There's bound to be conflict.
Already Trump, on the campaign trail a couple months before the election, accused Yellen of manipulating monetary policy to favor President Barack Obama and pave the way for Hillary Clinton, who nonetheless lost the election. For her part, Yellen has refused to be drawn into a public battle, rebuffing any and all efforts to get her to criticize Trump.
Where they go from here, now that the Yellen Fed has approved a rate hike in the central bank's first post-election meeting, will be something investors must watch closely.
"I'm sure that the president-elect would like to have his pro-growth agenda not be killed in the cradle by the Federal Reserve," said Ed Keon, portfolio manager and managing director at investment research firm QMA. "He's fortunate to have a chair at the Federal Reserve who shares a similar set of ideas about how the U.S. economy should perform to reach a greater percentage of people in the next couple of years."
At the root of Trump's fiscal plan is a rollback in tax rates and increased domestic infrastructure spending in the neighborhood of $1 trillion. Many evaluations of the plan figure that will tag on another $5 trillion or so to the $19.9 trillion national debt.
Whether that gaudy figure is manageable is fully dependent on the cost of debt service. In other words, low rates are going to be pivotal to make aggressive fiscal policy work.
Yellen has been unflinchingly dovish when it comes to her view on rates, though the Fed appears to be on the path back to policy that at least strives for normalization if not actual tightening.
"Actually there's a good chance that [it] will, despite some friction, be a positive relationship for both of them," Keon said. "They both have a similar agenda in trying to improve the lives of people who have been left behind by improvements in markets and the economy over the past few years."
Yellen and her predecessor Ben Bernanke have beseeched Washington lawmakers to come up with a cohesive, pro-growth fiscal policy. Instead, Congress and Obama have been deadlocked on spending priorities, resulting in budget-by-resolution that has come with little vision on promoting growth.
Given the opportunity on Wednesday during her quarterly news conference to praise or pan the president-elect, Yellen chose to do neither, at least overtly.
She did, somewhat surprisingly, dismiss the importance of fiscal spending to help with the jobs picture.
"I would say at this point that fiscal policy is not obviously needed to provide stimulus to help us get back to full employment," Yellen said. Then came the caveat: "Nevertheless, let me be careful that I am not trying to provide advice to the new administration and Congress on what is the appropriate stance of policy. There are many considerations that Congress needs to take account of."
Still, some of those considerations fall squarely in the Fed's lap.
Rising rates not only will push up the cost of debt and capital but also the value of the U.S. dollar. A strong greenback gives consumers more purchasing power and cuts the prices of imports. However, it also hurts multinationals by raising the cost of exports.
For his part, Trump has toned down the anti-Fed rhetoric.
Given the chance Wednesday to sound off on his prolific Twitter feed about the Fed decision, the president-elect was mum. He's made little noise about any changes he would implement on the central bank, though two and perhaps three vacancies at the Fed will give him the opportunity to remake it in whatever vision he chooses.
Peter Boockvar, chief market analyst at economic advisory firm The Lindsey Group, said that when it comes to the path of rates, "the Fed is still woefully behind the eight ball," which could complicate the Trump-Yellen relationship.
"I'm waiting for the Trump tweet complaining about the strong dollar," Boockvar said.