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Relationship between eigenvalues when the same change is applied to two different symmetric matrices with the same eigenvalues

I will disprove your conjecture in the case $n=2$. (A general suggestion: If you have a linear algebra conjecture, try to check it for $2\times 2$ matrices first.) Consider the following family of ...
Moishe Kohan's user avatar
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2 votes

Solving XOR matrix

XOR is addition mod $2\,.$ When you add the first row to the last row you get $$ \pmatrix{1&1&0&0&0\\0&1&1&0&0\\0&0&1&1&0\\0&0&0&1&1\\1&...
Kurt G.'s user avatar
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2 votes

Solving XOR matrix

First proof : based on associativity/commutativity property of $\oplus$ : By contradiction : Assume a solution $x_1,x_2,x_3,x_4,x_5$ exists. Let : $b=x_1\oplus x_2\oplus x_3\oplus x_4\oplus x_5\oplus ...
Jean Marie's user avatar
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Derivative of matrix exponential w.r.t. to each element of the matrix

It might be interesting to posit a more formal answer based on an application of the general chain rule. Consider this, if $A = \sum_{kl} A_{kl}E_{kl}$ then $$ \frac{\partial A}{\partial A_{ij}} = \...
James S. Cook's user avatar
1 vote

Is it possible to find matrix $A$ and matrix $B$ if you know $AB$ and $BA$?

$\def\ed{\stackrel{\text{def}}{=}}$ If $\ K_1\ $ and $\ K_2\ $ are two square matrices, then there exist invertible square matrices $\ A\ $ and $\ B\ $ satisfying the equations \begin{align} AB&=...
lonza leggiera's user avatar
1 vote

find transition matrix

If you want to transform $\mathbf{Z_1}\ (4\times 4)$ into $\mathbf{Z_2}\ (3\times 3)$ with a transition matrix $\mathbf A$, it'll have to be with this operation for sure: $$\mathbf{Z_2}=\mathbf{AZ_1A}...
Joan S. Guillamet F.'s user avatar
4 votes
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For $g\in\operatorname{SL}(2,q)$, do we have $\operatorname{tr}(g)=\operatorname{tr}(g^{-1})?$

Since $\det g = 1$, the inverse of $g$ is the adjugate of $g$. So, regardless of the ground field $\mathbb{K}$, for a $2 \times 2$ matrix in $\operatorname{SL}(2, \mathbb{K})$ $$ g = \begin{pmatrix} a ...
Sammy Black's user avatar
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0 votes

Any assured method for finding the derivative of p Euclidian norms?

Essentially coming back to the definition of the differential $$\|A(y_0+h)\|^2=\|Ay_0\|^2+2\langle Ay_0,Ah\rangle+\|Ah\|^2$$$$= \|Ay_0\|^2+2\langle A^TAy_0,h\rangle +o(h)$$ Thus the differential of $...
Letac Gérard's user avatar
1 vote

Any assured method for finding the derivative of p Euclidian norms?

The symbol $\frac{\partial L}{\partial \beta}$ will be a vector with entries $\frac{\partial L}{\partial \beta_k}$, in this case for $k=1,\ldots, p$. We can calculate $$\frac{\partial }{\partial \...
user469053's user avatar
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0 votes

Skew Symmetric Matrix for expressing a Rotation

More geometric approach. Suppose you have a vector $v=\begin {bmatrix} x &y & z \end{bmatrix}^T$ of unit length. The rotation matrix can be generated with so-called Rodrigues formula. With ...
Widawensen's user avatar
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2 votes

Does there exist at least four elements in $\operatorname{SL}(2,q)$ of different order?

As $\mathrm{SL}(2,2)\cong S_3$, it doesn't contain elements besides ones of oders $1,2,$ and $3$. In particular your claim is false for $q=2$. However for $q=2^n$ for $n>1$ you can prove the ...
Etropy's user avatar
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3 votes
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Does there exist at least four elements in $\operatorname{SL}(2,q)$ of different order?

