行列式(determinants)的几何意义

The determinant of a matrix is the factor by which areas are scaled by this matrix.

Because matrices are linear transformations it is enough to know the scaling factor for one single area to know the scaling factor for all areas. Let’s go back to our example:

 

The rectangle inscribed by the pink and blue unit vectors and has an area of 1. After applying our matrix transformation, this rectangle has turned into a parallelogram with base 2 and height 2. So it has an area of 4. This means, that our matrix scales areas by a factor of 4. Therefore, the determinant of our matrix is 4. Neat, isn’t it?

There is one caveat to the story: Determinants can be negative! If we start with an area of 1 and scale it by a negative factor, we would end up with a negative area. And negative areas are nonsense. So how can we make sense of our nice geometric definition in the presence of negative determinants? Luckily the fix is straightforward: If a matrix has a negative determinant, let’s say -2, areas are scaled by 2. The minus just means that space reversed its orientation. “What does that now even mean?”, you might rightfully ask. Let’s take a look:

We can see that the given matrix scales areas by a factor of 2. If we look closely we further notice that the blue vector was on the right of the pink vector but ended up on the left side. This is what’s meant by “space reversed its orientation”. That’s why the determinant of the matrix is not 2 but -2. Including negative determinants we get the full picture:

The determinant of a matrix is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

All our examples were two-dimensional. It’s hard to draw higher-dimensional graphs. The geometric definition of determinants applies for higher dimensions just as it does for two. In three-dimensional space, the determinant is the signed scaling factor for volumes and in even higher dimensions for hypervolumes.