# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### Calculating the matrix $M^{2006}$

Say you have the matrix $M$: $$\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$$ How do you find $M^{2006}$? My thinking was that you ...
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### Find a,b,c to match the linear transformation matrix?

P.S. Sorry for my bad explanation of the task, it was really hard to translate this into meaningful english For the given linear-transformation $A$ find all possible combinations of a,b,c for which ...
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### Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
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### Pearson Correlation

I have two matrices, which are square but of different size. I want to find correlation between data which is stored in these two matrices. It seems Pearson Correlation Coefficient is applicable for ...
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I have a variance matrix given by: $\boldsymbol{\Sigma}\boldsymbol{\Sigma}^{'}+\Omega$ where $\Omega=\left(\begin{array}{cccc} \sigma_{\varepsilon}^{2}\psi\left(\tau_{1}\right) & 0 & \... 0answers 24 views ### How to estimate the product of the$k$largest eigenvalues of a matrix Now I have a question which let me to prove that the product of the largest$k$singular values of a real matrix is always larger than the one of$k$largest eigenvalues. For$k=1$, I use the ... 3answers 365 views ### Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is neither diagonal nor triangular. Is there a term for such matrices, and have they been researched? 0answers 10 views ### Spectral norm of the matrix derivation I understand one possible way how to derive induced norm of symmetrix matrix M, i.e.$sup |M \tilde{x} |$, s.t.$|\tilde{x}|=\tilde{x}^T\tilde{x}=1$(i.e.$\tilde{x}$is lie in unit sphere) Here is ... 2answers 37 views ### is the trace of inverse of positive, positive definite matrix decreasing? Let$A, B$be non-negative, and symmetric positive definite matrices. If$A\le B$, i.e., all the entries of$B-A$are non-negative, is it true that$\mbox{trace}(A^{-1}) \ge \mbox{trace}(B^{-1})$? 1answer 56 views ### The maximal rotation matrix Let's consider two numbers calculated for a rotation matrix which are:$s_e=$the sum of all entries of a matrix$s_a=$the sum of absolute values of all entries for a given matrix. It ... 2answers 41 views ### If A is positive definite (but not necessarily symmetric) can you decompose it? If A is a$2 \times 2$matrix that is positive definite but may or may not be symmetric, does there exist another matrix B such that$A=B^TB$? 2answers 33 views ### How may I use a 3x3 matrix to simulate a larger square matrix? I am using a game engine where the library only provides 3x3 matrices with the multiplication and inverse operation. I could build my own matrix library to provide larger matrices, but it would be ... 2answers 77 views ### Prove that if$y=(y_1, \ldots, y_n)$is such that$y_1a_1 + \cdots + y_na_n = 0$, then$∀x ∈ \mathbb{R}^n$,$Ax · y = 0$I have no idea how to start the following question. Any help will be greatly appreciated. (a) Let$A$be a$n\times n$matrix and let$a_1,\ldots,a_n$be the rows of$A.$Suppose$y=(y_1, \ldots, ...
Matrix $M$ is given as \begin{bmatrix}3&-{\sqrt 7}\\{\sqrt 7}&3\end{bmatrix} I then am asked to describe the transformation, you are also told dis an enlargement followed by a rotation and you ...