How to check if a matrix is invertible without determinant? Find a unitary matrix which is not orthogonal, and find an orthogonal matrix which is not unitary. Determine whether the following matrix is Hermitian, unitary, and/or normal. A = \begin{bmatrix} 4 + 7i & 2 - i\\ 1 - 2i...
Show how to check if a matrix is unitary. Given A = \begin{pmatrix} 6 & -2\-4 & 1 \end{pmatrix} and I is the 2 by 2 identity matrix. (i) Prove that A^2 = 7A + 2I. (ii) Show that A^{-1} = \frac{1}{2}(A - 7I). ...
Since in upper triangular matrix, all elements under the principal diagonal are zeros, the eigenvalues are nothing but the diagonal elements of the matrix. What are the Eigenvalues of a Unitary Matrix? A unitary matrix is a complex matrix such that its inverse is equal to its conjugate transpos...
Interdisciplinarity is a polysemous concept with multiple, reasoned and intuitive, interpretations across scholars and policy-makers. Historically, quantifying the interdisciplinarity of research has been challenging due to the variety of methods used to identify metadata, taxonomies, and mathematical formulas...
An example of such a model is the superconformal quantum mechanics of Yang-Mills instantons, which has a proposed gravitational dual description as M-theory on a background of the form X7 × S4. This model arises in the strongly-coupled limit of the BFSS matrix model with additional ...
How to find the large-N critical point from strong coupling master fields We describe a new algorithm for the determination of the large-N critical point in lattice theories of unitary matrices (spin and gauge models). Our method is based on saddle-point equations but involves only standard st...
The structured-population model has been widely used to study the spatial transmission of epidemics in human society. Many seminal works have demonstrated the impact of human mobility on the epidemic threshold, assuming that the contact pattern of individuals is mixing homogeneously. Inspired by the ...
Intuitively, the output deconvolution layer can be written as a linear operator X=Ay where each column of matrix A represents an edge-color feature, and y is a sparse vector that determines which edge-color features to use to paint the output image X. It is known under standard...
If you need to understand the characteristics of a directional coupler, consider calculating the scattering matrix of the directional coupler to learn more.
Is it possible that a non symmetric matrix A be positive definite? If not, show a transformation that will make matrix A positive definite. Verify, in general, that if a matrix is positive definite, then its eigenvalues are positive.