determinants - analogy and linear transformations

1. Analogy of Determinants

In the context of matrices, the determinant can be thought of as analogous to the absolute value in the multiplication of real numbers. Here's why

• Scaling Factor The absolute value of a real number $a$ indicates how much $a$ scales other numbers when multiplied. For matrices, the determinant $\det(A)$ indicates how much the matrix $A$ scales areas (in 2 D), volumes (in 3 D), or hyper-volumes in higher dimensions.

• Sign of Determinant The sign of the determinant (positive or negative) can be compared to the sign of a real number. A positive determinant indicates preservation of orientation, while a negative determinant indicates a reversal of orientation. Similarly, a positive number maintains direction, while a negative number reverses it.

• Multiplicative Property Just like the product of absolute values $|a \cdot b| = |a| \cdot |b|$ for real numbers, the determinant of the product of two matrices equals the product of their determinants $\det(AB) = \det(A) \cdot \det(B)$.

2. Expansion on Linear Transformations and Volume Scaling

Linear Transformations and Dimensionality

Square matrices represent linear transformations that map an $n$ -dimensional space onto itself. This property is crucial for understanding why determinants are defined for square matrices

• Linear Transformations

  • A matrix $A \in \mathbb{R}^{n \times n}$ defines a transformation $T: \mathbb{R}^n \to \mathbb{R}^n$.
  • If $A$ is a square matrix, it transforms an $n$ -dimensional vector space back into the same $n$ -dimensional vector space.

• Volume Scaling

  • The determinant $\det(A)$ of a matrix $A$ provides a scalar value that indicates how volumes are scaled under the transformation.
  • For example, if $A$ represents a transformation in 2 D space, $\det(A)$ tells us how the area of a shape is scaled. If $\det(A) = 2$, a shape will have twice the area after the transformation.
  • In 3 D, if $\det(A) = 0.5$, the transformation shrinks volumes to half their original size.

Non-Square Matrices

Non-square matrices represent transformations between spaces of different dimensions, which makes volume scaling interpretation non-applicable

• Dimension Mismatch

  • A non-square matrix $B \in \mathbb{R}^{m \times n}$ defines a transformation $T: \mathbb{R}^n \to \mathbb{R}^m$.
  • Since the input and output spaces have different dimensions, the concept of preserving volume does not apply. For instance, a $2 \times 3$ matrix might map a 3-dimensional vector to a 2-dimensional vector.

• Geometric Interpretation

  • Consider a $2 \times 3$ matrix that maps 3 D space into a plane (2 D space). This transformation does not preserve volume, but instead projects the higher-dimensional space onto a lower-dimensional subspace.
  • Determinants are not defined for such transformations because the concept of volume scaling between spaces of different dimensions is not meaningful.

Example to Illustrate Volume Scaling

Consider a 2 D example for simplicity

• Transformation Matrix $$ A = \begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix} $$

• Area Scaling

  • The determinant of $A$ is $$ \det(A) = 2 \cdot 2 - 1 \cdot 1 = 4 - 1 = 3 $$
  • This means that any shape in 2 D space will have its area scaled by a factor of 3 under the transformation defined by $A$.

• Orientation

  • Since $\det(A) > 0$, the orientation of shapes is preserved. If the determinant were negative, it would indicate that the transformation includes a reflection, reversing orientation.

Summary

In summary, the determinant of a square matrix serves as a crucial measure of how linear transformations affect volumes and orientations within the same dimensional space. Non-square matrices, representing transformations between different dimensions, do not have determinants because the concept of volume scaling is inapplicable in those cases. The analogy to absolute values helps to intuitively grasp the significance of determinants in scaling transformations.