Two strong signs of recovery appeared last month as industrial production rose sharply and housing starts soared.

Led by a 10 percent rise in auto assemblies, industrial output climbed by 0.9 percent in January, after a revised 0.1 percent gain in December, the Federal Reserve reported. And housing starts increased by 36 percent to an annual rate of 1.7 million, the Commerce Department said.

"The housing recovery is developing much faster than we had expected," President Harry Pryde of the National Association of Home Builders commented, adding that the January increase in starts "is an indication of the strong pent-up demand for housing . . . all the housing statistics confirm that the recovery is on track."

A 16 percent increase in building permits between December and January also was reported yesterday by the Commerce Department. This gain suggests that the housing boom will continue in coming weeks, an NAHB analyst said. He warned, however, that if interest rates were to rise, the housing recovery could be choked off.

The housing industry has picked up much more than industry as a whole. Starts last month were 95 percent higher than a year earlier. By contrast, the January rise in industrial production still left output in the nation's factories 3.2 percent below the level of a year earlier, and 11 1/2 percent lower than its peak in July 1981, the Federal Reserve said in its release on industrial production.

A sharp rise in auto production led the nationwide increase in overall industrial output in January, according to the Federal Reserve report. Auto assemblies were up 10 percent from December to an annual rate of 5.6 million units, the release said. The industry has "scheduled another pretty good increase" in ouptut in February, Chief Commerce Department economist Robert Ortner said yesterday.

Lower interest rates are a major factor behind the revival in both the housing and auto industries, experts say. After having started in these sectors, the recovery should gradually spread across the economy.

However, analysts caution that the improvement will continue only if consumer spending picks up in coming months. Recent figures indicate that car sales dropped off somewhat in late January and early February, so that they are now running at about the same level as production, Ortner said. If sales drop below the February production levels, manufacturers may reduce assemblies again in order to stop a buildup of unwanted inventories, experts say.

The industrial production figures in January also were helped by the exceptionally mild weather, Ortner said. With last weekend's snow storm along the Eastern Seaboard, output in February likely will be distorted in the opposite direction, he added.

The unseasonable warmth in January may have contributed to the spectacular increase in housing starts but it was not the sole reason, NAHB's Pryde said yesterday.

The January increase in auto assemblies spilled over into higher steel production, the report indicated, while rising construction boosted output of supplies. "In January the production of construction supplies and of basic metals, particularly steel . . . advanced sharply," it said.

Consumer goods output rose by 0.9 percent last month, after a 0.6 percent increase in December. Durable-goods industries, including autos, increased production by 2.8 percent, while nondurables-which are less affected by cyclical changes in the economy-rose 0.2 percent. Business spending, however, remains weak. Output of business equipment edged down by 0.1 percent in January after a rise of 0.6 percent in December. Output in this sector is now 20 percent below its peak before the recession, the Federal Reserve said.

Intermediate-goods production was up by 0.7 percent last month from December, the report said. Within that sector, output of construction supplies was up by 1.4 percent.