Let T be a positive definite 2 ×2 matrix. Let En={T^n 𝐱 |𝐱=1}, n=0,1,2, …, be the image of the u

step 1 If we assume that the ellipse has been rotated to eliminate the xy term, then this implies that

Let T be a positive definite 2 ×2 matrix. Let En={T^n 𝐱...

Let $T$ be a positive definite $2 \times 2$ matrix. Let $E_n=\left\{T^n \mathbf{x} \mid\|\mathbf{x}\|=1\right\}, n=0,1,2, \ldots$, be the image of the unit circle under the $n^{\text {th }}$ power of $T$. (a) Prove that $E_n$ is an ellipse. True or false: (b) The ellipses $E_n$ all have the same principal axes. (c) The semi-axes are given by $r_n=r_1^n, s_n=s_1^n$. (d) The areas are given by $A_n=\pi \alpha^n$ where $\alpha=A_1 / \pi$.

Key Concepts

Matrix Powers and Invariance of Principal Axes

Raising a matrix to a power corresponds to applying the linear transformation repeatedly. In cases where the matrix has been diagonalized via its eigenvectors, these eigenvectors remain invariant under successive powers of the matrix. This invariance means that the directions of maximal and minimal stretching (the principal axes) are preserved, even though the lengths of these axes change according to the corresponding eigenvalues raised to increasing powers.

Spectral Decomposition and Eigenvalues

For symmetric (and by extension, positive definite) matrices, spectral decomposition allows the matrix to be expressed in a basis of its eigenvectors. The eigenvalues indicate the amount of stretching along these eigenvector directions. When the matrix is raised to a power, each eigenvalue is raised to that power, which directly affects the lengths of the semi-axes of the resulting ellipse. This principle is key to understanding how iterative applications of a linear transformation modify geometric objects.

Area Scaling Under Linear Transformations

The area of a transformed region under a linear map is scaled by the absolute value of the determinant of the transformation. When a matrix is raised to a power, its determinant is raised to that power, so the area of the transformed ellipse scales in an exponential manner. Understanding this scaling is crucial for connecting the algebraic properties of a matrix (like its determinant) with the geometric properties (such as area) of the shapes it transforms.

Positive Definite Matrices

A positive definite matrix is one for which the associated quadratic form is strictly positive for all nonzero vectors. In the context of real matrices, this property typically implies symmetry, and the matrix has all positive eigenvalues. This feature ensures well-behaved geometric transformations, such as preserving the orientation and ‘spreading out’ of vectors, which is crucial when analyzing how the unit circle transforms under repeated applications of the matrix.

Linear Transformations of the Unit Circle

Applying a linear transformation to the unit circle results in an ellipse in general. The shape and orientation of the ellipse are determined by how the transformation stretches and rotates the circle. This idea is central to understanding images of geometric objects under linear maps and is often explored through the transformation's action on the standard basis, linking geometric intuition with algebraic operations.

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