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Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n \det \left(A^2+B^2+C^2\right)$$

This problem is a test question from China (xixi). It is said one can use the equation

$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)$$

but I can't use this to prove it. Can you help me?

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    $\begingroup$ Doesn't your hint suggest $A$, $B$, and $C$ are commutative by multiplication? $\endgroup$
    – DanielV
    Jul 9, 2014 at 6:59
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    $\begingroup$ I believe the statement is likely to be true. It is easy to see the matrix $$\Delta \stackrel{def}{=} 6A\left(A-\frac{5}{12} I_n\right)^2 + 6B\left(B-\frac{5}{12} I_n\right)^2 + 6C\left(C-\frac{5}{12} I_n\right)^2$$ is positive semi-definite. This leads to $$\det\left(6(A^3+B^3+C^3)+\color{red}{\frac{25}{24}}I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ Since $A + B + C = I_n$, the $3$ squares in $\Delta$ cannot vanish at the same time. It sounds pausible to me $\Delta$ is actually "bounded below" by $\frac{1}{24}I_n$. If this is indeed the case, then the original inequality is true. $\endgroup$ Aug 21, 2014 at 19:19
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    $\begingroup$ @WillNelson I've carried out a numerical expt to verify the statement for $n$ up to 10, with 100,000 valid triples $(A,B,C)$ for each $n$. No counterexamples were found, but two things are worth notice. Let $X=6(A^3+B^3+C^3)+I$ and $Y=5(A^2+B^2+C^2)$. Numerical examples show that (1) $X-Y$ is not necessarily positive semidefinite. So, achille's conjecture in the previous comment that $\Delta\ge\frac{I}{24}$ is false. (2) The sorted eigenvalues of $X$ do not dominate their counterparts of $Y$. E.g. it can happen that $$\lambda_\max(X)>\lambda_\max(Y)>\lambda_\min(Y)>\lambda_\min(X).$$ $\endgroup$
    – user1551
    Sep 12, 2014 at 9:16
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    $\begingroup$ The analogous inequality for nonnegative real numbers, $$6(a^3+b^3+c^3)+1\ge5(a^2+b^2+c^2)$$ with $a+b+c=1$, can be solved straightforwardly: Rewrite the LHS as $6(a^3+b^3+c^3)+1=6(a^3+b^3+c^3)+(a+b+c)^3=7(a^3+b^3+c^3)+3\sum a^2b+6abc$, and the RHS as $5(a^2+b^2+c^2)=5(a^2+b^2+c^2)(a+b+c)=5(a^3+b^3+c^3)+\sum a^2b$, so that the inequality is equivalent to $2(a^3+b^3+c^3-\sum a^2b+3abc)\ge0$, which is Schur's inequality. Here, $\sum a^2b$ means $a^2(b+c)+b^2(c+a)+c^2(a+b)$. And this is enough to show that the original inequality works for positive definite diagonal or triangular matrices. $\endgroup$ Sep 13, 2014 at 8:01
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    $\begingroup$ Note. (1) $A,B,C$ do not necessarily commute. So, this problem does not immediately reduce to the scalar case. (2) $D=6(A^3+B^3+C^3)+I-5(A^2+B^2+C^2)$ is not always positive semidefinite. E.g. when $A=\frac1{60}\pmatrix{45&0\\ 0&20},\,B=\frac1{60}\pmatrix{3&6\\ 6&20},\,C=\frac1{60}\pmatrix{12&-6\\ -6&20}$, we have $D=\frac1{60^3}\pmatrix{113400&4860\\ 4860&-1080}=\frac1{400}\pmatrix{210&9\\ 9&-2}$. $\endgroup$
    – user1551
    Dec 31, 2016 at 15:39

3 Answers 3

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Here is a partial and positive result, valid around the "triple point" $A=B=C= \frac13\mathbb 1$.

Let $A,B,C\in M_n(\mathbb C)$ be Hermitian satisfying $A+B+C=\mathbb 1$, and additionally assume that $$\|A-\tfrac13\mathbb 1\|\,,\,\|B-\tfrac13\mathbb 1\|\,,\, \|C-\tfrac13\mathbb 1\|\:\leqslant\:\tfrac16\tag{1}$$ in the spectral or operator norm. (In particular, $A,B,C$ are positive-definite.)
Then we have $$6\left(A^3+B^3+C^3\right)+\mathbb 1\:\geqslant\: 5\left(A^2+B^2+C^2\right)\,.\tag{2}$$

Proof: Let $A_0=A-\frac13\mathbb 1$ a.s.o., then $A_0+B_0+C_0=0$, or $\,\sum_\text{cyc}A_0 =0\,$ in notational short form. Consider the

