# 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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### Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence

Consider the density matrices with the following spectral decompositions: $$\rho=\lambda_1|\nu_1\rangle+\lambda_{2}|\nu_2\rangle$$ and $$\sigma=\gamma_1|\omega_1\rangle+\gamma_2|\omega_2\rangle$$ such ...
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### Square root of a $3\times3$ matrix

Here is $3\times3$ matrix$$\begin{pmatrix} 0& 0& 1\\ 0 & -1 & 0\\ 1& 0 & 0\end{pmatrix}$$ How can I solve this by using Cayley-Hamilton? I know how to ...
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### Eigenvalues of a matrix formed by derivatives of a polynomial

Let $f=f(x,y)$ be a polynomial with $x\geq 0,\ -2x\leq y\leq -x/2$ (so when $x=1, -2\leq y\leq-1/2$). Denote \begin{equation*} \begin{split} f_1=&\frac{\partial \ln f}{\partial x}(1,y),\\ f_2=&...
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### Rotation matrix

HI I am wondering if there is a unique matrix that maps $(x_1,y_1,z_1)$ into $(x_2,y_2,z_2)$. These two vectors have equal magnitude and are defined in orthogonal 3-D basis. If there is a unique ...
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### How can we classify the eigenvalues of this 3x3 matrix?

Suppose we have the matrix $$\left( \begin{array}{c c c} -(A+B) & A & 0 \\ C & -(C+D) & 0 \\ 0 & E & -F \\ \end{array} \right)$$ where $0<A,B,C,D,E\in \mathbb{R}$. I am ...
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### Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
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### Verifying orthogonality between two binary sequences

I have studied that for orthogonality to exist between two binary sequences: [Number of bit agreements - Number of bit disagreements]/sequence length=0 Eg, for an orthogonal matrix X given by: \...
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### Matrix integration by parts

It seems to me that the integration by parts rule carries over simply to the matrix case. This can be seen by applying: $(AB)' = A'B + AB'$ and then integrating for square (time dependent) complex ...
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### Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
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### Proving that $L_{22}L_{22}^T=S$ is the Schur complement of a cholesky factorization

Let $A$ be an $n+m \times n+m$ symmetric positive definite matrix. $A=\begin{bmatrix}A_{11} & A_{12}\\ A_{12}^T & A_{22}\end{bmatrix}$ where $A_{11}$ is an $n \times n$ matrix, $A_{12}$ is an ...
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### Given a parametric solution $\vec{x}(t)$ to $Ax = b$, how can I choose the parameter $t$ so that all entries in $\vec{x}(t)$ is between 0 and 1?

Given a solution to the matrix equation $A\vec{x} = \vec{b}$ on the form $\vec{x}(t)$, how can I choose the parameter t such that all entries in $\vec{x}$ are squeezed between 0 and 1? That is, for ...
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### Decomposing a matrix as the product of rotations

I'm reading an article about joint diagonalization algorithms. The article states without proof that any nonsingular $n \times n$ matrix $Q$ can be decomposed as \begin{align*} Q = \prod_{1 \leq p <...
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### How to solve series of 8 equations with 8 unknowns?

In this article http://www.fmwconcepts.com/imagemagick/bilinearwarp/FourCornerImageWarp2.pdf they speak of solving for a0,a1,a2,a3,b0,b1,b2,b3 but I want to know ...
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I have a square and its 4 corner coordinates $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. And I have a quadrilateral with corner coordinates $(x_1',y_1'),(x_2',y_2'),(x_3',y_3'),(x_4',y_4')$ where $(x_i,... 2answers 64 views ###$rk(A^2)=rk(B^2) \implies rk(A)=rk(B)$is it true? The original statement is this: given A,B matrices$n \times n$, if$A^2$is "Left-Right equivalent" to$B^2$then A is LR equivalent to B (is it true or false?) I know that A is LR equivalent to B ... 1answer 65 views ### Find the minimum of a function for only positive values of the vector variable Let variable vector$\vec{q}$of size$m\times1$, and its diagonal counterpart$m\times m$matrix$Q=diag(\vec{q})$, for some$m\in\mathbb{N}$. Define fixed parameter$n\times1$vectors$\vec{p}, \...
Calculate the determinant of $M = \left( {\begin{array}{*{20}c} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \\ \end{array}} \right)\;$. How can one calculate this? Is there a ...