This is false for $q=2$ but true otherwise: For $q=2$ you can just directly check that $\mathrm{SL}_2(2)\simeq S_3$ which only has elements of orders 1,2 and 3. For $q\geq 8$, there are always ...
Tim Seifert's user avatar
2 votes

Skew Symmetric Matrix for expressing a Rotation

Rotations in $\Bbb{R}^3$ can be represented by the orthogonal matrices of the group $SO(3)$. Since it is a Lie group, the elements of the associated Lie algebra $\mathfrak{so}(3)$ play the role of ...
Abezhiko's user avatar
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3 votes
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If $B = x(xI-A)^{-1}$ for a generator matrix $A$, then $B-B^2$ has positive diagonal elements

Let me continue from @user1551's idea and show that indeed $\mathbf{B} - \mathbf{B}^2$ has non-negative diagonals. Step 1. We recap @user1551's reduction. Let $\mathbf{A}$ be a transition-rate matrix ...
Sangchul Lee's user avatar
1 vote

If $B = x(xI-A)^{-1}$ for a generator matrix $A$, then $B-B^2$ has positive diagonal elements

Not an answer, but some observations. Let $d$ be the maximum diagonal entry of $-\frac{1}{x}A$. Then $S=I+\frac{1}{xd}A$ is a stochastic matrix whose diagonal elements are less than $1$. Conversely, ...
2 votes
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Sum of positive semi-definite matrix and positive definite matrix?

A matrix $M$ is positive-definite (semidefinite) if and only if it is symmetric and $u^TMu>0\space (\geq)$ for all nonzero vectors $u$. If $M$ is positive definite and $N$ is positive semidefinite ...
Julio Puerta's user avatar
0 votes

Change of Basis over $\mathbb{Z}$

${}_C[Id_V]_B = {}_C[Id_V]_I \, {}_I[Id_V]_B=C^{-1}B$ = $\begin{bmatrix} 2 \:0\: 2 \\ 1 \:2\: 0 \\ 0 \:2\: 1\end{bmatrix}$ $\begin{bmatrix} 1 \:2\: 1 \\ 0 \:1\: 1 \\ 2 \:1\: 1\end{bmatrix}$ =$\...
False Equivalence's user avatar
1 vote

why this equality involving matrix holds true?

In the paper you have $A=zz^T+\sigma\Delta$ and $z\in\{\pm1\}^n$. Therefore $z^Tz=n$ and the matrix $zz^T$ has a diagonal of ones. It follows that $$ \operatorname{ddiag}(zz^Tzz^T) =\operatorname{...
user1551's user avatar
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0 votes

How can I solve "average" best rank-$1$ approximation?

$ \def\LR#1{\left(#1\right)} \def\op#1{\operatorname{#1}} \def\trace#1{\op{Tr}\LR{#1}} \def\frob#1{\left\| #1 \right\|_F} \def\qiq{\quad\implies\quad} \def\mt{\mapsto} \def\p{\partial} \def\grad#1#2{...
greg's user avatar
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1 vote
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An unparalleled problem on matrices ....

Consider taking the product of $d\in \mathcal{D}_n$ matrices from your set $\mathcal{M}$. Then, according to your problem we have; $${\bf{N}}=M_1M_2\cdots M_d$$ Additionally, we know that $d<n$ for ...
Volk's user avatar
  • 1,472
4 votes

The relation between the eigenvalue of a Hermitian matrix and the block matrix that composed by it real and imaginary part

Short calculations with block matrices show that the block matrix $U=\frac{1}{\sqrt{2}} \begin{pmatrix} iI & I \\ -iI & I \end{pmatrix}$ is unitary, and that $U^*\begin{pmatrix}A-Bi & 0\\0 ...
leslie townes's user avatar
2 votes

The relation between the eigenvalue of a Hermitian matrix and the block matrix that composed by it real and imaginary part

You can represent the matrix entries of a complex matrix as $2\times 2$ blocks with real components, where $a+b i$ then becomes the block: $$ \begin{pmatrix} a & -b \\ b & a \end{pmatrix} $$ ...
Jos Bergervoet's user avatar
4 votes

The relation between the eigenvalue of a Hermitian matrix and the block matrix that composed by it real and imaginary part

First of all, it's well known (from the min-max theorem for example) that for a Hermitian matrix $H$ the maximum eigenvalue is given by $\sup\{\mathbf{z}^* H \mathbf{z}: \mathbf{z}^*\mathbf{z}=1\}$ ...
Giorgos Giapitzakis's user avatar
1 vote
Accepted

Change of Basis over $\mathbb{Z}$

What a strange notation. Note that ${}_I[Id_V]_B=B$, where $I$ is the unit matrix, and then ${}_C[Id_V]_I=C^{-1}$ and ${}_C[Id_V]_B = {}_C[Id_V]_I \, {}_I[Id_V]_B=C^{-1}B$.
colt_browning's user avatar
0 votes

Proving convexity of a function using Linear Algebra

You have a function $f : \mathbb R^n \to \mathbb R$ which can be written in the form $f(x) = x'Hx$ where $H$ is a symmetric matrix. The intuition is that $f$ is convex if and only if the restriction ...
Digitallis's user avatar
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1 vote