  • Sum of squares $$\sum_\text{cyc}\big(A_0 + \tfrac13\mathbb 1\big)^2 \:=\: \sum_\text{cyc}\big(A_0^2 + \tfrac23 A_0+ \tfrac19\mathbb 1\big) \:=\: \sum_\text{cyc}A_0^2 \:+\: \tfrac13\mathbb 1$$
  • Sum of cubes $$\sum_\text{cyc}\big(A_0 + \tfrac13\mathbb 1\big)^3 \:=\: \sum_\text{cyc}\big(A_0^3 + 3A_0^2\cdot\tfrac13 + 3A_0\cdot\tfrac1{3^2} + \tfrac1{3^3}\mathbb 1\big) \\ \;=\: \sum_\text{cyc}A_0^3 \:+\: \sum_\text{cyc}A_0^2 \:+\: \tfrac19\mathbb 1$$ to obtain $$6\sum_\text{cyc}\big(A_0 + \tfrac13\mathbb 1\big)^3+\mathbb 1 \;-\; 5\sum_\text{cyc}\big(A_0 + \tfrac13\mathbb 1\big)^2 \:=\: \sum_\text{cyc}A_0^2\,(\mathbb 1 + 6A_0) \:\geqslant\: 0$$ where positivity is due to each summand being a product of commuting positive-semidefinite matrices. $\quad\blacktriangle$

Two years later observation:
In order to conclude $(2)$ the additional assumptions $(1)$ may be weakened a fair way off to $$\tfrac16\mathbb 1\:\leqslant\: A,B,C\tag{3}$$ or equivalently, assuming the smallest eigenvalue of each matrix $A,B,C\,$ to be at least $\tfrac16$.

Proof: Consider the very last summand in the preceding proof. Revert notation from $A_0$ to $A$ and use the same argument, this time based on $(3)$, to obtain $$\sum_\text{cyc}\big(A-\tfrac13\mathbb 1\big)^2\,(6A -\mathbb 1)\:\geqslant\: 0\,.\qquad\qquad\blacktriangle$$

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Not a full proof, but a number of thoughts too long for a comment. This post aims at finding alternative (yet harder) criteria for proving the conjecture. Please discuss.

As in previous comments, let's denote $ X=6(A^3+B^3+C^3)+I $ and $ Y=5(A^2+B^2+C^2) $ and $D = X-Y$.

The question is to show $\det(X) \ge \det(Y)$ or $1 \ge \det(X^{-1}Y)$. Write $Q = X^{-1}Y$, then $Q$ is positive definite, since $X$ and $Y$ are positive definite. Now it is known for a positive definite matrix $Q$ (see e.g. here) that the trace bound is given by $$ \bigg(\frac{\text{Tr}(Q)}{n}\bigg)^n \geq \det(Q) $$

So a second (harder) criterion for the conjecture is $n \ge \text{Tr}(X^{-1}Y)$ or $ \text{Tr}(X^{-1}D) \geq 0$. I wouldn't see how I can compute this trace or find bounds, can someone?

Let's call $d_i$ the eigenvalues of $D$, likewise for $X$ and $Y$. While $x_i > 0$, this doesn't necessarily hold for $d_i$ since we know from comments that $D$ is not necessarily positive definite. So (if $X$ and $D$ could be simultaneously diagonalized) $ \text{Tr}(X^{-1}D) = \sum_i \frac{d_i}{x_i} = r \sum_i {d_i}$ where there exists an $r$ by the mean value theorem. Where $r$ is not guaranteed to be positive, it is likely that $r$ will be positive, since $r$ will only become negative if there are (many, very) negative $d_i$ with small associated $x_i$. Can positivity of $r$ be shown? If we can establish that a positive $r$ can be found, a third criterion is $ \text{Tr}(D) \geq 0$.

Now with this third criterion, we can use that the trace is additive and that the trace of commutors vanishes, i.e. $\text{Tr} (AB -BA) = 0$. Using this argument, it becomes unharmful when matrices do not commute, since under the trace their order can be changed. This restores previous solutions where the conjecture was reduced to the valid Schur's inequality (as noted by a previous commenter), which proves the conjecture.


A word on how hard the criteria are, indicatively in terms of eigenvalues:

(hardest) positive definiteness: $d_i >0$ $\forall i$ or equivalently, $\frac{y_i}{x_i} <1$ $\forall i$

(second- relies on positive $r$) $ \text{Tr}(D) \geq 0$: $\sum_i d_i \geq 0$

(third) $n \ge \text{Tr}(X^{-1}Y)$: $\sum_i \frac{y_i}{x_i} \leq n$

(fourth - least hard) $\det(X) \ge \det(Y)$: $\prod_i \frac{y_i}{x_i} \leq 1$

Solutions may also be found by using criteria which interlace between those four.


