Chapter 1: The Discussion Section
"If the characteristic polynomial is zero at λ equals three—with multiplicity two—does that guarantee we can diagonalize this matrix over the reals?"
Evans Hall Room 106 smelled faintly of dry-erase markers and old coffee. Ethan stood at the whiteboard, marker in hand, while twenty undergraduates scribbled notes or stared at the board.
Ethan was twenty-six. Despite being a third-year PhD student, he hadn't yet succumbed to long nights of paper-reading and proof-solving that would cause him to lose the athletic build years of diligence at the gym had built. His fitted button-downs hinted at the body underneath without trying too hard -- and students noticed.
Two of them noticed more than most.
"What's diagonalization again?"
Chris, a twenty-year-old with an easy grin that immediately made people want to talk to him, chirped up in the second row. Chris was the type who asked the questions in section that everyone wanted to know the answer to but were too afraid to ask; Linear Algebra was hard, but not everyone wanted to admit it. He liked watching Ethan's hands move across the board and would've been content to watch that button-down contort around Ethan's frame as he sketched equations, but liked the way Ethan's voice got animated with excitement when responding to questions even more.
"No, a matrix is diagonalizable over the reals only if, for every eigenvalue, its geometric multiplicity equals its algebraic multiplicity. The algebraic multiplicity being 2 does not guarantee the geometric multiplicity is also 2."
In the back row, Tom spoke up. Tom was twenty-one, a senior, slim and dark-haired, with green eyes behind thinner frames than Ethan's. Until now, he'd kept his head down over his notebook, tuning the class out while he finished that week's problem set. Tom didn't need to participate -- he knew the material already -- but wanted Ethan to see he knew it as well.
"That's correct! While my statement was true for the identity-scaled matrix 3I, it wouldn't be true for... okay, the blank stares in the room tell me I've lost all of you. Let's stop here for today. Read ILA Section 6 and let's try again on Thursday."
As the class rushed out the door, happy to be done with this discussion section for a math class they had to take for their major but knew they would never use, Chris lingered.
"Hey, Ethan -- got a minute?"
Ethan capped the marker, turned, and smiled. “Always. What’s up?”
Chris shouldered his backpack and stepped closer: "That last problem set crushed me. The eigenvalue stuff. I could really use some pointers."
They ended up in Ethan's office in Evans Hall -- small, cluttered with textbooks and printouts, shared with 2 other graduate students. Chris sat across the desk as Ethan walked him through canonical forms. It took a while; as Ethan leaned forward, reaching across the desk to annotate Chris's work, his unbuttoned collar revealed his well-defined pecs and subtle abs built from a combo of late-night gym sessions after frustrating advisor meetings and grad student budgets that funded lean proteins and not much else, causing Chris to spend more time thinking of ways to look down that collar without attracting attention than on the problem set.
They finished late in the evening; as they walked out of Evans, the sun was setting and cold air from the San Francisco Bay had started rolling in.
"You're a really good teacher, you know that?" Chris said, concentrating hard on making eye contact with Ethan instead of trying to figure out if he saw Ethan's nipples poking out from behind his shirt's thin linen.
Ethan flushed slightly. "Thanks, I try."
“You should let me buy you coffee sometime. As thanks.”
Ethan hesitated — TA–student boundaries — but coffee felt harmless. He grinned: “Sure. If you pay a bit more attention during office hours next time, we'll finish when it's still light out. Northside cafe should be open.”
Chris chuckled and quickly turned to walk away, acting like he was in a rush to cover up the panic that was quickly, viscerally, setting in. "Fuck! Did he see me?"
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