Even simpler without the math: Let’s say you eat a piece of cake. You say, “hmm, salty, sweet, fruity, nice texture.” Let’s call those attributes the “princi

Even simpler without the math: Let’s say you eat a piece of cake. You say, “hmm, salty, sweet, fruity, nice texture.” Let’s call those attributes the “principal components” of the cake from your point of view. Are there others? Maybe you could have also said “moist, or fluffy,” but you didn’t because those weren’t as obvious, so not as important. When the cake was made, according to the recipe, there were no instructions on how to add “sweet” or “fruity.” Instead, there was a list of ingredients: sugar, vanilla, lemon juice, flour, water, baking powder, etc. The mixture of these ingredients in the quantities dictated (plus the baking) resulted in the cake having the characteristics that you described. Some of the characteristics have a strong reliance on just one or two ingredients, e.g. “sweet” with “sugar,” and some characteristics are the result of subtle combinations of many ingredients, e.g. “texture.” The list of characteristics (principal components) definitely describe the cake, but in a more convenient and relevant way. You don’t need the whole list of ingredients to describe the cake. This is what makes principal component analysis useful.