Intereting Posts

Factorization, simple expression!
Are there arbitrarily large gaps between consecutive primes?
There are 3 points in the spectrum of some self-adjoint element of a non-unital C*-algebra.
This one weird trick integrates fractals. But does it deliver the correct results?
How to read letters such as $\mathbb A$, $\mathbb B$, etc., or $\mathfrak A$, $\mathfrak B$, etc.?
Which of the following is not true?
How to prove floor identities?
Does Stirling's formula give the correct number of digits for $n!\phantom{}$?
Sanity check on example 6.5 from “Counterexamples in probability and real analysis” by Wise and Hall
How to prove that equality is an equivalence relation?
probability circle determined by chord determined by two random points is enclosed in bigger circle
Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)
Prove that intervals of the form $(a,b]$, $$, $[a,\infty)$ do not have the fixed point property.
Solving $ \sin x + \sqrt 3 \cos x = 1 $ – is my solution correct?
Finding the Jordan canonical form of this upper triangule $3\times3$ matrix

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix with positive entries $\lambda_i > 0$ (some of them might coincide). If we have the matrix equation $A D A^t = D$, does it follow that $A$ is invertible with $A^t = A^{-1}$?

It is obviously true if $A$ and $D$ commute. ~~For $(2\times 2)$-matrices one can check by hand that the claim is true.~~ I expect a counter-example for bigger matrices, but can not come up with one.

- Are one-by-one matrices equivalent to scalars?
- Why teach linear algebra before abstract algebra?
- Bases are looping using simplex method
- Eigenvalues of a Permutation?
- Jordan canonical form of an upper triangular matrix
- row operations, swapping rows

- Counting symmetric unitary matrices with elements of equal magnitude
- $AB-BA$ is a nilpotent matrix if it commutes with $A$
- Orbits of $SL(3, \mathbb{C})/B$
- Is there a name for the group of complex matrices with unimodular determinant?
- Action of a matrix on the exterior algebra
- Finding number of matrices whose square is the identity matrix
- Winning strategies in multidimensional tic-tac-toe
- Show $\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A,A\in M(2,\mathbb{C})$
- Proving: “The trace of an idempotent matrix equals the rank of the matrix”
- Proving linear independence

Presumably all matrices here are real square matrices. Let $Q=D^{-1/2}AD^{1/2}$. Then the equation in question becomes $QQ^T=I$. Hence $Q$ must be real orthogonal and $A$ must be in the form of $D^{1/2}QD^{-1/2}$. Therefore, $A$ in invertible but it’s not necessarily orthogonal. For instance, consider

$$

D=\pmatrix{1\\ &2},\ Q=\frac1{\sqrt{2}}\pmatrix{1&-1\\ 1&1},

\ A=D^{1/2}QD^{-1/2}=\pmatrix{\frac1{\sqrt{2}}&-\frac12\\ 1&\frac1{\sqrt{2}}}.

$$

- Inequality with Complex Numbers
- Distances between randomly distributed points in a ball
- How does rounding affect Fibonacci-ish sequences?
- Number of $2n$-letter words using double $n$-letter alphabet, without consecutive identical letters
- Fibered products in $\mathsf {Set}$
- Invariance of subharmonicity under a conformal map
- How can LU factorization be used in non-square matrix?
- Don't we need the axiom of choice to choose from a non-empty set?
- $\frac{\partial^2 u}{\partial t^2}-c^2\frac{\partial^2u}{\partial x^2}=0\text{ implies } \frac{\partial^2 u}{\partial z \partial y}=0$
- Calculating the length of the semi-major axis from the general equation of an ellipse
- “Binomial theorem”-like identities
- Determinating the angle in a triangle
- Borel subalgebras contain solvable radical
- What's application of Bernstein Set?
- Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds