Preheat oven to 350 degrees. Grease a 9 x 13 inch cake pan, line with parchment paper, grease parchment, and flour pan.
Drain pineapple in fine-mesh strainer set over bowl, pressing to remove juice.
Pour juice into small saucepan and cook over medium heat until reduced to ⅓ cup, about 5 minutes; set aside.
Whisk flour, baking powder, baking soda, cinnamon, and salt together in bowl.
Whisk sugar and eggs together in separate large bowl. Once combined, whisk in oil.
Stir in bananas, pecans, vanilla, drained pineapple, and reduced pineapple juice.
Stir in flour mixture until just combined.
Pour batter into prepared pan and smooth top with rubber spatula. Bake until dark golden brown on top and toothpick inserted in center comes out clean, 45 to 50 minutes, rotating pan halfway through baking.
Allow cake to cool in pan on wire rack for 20 minutes. Remove cake from pan, discarding parchment, allow to cool completely on rack, about 2 hours.