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How to find eigenvalues of the following block circulant matrix

I have a block matrix of size PN x PN of the form: Where A and C are P x P matrices. I would like to find the eigenvalues of the matrix B, that is where
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Transpose of higher dimension matrices

We all know transpose of 2D matrix A Old $A_{ij}$ will be replaced by $A_{ji}$ in the transpose matrix and vice versa Question If A is 3D matrices of $3\times 3 \times 8$ then what is old ...
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Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK))$$ $$K = 0:dK:\pi$$ Where K is a ...
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Is this symmetric, block-diagonal matrix positive semi-definite?

I have a matrix of the following form, where $a,b,c>0$ \begin{align*} A = \left[ \begin{array}{cccccc} aM_{12}^2 & aM_{12}M_{13} & 0 & 0 & 0 & 0 & 0 \\ aM_{13}M_{12} ...
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Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
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I'm trying to expand the following product of $\pm1$ matrices $H_1, H_2, K_1, K_2$: (\frac12 (H_1+H_2)\otimes K_1^T+\frac12 (H_1-H_2)\otimes K_2^T)(\frac12 (H_1+H_2)\otimes K_1^T+\frac12 ... 5answers 121 views Positive semi-definite of a matrix composed of semi-definite blocks Say a matrix A is positive semi-definite. Let B be a square matrix composed of replicas of A as sub-blocks, s.t. $$B=\begin{pmatrix} A & A \\ A & A \\ \end{pmatrix},$$ or \begin{pmatrix} A ... 2answers 19 views Problem solving for linear algebra's matrix R i found "U" but failed to find "R". What I did is add row3 to row2, then divide by pivots, which gave me my "R". i thought that;s how its done, why did i fail it. How is it done? 0answers 19 views Block Circulant matrix Let \begin{align} A=\left(\begin{array}{cc} B & C \\ -C & B \end{array}\right), \end{align} where B=\left(\begin{array}{ccc} \alpha^R & \beta^R & \gamma^R \\ \gamma^R &\alpha^R ... 1answer 25 views Block partitioned symmetric orthogonal matrix Let A\in\mathcal{M}_n(\mathbb{R}),\ n\in\mathbb{N} and X\in\mathbb{R}^n. For any \lambda\in\mathbb{R}, define M(\lambda)=\left( \begin{array} {c|c}A & X\\ \hline\phantom{}^tX & \lambda ... 1answer 20 views The similarity of the block matrices Let \mathbb{F} be a field,and let A,B,C be matrices over \mathbb{F} of respective sizes n\times n , k\times k, and n\times k. put M=\begin{bmatrix} A&0 \\ 0&B ... 1answer 40 views Anyone worked with this particular orthogonal matrix In my recent studies of quaternions, the following orthogonal matrix has come up. For example, it is related to the matrix representation of quaternion multiplication. Has anyone seen it come up in ... 0answers 23 views Block Diagonalization related to Direct Sum and Single Eigenvalue? I'm just a beginner in Linear Algebra, and I've proved myself the following: A matrix A^{n \times n} is block diagonalizable if and only if the base field F^n can be divided into at least two ... 3answers 76 views Prove or disprove that trace of matrix X is zero I was trying to solve a question from a competitive exam paper. This is a part of that question. Let I_n and O_n be n\times n identity and null matrices respectively.Let S be 2n\times ... 1answer 70 views Does there exist B for which BB^T=I? My question is Does there exist a real matrix B_{n\times m} with m<n for which BB^T=I_n? Why do I need this? Suppose we are given a real matrix Q_{m\times n} (again, with ... 0answers 30 views Canonical Form of Nilpotent Matrices Given the matrix\hat{S}=\begin{bmatrix} S & *& *&* \\ 0& S &* &* \\ 0& 0& S &* \\ 0&0&0&S\\ \end{bmatrix} $$where S is an n \times n ... 0answers 22 views Design feedback control law to make the whole matrix Hurwitz Suppose (A_1, B_1) and (A_2, B_2) are both stabilizable. Then we know that we can find some K_1 and K_2 to make A_1+B_1K_1 and A_2+B_2K_2 Hurwitz, respectively. Now, for non-zero constant ... 1answer 40 views Solve Matrix: How many trips can be made? We have the matrix:$$\mathbf{M}=\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatrix}$$The matrix ... 4answers 115 views Find the axis of rotation of a rotation matrix by INSPECTION (NOT by solving Kv=v)$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$Find the axis of rotation for the rotation matrix K by INSPECTION. This is from my other thread ... 1answer 30 views Find the Axis of rotation of rotation matrix K after solving (K-I)v=0$$K=\ \begin{pmatrix} 0 & 0 & 1\\ -1 & 0 & 0\\ 0 & -1 & 0 \end{pmatrix}$$Find the axis of rotation for the rotation matrix K. This is from my previous thread click here ... 2answers 64 views Find the axis of rotation from the rotation matrix. This is a problem from the book "Mathematical Methods in the Physical Sciences" Third Edition by author Mary L. Boas. on page 129, Example 5, just in case any of you are familiar with it. So I ... 0answers 13 views generalisation of Knonecker matrix product In the Kronecker matrix product C = A\otimes B we have that C(i,j)=A(i,j)*B where the elements A(i,j) are just numeric scalar values. What if the A(i,j) are matrix operators which act on ... 2answers 48 views Notation - Transpose of Block Matrices [Lay P121 Q2.4.12] Definition of Transpose is (A^T)_{ij} = A_{ji} 1. Why \begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}, and NOT \begin{bmatrix} M \\ N\end{bmatrix}? ... 1answer 25 views Solving Ax = b where A is composed of diagonal blocks I would like to solve the equation Ax=b where x\in\mathbb{R}^n and A is of the form:$$A= \begin{bmatrix} D_1 & D_2 &D_3 \\ D_2 & D_4 & D_5 \\ D_3 & D_5 & D_6 ... 0answers 43 views Separating the eigenvalues of a Hermitian matrix with a special block structure I have a square matrixJ \in \mathbb{C}^{2n \times 2n}$where,$J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}A \in \mathbb{R}^{n \times n}$and is${\bf diagonal}B \in ...
A self-adjoint matrix $A$ in a complex linear space has real eigenvalues bounded between $0$ and $1$. If $A$ is projected onto an arbitrary two-dimensional subspace, what would be the bounds on the ...