The SVD decomposition of a 2 by 2 matrix.

Note: This interact and it's description was generated by Gemini with the prompt below, and then modified a bit by me.

Prompt: An interact which takes a vector x, 2 by 2 matrix A, finds its SVD decomposition U,D,V, and draws the row space eigenvectors in black/grey and column space vectors in red/pink, then draws x and Ax in blue and magenta, and shows the eigenvalues of A in the title.

The Singular Value Decomposition (SVD) of the matrix $A = \begin{pmatrix} 2 & -3 \\ 1 & -3/2 \end{pmatrix}$ decomposes $A$ into three matrices such that $A = U \cdot \Sigma \cdot V^T$ (or $V^*$)

1. Unitary Matrix $U$ (Left Singular Vectors) $$U \approx \begin{pmatrix} -0.8944 & -0.4472 \\ -0.4472 & 0.8944 \end{pmatrix}$$ The columns of $U$ are basis for the output space (the column space/range of $A$).

Properties: The columns of $U$ are orthogonal unit vectors that define the principal directions in the output space after transformation.

2. Diagonal Matrix $\Sigma$ or $D$ (Singular Values) $$D \approx \begin{pmatrix} 4.0311 & 0 \\ 0 & 0 \end{pmatrix}$$

Role: Contains the non-negative singular values $(\sigma_1 \approx 4.0311, \sigma_2 = 0)$ along its main diagonal, sorted in descending order.

Interpretation: $\sigma_1 \approx 4.0311$ represents the primary scaling/stretching factor along the first principal axis.$\sigma_2 = 0$ indicates that matrix $A$ is singular (rank 1). The 2D input space is flattened onto a 1D line in the output space, meaning $\det(A) = 0$.

3. Unitary Matrix $V$ or $V^T$ (Right Singular Vectors)$$V \approx \begin{pmatrix} -0.5547 & 0.8321 \\ 0.8321 & 0.5547 \end{pmatrix}$$

Role: Represents the orthonormal basis for the input space (domain of $A$).

Properties: The columns of $V$ are the directions in the domain that map to the principal directions in $U$, scaled by the respective singular values in $D$.

Here's another Gemini generated modified by me interact. This one takes your 2 by 2 or 3 by 3 matrix and computes eigenvalues/eigenvectors and right and left singular values and singular vectors and plots the vectors.