linear algebra - What's the interpretation of a unitary matrix ...
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 .
Show that the eigenvalues of a unitary matrix have modulus $1$
$\begingroup$ 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. $\endgroup$
linear algebra - How to convert a random matrix to Unitary Matrix ...
$\begingroup$ The Gram-Schmidt procedure gives you an orthonormal basis, which when arragned into a matrix gives a unitary matrix. I included the normalization step in the procedure, so the determinant of the resulting matrix will have absolute value $1$. $\endgroup$ –
linear algebra - Norm preservation properties of a unitary matrix ...
Let $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ be either the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$.
Eigenvalues of a unitary matrix - Physics Forums
2007年12月11日 · If a unitary matrix has all its eigenvalues equal to 1, it is called a special unitary matrix. This means that the matrix is not only unitary, but it also has a determinant of 1. Special unitary matrices are important in physics, particularly in quantum mechanics. Can a unitary matrix have complex eigenvalues? Yes, a unitary matrix can have ...
Given a unitary matrix $U$, how do I find $A$ such that $U=e^{iA}$?
2015年1月1日 · A unitary matrix $U \in \mathbb C^{n \times n}$ can always be written in exponential form $$U = e^{iA} \tag{1}$$ where $A$ is Hermitian.
Proving the Product of Unitary Matrices is also Unitary
2016年10月5日 · Convex combination of unitary matrices equal to another unitary matrix. 0. Question on unitary matrices.
Powers of a unitary matrix - Mathematics Stack Exchange
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linear algebra - Singular value decomposition for unitary matrices ...
2021年1月13日 · I know the title is strange, but there are many instances in quantum information in which one is interested not in diagonalizing a unitary matrix, but instead in finding its singular value decomposition (SVD) — the theorem of existence is Cor. 2.4 in Nielsen & Chuang.
What is the general expression for a unitary $3 \\times 3$ matrix?
2017年6月19日 · $\begingroup$ It seems unlikely; 2x2 orthogonal (real) matrices have a similarly nice form, but there's not really any such for the 3x3 because the shape is just substantially more complicated than $\mathcal{S}^1$, and I can't imagine …