Intereting Posts

Connected, locally connected, path-connected but not locally path-connected subspace of the plane
When is a recurrence relation linear
Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$
A question about the fixed point and group action.
The convergence of a sequence of sets
Tough quadrilogarithm integral
Functions defined by integrals (problem 10.23 from Apostol's Mathematical Analysis)
Product of perfect maps is perfect
Computing matrix-vector calculus derivatives
Why is the decimal representation of $\frac17$ “cyclical”?
Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$
$C(S^1)$ does not have a single generator
Can one differentiate an infinite sum?
Notions of equivalent metrics
Are there any interesting semigroups that aren't monoids?

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices.

The geodesic distance between $A,B\in\Bbb{S}_{++}^n$ is given by the following Riemannian metric

$$

d(A,B):= \Bigg(\operatorname{tr}\bigg(\ln^2\big(\sqrt{A^{-1}}B\sqrt{A^{-1}}\big)\bigg)\Bigg)^{\frac{1}{2}}.

$$

**EDIT**

It could also be defined as follows

$$

d(A,B):= \lVert\log(A^{-1}B)\rVert_{F}

$$

- Drawing a tetrahedron from a parellelepiped to convince myself it is 1/6th the volume,
- How to randomly construct a square full-ranked matrix with low determinant?
- Upper Triangular Form of a Matrix
- Find three real orthogonal matrices of order $3$ having all integer entries.
- Why $\bigwedge^{d-1}A=\bigwedge^{d-1}B \Rightarrow A= \pm B$
- Sparseness of a Vector

**EDIT II** I could show the negative-definiteness of $d^2$ if I could write $d$ as follows

$$

d(A,B)=\lVert P-Q\rVert_{F},

$$

where $P$ depends solely on $A$, and $Q$ depends only on $B$. So, is there any way of expressing $d$ as above?

I would like to prove that $d^2$ is a negative-definite function.

For the negative-definiteness of a function $f\colon\mathcal{X}\times\mathcal{X}\to\Bbb{R}$, it suffices to show that, for all $m\in\Bbb{N}$, $\{x_1,\ldots,x_m\}\subset\mathcal{X}$, $m\in\Bbb{N}$, $\{c_1,\ldots,c_m\}\subset\Bbb{R}$, $\sum_{i=1}^{m}c_i=0$, the following holds true

$$

\sum_{i,j=1}^{m}c_ic_jf(x_i,x_j)\leq0.

$$

- On a non-standard approach to the classification of conics?
- Prove $A^tB^t = (BA)^t$
- If $A,B\in M(2,\mathbb{F})$ and $AB=I$, then $BA=I$
- Span of Permutation Matrices
- How can we parametrise this matricial hypersphere?
- Fredholm alternative theorem for matrices
- $A \in {M_n}$ is normal.why the range of $A$ and ${A^*}$ are the same?.
- Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?
- Is axiom of choice required for there to be an infinite linearly independent set in a (non-finite-dimensional) vector space?
- Order of general linear group of $2 \times 2$ matrices over $\mathbb{Z}_3$

- Closed form for $\sum_{n=1}^\infty \left(e-\left(1+\frac{1}{n}\right)^n \right)^2$?
- Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?
- Proof of Existence of Algebraic Closure: Too simple to be true?
- Solving for $A$ in $Ax = b$
- Exact sequences of $SU(N)$ and $SO(N)$
- Classify orbits of conjugating action on $GL_2(\mathbb{C})$
- How to obtain the series of the common elementary functions without using derivatives?
- Why $X$ independent from $(Y,Z)$ implies that $E(XY^{-1} | Z)= E(X) E(Y^{-1}|Z)$?
- Examples proving that the tensor product does not commute with direct products
- If $(a_n)$ is increasing and $\lim_{n\to\infty}\frac{a_{n+1}}{a_1\dotsb a_n}=+\infty$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is irrational
- Every $k$ vertices in an $k$ – connected graph are contained in a cycle.
- Correlation between three variables question
- $f$ strictly increasing does not imply $f'>0$
- Fastest way to check if $x^y > y^x$?
- Adding relations to a partial order