EnglishLinear AlgebraDeterminants
The determinant of a square matrix is a single number which captures some important information about how the transformation behaves. In this section, we will develop a geometrically-motivated definition of the determinant.
Exercise
Suppose that is a region in and that is an matrix. Consider the singular value decomposition .
- Let . By what factor does transform volumes?
- Let . In terms of the entries of , by what factor does transform volumes?
- Let . By what factor does transform volumes?
Solution. Since and are orthogonal, and both preserve volumes. So they multiply volumes by a factor of 1. Since scales volumes by a factor of along the first axis, along the second, and so on, it scales volumes by a factor of .
Volume scale factor
We see from this exercise that a linear transformation from to scales the volume of any -dimensional region by the same factor: the volume scale factor of .
Exercise
Find the volume scale factor of the matrix by describing how the matrix transforms a region in .
Solution. Since , we see that stretches (or compresses) regions in by a factor along the -axis and then reflects across the plane . For example, the unit cube is mapped to a box Since such a box has volume the volume scale factor of is
Orientation factor
Another geometrically relevant piece of information about is whether it preserves or reverses orientations. For example, rotations in are orientation preserving, while reflections are orientation reversing. Let's define the orientation factor of to be if is orientation preserving and if is orientation reversing.
Definition
We define the determinant of a transformation to be the product of its orientation factor and its volume scale factor.
We define the determinant of a matrix to be the determinant of the corresponding linear transformation .
Exercise
Interpret geometrically and use this interpretation to find , the determinant of .
Solution. Since , reflects points in across the line . Therefore, it preserves areas and reverses orientations. So its determinant is .
There is relatively simple formula for in terms of the entries of . For example,
is the determinant of a matrix. However this formula is terribly inefficient if has many entries (it has terms for an matrix), and all scientific computing environments have a det function which uses much faster methods.
Exercise
For various values of , use the expression np.linalg.det(np.random.randint(-9,10,(n,n))) det(rand(-9:9, n, n)) to find the determinant of an matrix filled with random single-digit numbers. How large does have to be for the determinant to be large enough to consistently overflow?
import numpy as np np.linalg.det(np.random.randint(-9,10,(n,n)))
using LinearAlgebra det(rand(-9:9, n, n))
Solution. Trial and error reveals that this determinant starts to consistently return inf Inf at .
Exercise
Suppose that and are matrices, with determinant and respectively. Suppose that is a 3D region modeling a fish whose volume is 14. What is the volume of the transformed fish ?
Solution. The volume of is . The volume of is .
Exercise
Let be 3D region modeling a fish, and suppose an invertible matrix. If has volume and has volume , then the determinant of is equal to
Solution. We can see that the matrix scales volumes by , and hence . This implies that .
Determinants can be used to check whether a matrix is invertible, since is noninvertible if and only if it maps to a lower-dimensional subspace of , and in that case squishes positive-volume regions down to zero-volume regions.
Exercise
Let Find the values of for which the equation has nonzero solutions for .
Solution. We can rewrite as , where is the identity matrix. We can rearrange this to give the equation . This has a nontrivial solution if has a nonzero nullspace. Since is a square matrix, this is equivalent to it having determinant zero.
Setting this equal to zero gives
The left-hand side can be factored
Thus our two solutions are .
Exercise
For an square matrix, which of the following is the relationship between and ?
Solution. The answer is (4) . There are two ways to see this,
To check that this is the right answer using algebra, let be the identity matrix, with determinant . The matrix is diagonal, with threes on the diagonal. Its determinant is the product of the entries on its diagonal, .
Geometrically, we know that the determinant of measures how much scales volume. The matrix scales by a factor of three more in each dimension. Since there are dimensions, the total scaling of volume is multiplied by a factor .
Exercise
Is every matrix with positive determinant positive definite?
Solution. No. Consider the negation of the identity matrix. It has determinant 1, yet its eigenvalues are both negative.
Congratulations! You have completed the Data Gymnasia Linear Algebra course.
