z)$ ; they are unit-length vectors along each of the axes in a new coordinate system, represented in the former coordinate system.
We have that the DFT Matrix is: $$ W = \frac{1}{\sqrt{N}} \begin{bmatrix} 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^{N-1} \\ 1&\omega^2&\omega^4 ...
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Prove this matrix to be unitary. 0. Commutativity in Binomial Theorem. 1.
What I understand about Unitary matrix is : If we have a square matrix (say 2x2) with complex values. We can say it is Unitary matrix if its transposed conjugate is same of its inverse. One example is provided in the above mentioned page, where it says it depends on 4 parameters:
An isometry, on the other hand, only requires that the columns are orthonormal, but not that they form a basis. It trivially follows that any unitary is also an isometry. In other words, an isometry is a matrix whose columns are orthonormal, while a unitary is a squared matrix whose columns are orthonormal.