Specific basis for the space of symmetric matrices

Peliminary result : Let $(S_n)$ be the vector space of real symmetric $n \times n$ matrices. The set of all matrices of the form $xx^T-yy^T$ (which belong to $(S_n)$) generate $(S_n)$. General idea ...
Jean Marie's user avatar
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2 votes
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Sublinear convergence of a fixed point iteration

I want to prove the following (weaker) claim: there is $\rho\in (0,1)$ and $c>0$ such that $\|x_k - x^*\|_2 \le c \rho^k$. This is R-linear convergence, but not a contradiction to Q-sublinear ...
daw's user avatar
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1 vote
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"Coordinate-free" minors, submatrices and cofactors.

For the minors and the submatrices, they inherently rely on the way we picked our bases for the domain and the target space, so I would say it is impossible to completely do away with the coordinates, ...
BigbearZzz's user avatar
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0 votes

Looking at all permutations of a product of matrices

In the $2 \times 2$ case, with $M = \pmatrix{a_{11} & 1-a_{11}\cr a_{21} & 1 - a_{21}}$. your sum is $$\left(\begin{array}{cc} -a_{2,1} a_{1,1}+2 a_{1,1}+a_{2,1} & \left(a_{2,1}-2\right) \...
Robert Israel's user avatar
1 vote
Accepted

Verify Sherman–Morrison–Woodbury rank-one update inverse

I believe the key is to note that $$ab^T A^{-1} + ab^T A^{-1}ab^TA^{-1} = a(1 + b^TA^{-1}a)b^TA^{-1}$$ so then you get \begin{align*} \overline{A} \cdot \overline{A}^{-1} &= (A + ab^T) A^{-1} -...
PhysicsKid's user avatar
1 vote
Accepted

Matrix construction of Dorroh extension

With $\phi:(a,b)\mapsto \begin{bmatrix}a&0\\n&a+n\end{bmatrix}$ You have an injective additive map that also satisfies $$ \phi((a,n))\phi((b,m))=\begin{bmatrix}a&0\\n&a+n\end{bmatrix}\...
rschwieb's user avatar
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0 votes

Classify the conic $x^2+xy+3y^2+5x$ and determine its cartesian equation

The conic matrix $$A=\begin{pmatrix}1 & \frac{1}{2} & \frac{5}{2}\\ \frac{1}{2} & 3 & 0\\ \frac{5}{2} & 0 & 0 \end{pmatrix} \tag{1}$$ can be transformed into the following ...
John Alexiou's user avatar
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0 votes

Classify the conic $x^2+xy+3y^2+5x$ and determine its cartesian equation

The rotation angle for $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ is $\theta$ where $\tan{2\theta}=\frac{B}{A-C}$ which make $\cos{\theta}=\frac1{\sqrt{(2-\sqrt5)^2+1}},\sin{\theta}=-\frac1{\sqrt{(2+\sqrt5)^2+1}},$ ...
Jan-Magnus Økland's user avatar
1 vote
Accepted

Prove that the set of real symmetric matrices is closed

Decomposing into symmetric and antisymmetric parts we have $A= \frac12 (A+A^T)+\frac12 (A-A^T)$. We also know that $A_n = \frac12 (A_n + A_n^T)$ since it was assumed symmetric. Now if $A \neq A^T$ ...
CyclotomicField's user avatar
2 votes
Accepted

prove that if $A^k =0$ then there is $v \in \mathbb{F}^{n}$ such that the group $\{v,vA,vA^2,\dots,vA^{k-1}\}$ is linearly independent in induction

Hint: To prove the linear independence of $\{v, \dots, A^{k-1}v\}$, all you need to do is show that $\alpha_0 v + \cdots + \alpha_{k-1}A^{k-1}v = 0$ implies that $\alpha_0 = \cdots = \alpha_{k-1} = 0$....
BBBBBB's user avatar
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5 votes
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Is there a symmetric matrix with integer coefficients of order n?

Hint Since $A^n = {\bf 1}$, all of its eigenvalues are $n$th roots of unity. But all of the eigenvalues of a real symmetric matrix are real. (It could be helpful, too, to remember that symmetric ...
Travis Willse's user avatar
1 vote

Question about determinant formula.

