Penelope's loyalty to Odysseus can be seen throughout the epic poem. She remains a devoted wife to her husband. She is the epitome of a devoted wife. Examples of this can be seen in her creative ways of putting off the many suitors who've taken over her house in Ithaca. In Odysseus' absence, many suitors have taken over her house wanting to marry her so they can inherit Odysseus' kingdom and all that comes with it. Penelope wants nothing to do with them. She still lovers her husband and believes he is alive and will return home. She tells the suitors that she is weaving a shroud for Odysseus' dead father Laertes. She tell them that when she finishes the shroud, she will marry one of them. However, what she weaves during the day, she unweaves at night hoping to buy herself some time until Odysseus comes home. When the suitors find out about her plan, she must come up with another one. She tells the suitors that whichever one can string Odysseus' bow and shoot an arrow through 12 ax-helve sockets will win her hand in marriage. However, what the suitors do not know that Penelope does know is that only Odysseus himself can accomplish this feat. Again, she is hoping to buy herself some time until Odysseus can come home and reclaim his kingdom.
There are several places throughout the Odyssey where Penelope shows herself to be a loyal wife to Odysseus. Most obviously is her deriving numerous stratagems to allow her to refuse to marry one of the suitors, the most clever being weaving during the day and undoing her weaving at night. She has told the suitors that she will choose one of them when she has finished weaving the shroud for burial of Laertes, but by undoing the weaving she guarantees that she will not complete it (Book 2). She also displays loyalty in questioning travelers (including Odysseus himself disguised as a beggar) to find news of her husband.
Odysseus' belief in Penelope's loyalty motivates him on his travels, including stiffening his resolve to leave Calypso and Circe, but this belief, although it proves correct, is not based on any actual communication with Penelope. Instead, rather than Penelope actively helping Odysseus in his travels (something she cannot do), her main function is to ensure that he has a home to which to return and to serve symbolically as a motivating force on his journey.