Let A be a nonsingular 2 ×2 matrix with singular value decomposition A=P ΣQ^T and singular value

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Let A be a nonsingular 2 ×2 matrix with singular value...

Let $A$ be a nonsingular $2 \times 2$ matrix with singular value decomposition $A=P \Sigma Q^T$ and singular values $\sigma_1 \geq \sigma_2>0$. (a) Prove that the image of the unit (Euclidean) circle under the linear transformation defined by $A$ is an ellipse, $E=\{A \mathbf{x} \mid\|\mathbf{x}\|=1\}$, whose principal axes are the columns $\mathbf{p}_1, \mathbf{p}_2$ of $P$, and whose corresponding semi-axes are the singular values $\sigma_1, \sigma_2$. (b) Show that if $A$ is symmetric, then the ellipse's principal axes are the eigenvectors of $A$ and the semi-axes are the absolute values of its eigenvalues. (c) Prove that the area of $E$ equals $\pi|\operatorname{det} A|$. (d) Find the principal axes, semi-axes, and area of the ellipses defined by $A$ is singular? (i) $\left(\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right)$, (ii) $\left(\begin{array}{rr}2 & 1 \\ -1 & 2\end{array}\right)$, (iii) $\left(\begin{array}{ll}5 & -4 \\ 0 & -3\end{array}\right)$. (e) What happens if

Let $A$ be a nonsingular $2 \times 2$ matrix with singular value decomposition $A=P \Sigma Q^T$ and singular values $\sigma_1 \geq \sigma_2>0$. (a) Prove that the image of the unit (Euclidean) circle under the linear transformation defined by $A$ is an ellipse, $E=\{A \mathbf{x} \mid\|\mathbf{x}\|=1\}$, whose principal axes are the columns $\mathbf{p}_1, \mathbf{p}_2$ of $P$, and whose corresponding semi-axes are the singular values $\sigma_1, \sigma_2$. (b) Show that if $A$ is symmetric, then the ellipse's principal axes are the eigenvectors of $A$ and the semi-axes are the absolute values of its eigenvalues. (c) Prove that the area of $E$ equals $\pi|\operatorname{det} A|$. (d) Find the principal axes, semi-axes, and area of the ellipses defined by $A$ is singular?
(i) $\left(\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right)$,
(ii) $\left(\begin{array}{rr}2 & 1 \\ -1 & 2\end{array}\right)$,
(iii) $\left(\begin{array}{ll}5 & -4 \\ 0 & -3\end{array}\right)$.
(e) What happens if

Key Concepts

Principal Axes and Ellipse Geometry

The principal axes of the ellipse are the directions along which the ellipse stretches the most and the least. In the context of SVD, these axes correspond to the columns of one of the orthogonal matrices (usually denoted as P), aligning with the directions of maximum and minimum stretching defined by the singular values. This concept is central when analyzing the geometry of ellipses resulting from linear maps.

Determinant and Area Scaling

The determinant of a matrix encodes the scaling factor by which the area (or volume in higher dimensions) is transformed under the linear transformation. In the context of mapping a circle to an ellipse, the area of the ellipse is exactly the area of the circle scaled by the absolute value of the determinant. This relationship provides a clear geometric interpretation of the determinant in terms of area distortion.

Eigenvalues and Eigenvectors in Symmetric Matrices

For symmetric matrices, the singular value decomposition simplifies to eigen decomposition because the matrix can be diagonalized with an orthogonal basis of eigenvectors. In such cases, the eigenvectors serve as the principal axes of the transformed ellipse, and the absolute values of the eigenvalues correspond to the lengths of the semi-axes. This connection is a key concept in linear algebra, highlighting the special structure and properties of symmetric matrices.

Singular Value Decomposition

Singular Value Decomposition (SVD) is a factorization technique for matrices wherein any matrix is written as the product of three matrices: two orthogonal matrices and a diagonal matrix consisting of nonnegative real numbers called singular values. SVD is fundamental in understanding how a matrix transforms space, particularly in decomposing its action into rotations and scaling along orthogonal directions.

Linear Transformation of a Circle into an Ellipse

A linear transformation defined by a matrix maps the unit circle into an ellipse. The process involves scaling and rotating the circle, where the scaling factors (the singular values) determine the lengths of the semi-axes of the ellipse and the rotation given by the orthogonal matrices provides the orientation. This concept helps in visualizing how any linear operator distorts geometry.

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