$\mathbf{Solve}$ \begin{align} det(kA) &= \sum_{(i_1,i_2,\cdots,i_n)}(k\cdot a_{1i_1})(k\cdot a_{2i_2})\cdots(k\cdot a_{ni_n})\\ &=k^n\cdot\sum_{(i_1,i_2,\cdots,i_n)}a_{1i_1}\cdot a_{2i_2}\...
Kevin027's user avatar
24 votes
Accepted

Every matrix is a product of two symmetric matrices

This is true over all fields, including those of characteristic two. Moreover, one of the two symmetric matrices can be taken to be nonsingular. Let $A=P^{-1}CP$ where $C$ be the Frobenius normal form ...
0 votes

Rotation a 3D frame of a reference to match the X-axis with the direction of a unite vector.

I found the correct answer. In reference [1]. It is almost what @of course answered earlier. But they forgot that we need to take an inverse problem: Once the normal vector has been found, the angles (...
Khoder Alshaar's user avatar
0 votes

Normal block upper triangular matrix proof

I think the following may be an argument which is roughly of the kind the OP requests: suppose that $T$ is a normal operator on $\mathbb C^n$ which has matrix $$ \left(\begin{array}{cc} A & B\\ 0 &...
krm2233's user avatar
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2 votes
Accepted

Computing partial trace of a given kronecker product matrix with respect to the first component

Let $M$ denote the block matrix $$ M = \pmatrix{M_{11} & M_{12}\\ M_{21} & M_{22}}, $$ with each $M_{ij}$ of size $n \times n$. Then the partial trace with respect to the second component is ...
Ben Grossmann's user avatar
2 votes

$(A+B)^2\|B^{-1}\|-A$ is positive definite?

If $A\ge 0$, $B\ge 0$ then $$A^{\frac12} B A^{\frac12} \ge 0$$ PS: this property does not require $A$ and $B$ to commute. Since $B \ge \lambda_n I$ then $A+B \ge \lambda_n I$ then, \begin{align} \...
Kroki's user avatar
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0 votes

To find unknown rows in a unitary matrix

To complete a unit row $u_1=(u_{11},u_{12},u_{13})$ with $(u_{11},u_{12})\ne 0\ne u_{13}$ to a unitary matrix, we can choose a row $u_2$ proportional to $(\overline{u_{12}},-\overline{u_{11}},0)$ and ...
Alex Ravsky's user avatar
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Rotation a 3D frame of a reference to match the X-axis with the direction of a unite vector.

Let $T_y = \begin{bmatrix} c_1 && 0 && s_1 \\ 0 && 1 && 0 \\ -s_1 && 0 && c_1 \end{bmatrix} $ where $c_1 = \cos(\theta_y) , s_1 = \sin(\theta_y) $ And ...
of course's user avatar
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5 votes

Inverse of a special triangular matrix

The matrix $A$ in question is an example of an upper triangular Toeplitz matrix. As noted in the other answer, it can be expressed as $p(J)$ for some polynomial $p$ in the upper triangular nilpotent ...
user1551's user avatar
  • 138k
4 votes
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Inverse of a special triangular matrix

Here's a constructive approach, i.e., one that gives you an algorithm for computing the inverse $A^{-1}$. Denote the size of $A$ by $n \times n$, and denote by $$J := \pmatrix{\cdot&1\\&\cdot&...
Travis Willse's user avatar
0 votes
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Elementary Row Operations hoffman

The first row is simply a scaled-up version of the second row with the scalar being $(1-i)$. Since it is redundant information, it seems that they just made everything zero. More specifically, $r_1\...
Lucien Jaccon's user avatar
4 votes
Accepted

Solve Determinant Equation

Let $X:=\begin{pmatrix} \mathbf{x}& \mathbf{B} \\ \mathbf{0}& \mathbf{D} \\ \end{pmatrix}$ and let $Y:=\begin{pmatrix} \mathbf{y}& \mathbf{B} \\ \mathbf{c}& \mathbf{D} \\ \end{pmatrix}$...
Desperado's user avatar
  • 1,948
1 vote

Prove that $f(u+v)\le f(u) +f(v)$ being $f(u)= Au\cdot u\in\mathbb R$

If $a,b$ are norms then $n=(a^2+b^2)^{1/2}$ is as well. For this question we will only bother to check triangle inequality. Since $(x^2+y^2)^{1/2}$ is the standard norm on $\mathbb R^2$, it satisfies ...
Calvin Khor's user avatar
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1 vote

Closed form solution to matrix equation

This is the Sylvester-Transpose matrix equation sometimes called the T-Sylvester equation, analyzed in The solution of the equation $AX + X^* B = 0$ by Teran,Dopico If you want to implement a solver ...
Yaroslav Bulatov's user avatar

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