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Singular Value Decomposition (SVD) Calculator: Advanced Linear Algebra

SVD Calculator & Visualizer

SVD Calculator

Decompose any $2 \times 2$ matrix into its singular value components instantly.

Input Matrix (A)

A =

Results

$$ A = U \cdot \Sigma \cdot V^T $$
Matrix U (Rotation)
Left Singular Vectors
Matrix Σ (Scaling)
Singular Values (Diagonal)
Matrix Vᵀ (Rotation)
Right Singular Vectors Transposed

Geometric Visualization

Visualizing how the unit circle (blue) transforms into an ellipse (red) under matrix A.

Original Unit Circle
Transformed Ellipse
Major Axis (σ₁)
Minor Axis (σ₂)

Understanding SVD

Singular Value Decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any $m \times n$ matrix via an extension of the polar decomposition.

The Components

  • U (Left Singular Vectors): An orthogonal matrix representing a rotation or reflection.
  • Σ (Sigma): A diagonal matrix with non-negative real numbers on the diagonal. These are the singular values, representing the stretching factor along specific axes.
  • Vᵀ (Right Singular Vectors): Another orthogonal matrix representing the initial rotation of the input space.

Applications

SVD is used extensively in signal processing and statistics. In machine learning, it powers Principal Component Analysis (PCA), helping to reduce the dimensionality of data while preserving its variance. It is also used in image compression, where keeping only the largest singular values allows for reconstructing an image with significantly less data.

FAQ

What if Sigma has zeros?
If a singular value is zero, the matrix is singular (non-invertible). Geometrically, this means the transformation collapses the space along that dimension (e.g., a 2D plane becomes a line).
Is SVD unique?
The singular values $\Sigma$ are unique. However, the matrices $U$ and $V$ are not always unique; columns corresponding to repeated singular values can be rotated.
How is this useful for students?
Understanding SVD provides a geometric intuition for matrix operations. It connects linear algebra concepts like rank, range, and null space in a unified framework.
Tip: Try entering a matrix like `[[1, 1], [1, 1]]` (singular matrix) to see how the ellipse collapses into a line!

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