Intereting Posts

Find: $ \int^1_0 \frac{\ln(1+x)}{x}dx$
Two fundamental questions about convexity of a function (number1)
Cyclotomic polynomials explicitly solvable??
Proof about a Topological space being arc connect
Proving that Tensor Product is Associative
Description of SU(1, 1)
For a Planar Graph, Find the Algorithm that Constructs A Cycle Basis, with each Edge Shared by At Most 2 Cycles
A confusing excersice about Bayes' rule
How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $
Is it wrong to say $ \sqrt{x} \times \sqrt{x} =\pm x,\forall x \in \mathbb{R}$?
$n$th derivative of $e^x \sin x$
diophantine equation $x^3+x^2-16=2^y$
Probability distribution and their related distributions
Question regarding partial derivatives 1
Finitely many prime ideals lying over $\mathfrak{p}$

I tried to find matrices A, which are both orthogonal and symmetric, this means

$A=A^{-1}=A^T$. I only found very special examples like I, -I or the matrix

$$\begin{pmatrix}

0 &0& -1\\

0& -1& 0\\

-1& 0& 0

\end{pmatrix} $$

Can a matrix with the desired properties only contain the values -1,0 and 1 ?

Which matrices of a given size have the desired property ?

- Find the determinant of the linear transformation $L(A)=A^T$ from $\mathbb{R}^{n\times n}$ to $\mathbb{R}^{n\times n}$
- Is the trace of inverse matrix convex?
- Is there a 3-dimensional “matrix” by “matrix” product?
- Proving that the coefficients of the characteristic polynomial are the traces of the exterior powers
- A matrix is similar to its transpose
- Calculating the eigenvalues of a matrix
- Inner product exterior algebra
- $\operatorname{Adj} (\mathbf I_n x-\mathbf A)$ when $\operatorname{rank}(\mathbf A)\le n-2$
- Finding the rotation matrix in n-dimensions
- Matrices whose Linear Combinations are All Singular

For your first question, the answer is no. Every real Householder reflection matrix is a symmetric orthogonal matrix, but its entries can be quite arbitrary.

In general, if $A$ is symmetric, it is orthogonally diagonalisable and all its eigenvalues are real. If it is also orthogonal, its eigenvalues must be 1 or -1. It follows that every symmetric orthogonal matrix is of the form $QDQ^\top$, where $Q$ is a real orthogonal matrix and $D$ is a diagonal matrix whose diagonal entries are 1 or -1.

$A$ is orthogonal and symmetric, so $A=A^{-1}$ and $A=A^{T}$. More

general, let $A$ be a unitary and self-adjoint operator with discrete

spectrum in a separable Hilbert space. Then $A=\exp [iW]$ with $W$

self-adjoint and $A=A^{\ast }=\exp [-iW]$. Thus $W=\sum_{n}\lambda _{n}P_{n}$

with $\lambda _{n}\in \mathbb{R}$ and the $P_{n}$ are orthogonal projectors,

$\lambda _{m}\neq \lambda _{n}$, $m\neq n$ and $P_{m}P_{n}=\delta _{mn}P_{m}$

. Now

\begin{equation*}

A=\sum_{n}\exp [i\lambda _{n}]P_{n}=A^{\ast }=\sum_{n}\exp [-i\lambda

_{n}]P_{n},

\end{equation*}

so $\exp [2i\lambda _{n}]=1$ leading to $\lambda _{n}=k_{n}\pi $, $k_{n}\in

\mathbb{Z}$, which is either $+1$ or $-1$.

Can a matrix with the desired properties only contain the values -1,0 and 1 ?

For this part of your question every 3-D rotation matrix (it’s orthogonal) about any axis ( defined by a unit vector $v$) by angle $\pi$ is symmetric.

You can generate plenty of them with Rodrigues’ rotation formula which for a $\pi$ case takes simpler form $rot(v, \pi)= 2vv^T-I$ and they are not necessary consist only of $-1, 0, 1 $.

- What happens to the frequency-spectrum if this sine-signal gets reset periodically?
- Examples of compact sets that are infinite dimensional and not bounded
- How to solve $\frac{2}{3\sqrt{2}}=\cos\left(\frac{x}{2}\right)$?
- constructive canonical form of orthogonal matrix
- Reference request: compact objects in R-Mod are precisely the finitely-presented modules?
- Is $2^k = 2013…$ for some $k$?
- Support of a vector
- $p$-group and normalizer
- How to integrate a three products
- Proof of Non-Convexity
- Definition of the Infinite Cartesian Product
- How to prove unitary matrices require orthonormal basis
- Computing the Integral $\int\tanh\cos\beta x\,dx$
- Fixed point combinator (Y) and fixed point equation
- $f: \to \mathbb{R}$ is differentiable and $|f'(x)|\le|f(x)|$ $\forall$ $x \in $,$f(0)=0$.Show that $f(x)=0$ $\forall$ $x \in $