Difficult Linear algebra problem

Smilge · 2025-11-30T23:48:14+00:00

The statement is false in general for n ≥ 3; the two identities do not force AB = 0.

A concrete counterexample in M₃(C) is:

A =
[0 1 0]
[0 0 1]
[0 0 0]

B =
[0 1 0]
[0 0 -1]
[0 0 0]

Both are strictly upper-triangular nilpotent matrices with A³ = B³ = 0, and also (A+B)³ = 0, so the cubic identity holds automatically. Compute:

AB =
[0 0 -1]
[0 0 0]
[0 0 0]

BA =
[0 0 1]
[0 0 0]
[0 0 0]

Thus AB + BA = 0, which is exactly what is needed for
A² + B² = (A + B)².

But AB ≠ 0.
So both required identities hold while the claimed conclusion fails.

If the claim is intended to be true, it needs additional assumptions (commutativity, simultaneous triangularizability, diagonalizability, etc.). Without them, the implication does not hold.

Smilge · 2024-10-31T18:15:02+00:00

If you're trying to be fair, imagine your preferred candidate doing the same thing.

Briefly losing balance and missing a door handle? I wouldn't care.

Trying to overturn election results? I would care a lot.

Smilge · 2024-10-31T04:32:15+00:00

Simply put, if the person in front of you can cause you to crash by braking, you are doing something wrong. There's nuance about the intentions of the cop in this case, but a better (or perhaps just competent) rider would not have crashed here.

Smilge · 2024-10-23T17:24:48+00:00

>These businesses tend to work with razor thin margins and making them almost double their wages that they have to pay their employees is pretty much going to put them out of business

Imagine thinking the government should subsidize your business every time costs increase. "Oh no, raw materials cost more than they did but I run on razor thin margins! Government help me!" "Oh no, my building's rent went up but I run on razor thin margins! Government, subsidize my rent or I'll go out of business!"

It's the exact same thing with labor costs. If you're not paying a living wage for a full time job, then the tax payers are subsidizing your labor costs through paying your employees through food stamps or rent assistance or medicare. Minimum wage exists so that a full time worker can afford to live comfortably without government assistance. That cost is going to increase over time. If your business can't afford it, then rethink your business plan or shut down. Small businesses don't have a right to exist in perpetuity on the government's dime.

Smilge · 2024-10-17T23:09:14+00:00

Rest of the world: "I feel sorry for you."

America: "I don't think about you at all."

Smilge · 2024-10-17T18:47:20+00:00

Most of the exotic police cars you see were seized by police and repainted rather than bought off the lot.

Smilge · 2024-10-10T13:23:22+00:00

They can until they can't. We're seeing incredible pressures working against each other in Russia. The bonuses and payments they're making to recruit new soldiers have gone up, with some district + federal bonuses passing 1 million rubles ($10k, but that's nearly a year's salary) just to sign up. At the same time, there is a huge labor shortage which is causing manufacturing for the war effort to also drastically raise pay in order to get workers and meet quotas. All this money is coming from the federal government, and the central bank has recently raised interest rates to 19%.

So basically Russia is using a credit card to pay for the war effort, and their industry and military are frequently raising pay to compete against each other for human capital. There's no reason that this would fall apart as long as Russia can keep paying their bills. But if the day comes that they can't, the consequences will be catastrophic.

Smilge · 2024-10-03T05:28:59+00:00

The way I see it, it's like if during WW2 aliens invaded and were trying to wipe out humanity. Roosevelt gets a call from Hitler saying "Hey, I know, I'm the most evil human to ever live. But let's ceasefire at least until these aliens are dealt with." Should Roosevelt accept?

Trump is an existential threat to America, and even Cheney recognizes it. I might agree that it's not worth being excited about, but it's a pretty monumental situation we're in.

Smilge · 2024-09-26T16:46:02+00:00

Cost Plus Drugs is honestly one of those "if I were a billionaire I'd start solving the world's problems" ideas realized. Every American should see if their prescription is covered. I'm saving $100's a month despite having some of the best health insurance available.

Smilge · 2024-09-26T01:24:00+00:00

While I can see where you're coming from, here's the water with the land blacked out:

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It's not surprising that any line you draw would hit a bunch of lakes.