A word on simulations and non-positive-definiteness:

I checked the above criteria for the non-positive definite example given by @user1551 in the comments above, and the second, third and fourth criteria hold.

Note that equality $\det(X) = \det(Y)$ occurs for (a) symmetry point: $A=B=C=\frac13 I$ and for (b) border point: $A=B=\frac12 I$ and $C=0$ (and permutations). I checked the "vicinity" of these equality points by computer simulations for real matrices with $n=2$ where I extensively added small matrices with any parameter choices to $A$ and $B$ (and let $C = I - A-B$), making sure that $A,B$ and $C$ are positive definite. It shows that for the vicinity of the symmetry point, the second, third and fourth criteria above hold, while there occur frequent non-positive-definite examples. For the vicinity of the border point all four criteria hold.

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Faulty Proof

In the comments we have a proof over positive real numbers $a,b,c \geq 0$ that the inequality holds

$$ 6 \big( a^3 + b^3 + c^3 \big) + 1 \geq 5(a^2 + b^2 + c^2) $$

We can lift this to diagonal matrices, but how to we lift this to Hermitian positive definite matrices? Hermitian already gives you $\lambda \in \mathbb{R}$. I think that is missing is that

$$ X = \Big[ 6 \big( A^3 + B^3 + C^3 \big) + I_n \Big]- \Big[ 5(A^2 + B^2 + C^2) \Big] \geq 0 $$

is also positive definite. In particular one shows $\mathbf{v}^T X \mathbf{v} \geq 0$ for all unit vectors $|| \mathbf{v} || = 1$

$$ \lambda_{min} = \inf_{||v|| =1} \mathbf{v}^T X \mathbf{v} $$

This is the variational characterization of the smallest eigenvalue. Through elaborate checks and balances one can obtain * all * eigenvlues. [1] (Not needed here, but one time I needed to compute $\lambda_2$)

There are three positive definite matrices. $X = X_1 - X_2$ where

  • $X_1 = 6 \big( A^3 + B^3 + C^3 \big) + 1 \geq 0$
  • $X_2 = 5 \big( A^2 + B^2 + C^2 \big) \quad \;\, \geq 0$

all of this follows from the variational inequality. Last we can take the determinant of both sides:

$$ \det \bigg[ 6 \big( A^3 + B^3 + C^3 \big) + 1\bigg] \geq \det (X_1 - X_2) + \det X_2 \geq 5^n \det \bigg[ \big( A^2 + B^2 + C^2 \big) \bigg] $$

There are a few discussions of inequalities of this type on MathOverflow

$\det (X+Y) \geq \det X + \det Y $ follows from Minkowski inequality.


Other Ideas

I have thought of several faulty proofs over night but here is one observation:

  • all eigenvalues $0 \leq \lambda_A, \lambda_B, \lambda_C\leq 1 $

I think this is right. Certainly they are all $\geq 0$. And then we can use variational inequality

$$ 0 \leq \lambda_A + \lambda_B + \lambda_C \leq v^T ( A+B+C) v = v^T v = 1$$

There is even easier argument if we take the trace and there is equality.

$$ \text{Tr}(A+B+C) = \sum \lambda_A + \sum \lambda_B + \sum \lambda_C = n $$

This is very restrictive (as Hans said) we get that $|\lambda_i - \lambda_j| \leq 1$ always and a fair bit more. In particular, the max and the min are separated by less than 1.

Remember: in many cases faulty proofs can be turned into correct ones! Thanks.

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  • $\begingroup$ according to user1551's comment two years ago (Sep 12'14), numerical simulation indicate $X_1 - X_2$ need not be positive semi-definite. $\endgroup$ Jan 3, 2017 at 17:41
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    $\begingroup$ @achillehui did anyone ever find the original test question? $\endgroup$
    – cactus314
    Jan 3, 2017 at 18:05
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    $\begingroup$ Nope. aside from OP's claim it is one of china (xixi) test. no one here have the original test question. $\endgroup$ Jan 3, 2017 at 18:29
  • $\begingroup$ What's missing in almost everybody's attempt is the importance of the (very restrictive) condition $A+B+C=I_{n}$ . That's one reason why I give this answer, too, little (+0) credit. $\endgroup$ Jan 4, 2017 at 19:21
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    $\begingroup$ @cactus314: Yesterday, I've voluntarily removed a rather elaborate question and my own answer to it : How to prove this enduring conjecture for diagonal matrices only? . I'm in good company: this question seems to be top 3 with respect to the number of deleted answers. I'm not suggesting anything, though :-( $\endgroup$ Jan 6, 2017 at 16:14

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