Preheat oven to 350°F (175°C). Grease and flour a 9x9 inch pan.
In a large bowl, beat together eggs, oil, sugar, and vanilla. Mix in flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Stir in carrots and fold in walnuts. Pour into prepared pan.
Bake in preheated oven for 45 minutes, or until a toothpick inserted into the center of the cake comes out clean. Let cool in pan for 10 minutes, then turn out onto a wire rack and cool completely.