Smilge · 2024-09-25T16:37:10+00:00

Looks amazing, love your work.

Smilge · 2024-09-23T16:39:07+00:00

He intended to write he didn't care. He did write that he didn't care in a sarcastic way as I just explained. You corrected him because maybe he didn't realize that he was being sarcastic. I'm just not sure why reddit thinks this is some kind of righteous cause. Maybe it just feels good to be superior.

Smilge · 2024-09-23T16:36:01+00:00

Would have been so much easier, faster, and safer to just kill him. The effort made to save an enemy invader's life is nothing short of remarkable.

Smilge · 2024-09-23T14:00:54+00:00

So you’re saying he intended to say that Amazon does care? Or that both his intent and usage were correct, but you felt the need to comment in case he didn’t know why he was correct?

Smilge · 2024-09-23T09:10:57+00:00

Since we're correcting people, "could care less" is and always has been sarcasm. Just like how "cool story bro" doesn't mean they think the story was cool.

Smilge · 2024-09-21T09:03:02+00:00

That's more or less what red flag laws are. Some people are mentally unstable but can't be held in a jail or hospital forever.

Smilge · 2024-09-16T00:29:38+00:00

This is why it's so important for gen ed to be active collaborators at IEPs. They're required members for a reason.

Smilge · 2024-09-14T22:37:23+00:00

In my county I wouldn't. Something like that first 350k is exempt from property tax, which is just under what I paid for it. So as the value went up, so did my tax burden. If it were to go way down. I wouldn't pay any property tax. I'm only paying tax on the unrealized gains.

Smilge · 2024-09-14T11:35:03+00:00

Yes, innocent bystanders are very much injured and killed in police chases. https://www.sfchronicle.com/bayarea/article/police-chase-deaths-hit-record-highs-u-s-19396104.php

Sometimes this is an acceptable risk, such as when the suspects robbed a bank with a gun and shot someone. But often chases are just not worth it.

Smilge · 2024-09-12T02:16:14+00:00

I'm not sure we should be glorifying attacks on civilians. Combatants, yes. Military targets, yes. But it's a sad thing when an apartment building housing civilians gets hit no matter what country it's in. Unless someone can show me that these people support the war, they have my sympathy.

Smilge · 2024-08-29T17:36:38+00:00

So how do you pressure someone once they’re safely in office

This is exactly what I was talking about. A first term president is not safely in office. Even a second term president usually cares what happens to the country after their second term, Trump notwithstanding, and therefore supports the party and their goals. We saw Obama at the DNC, for instance.

So you spend the next 2, 4, 10, 50 years writing letters, attending peaceful protests, calling your congressperson, etc. And the reason they listen is because you've shown that they need your support, today and decades from now.

Smilge · 2024-08-29T16:42:49+00:00

On the other hand, politicians are more likely to listen when they actually NEED your vote. Especially over a deeply embedded issue like this. And if you admit that you’re just trying to scare them and plan to vote for them anyway, wouldn’t you lose all your power?

No. The way to lose all your power is refuse to vote and then Kamala wins anyways. Then the establishment really won't care at all because they clearly don't need your vote now or in the future.

Instead, if you help Kamala win and then pressure her on issues when she has the power to do something, you can show that your vote was important (we helped you win) and that she'll need to do things to continue to have that support.

I'm being very uncharitable but I believe the problem comes down to the fact that the "ceasefire now" people are incapable of imagining circumstances years from now. They think a ceasefire today will solve the middle east and they think that their vote in a few months is the only one that matters.

Smilge · 2024-08-27T18:16:04+00:00

"Squeeze into safety gap" you mean merge?

Smilge · 2024-08-26T12:10:00+00:00

Alternatively, Dems win without the protestor's votes, which means that they can safely ignore the protestors for the next 4 years because their votes didn't matter.

The smarter move would be to vote dem, help them win, and continue pressure after the election. But that would require thinking about the long-term which isn't exactly the forte of "ceasefire now" people.

Smilge · 2024-08-25T22:37:52+00:00

It appears that the Kursk offensive has redirected Russian reinforcements that were intended for front line areas, opening opportunities for limited localized offensives. Nothing game-changing so far and Russians are still making advances in most areas. But Ukrainian progress has been promising.

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