Singular Values Calculator Normal Distribution

Marchenko-Pastur Distribution for Singular Values:

\[ f(\lambda) = \frac{1}{2\pi \gamma \lambda} \sqrt{(\lambda_{\text{max}} - \lambda)(\lambda - \lambda_{\text{min}})} \] \[ \lambda_{\text{min}} = (1 - \sqrt{\gamma})^2, \quad \lambda_{\text{max}} = (1 + \sqrt{\gamma})^2 \] \[ \gamma = \frac{m}{n}, \quad \sigma = \sqrt{\lambda} \]

Probability Density f(λ):

Corresponding Singular Value (σ):

λ Range: [λ_min, λ_max]:

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1. What is the Marchenko-Pastur Distribution?

The Marchenko-Pastur distribution describes the asymptotic behavior of singular values (or eigenvalues) of random matrices with independent identically distributed (i.i.d.) entries. It's fundamental in random matrix theory and has applications in signal processing, statistics, and machine learning.

2. How Does the Calculator Work?

The calculator uses the Marchenko-Pastur density formula:

Where:

\( \lambda \) — Eigenvalue of the Wishart matrix
\( \gamma \) — Aspect ratio (m/n) of the matrix
\( \sigma \) — Singular value of the original matrix
\( m \) — Number of rows in the random matrix
\( n \) — Number of columns in the random matrix

Explanation: The distribution describes the limiting spectral density of large random matrices with normal distributed entries.

3. Importance of Singular Value Distribution

Details: Understanding the distribution of singular values is crucial for principal component analysis (PCA), signal detection, and understanding the behavior of large dimensional data.

4. Using the Calculator

Tips: Enter the matrix dimensions (m, n) and an eigenvalue λ. The calculator will show the probability density and corresponding singular value. Note that f(λ) = 0 outside [λ_min, λ_max].

5. Frequently Asked Questions (FAQ)

Q1: What types of matrices does this apply to?
A: The distribution applies to m×n matrices with i.i.d. entries having zero mean and finite variance (often normal distribution).

Q2: What is the significance of γ?
A: γ = m/n determines the shape of the distribution. When γ = 1 (square matrix), the distribution has a special symmetric form.

Q3: How does this relate to PCA?
A: In PCA of large random data matrices, the bulk of eigenvalues follow this distribution, with possible outliers signaling true structure.

Q4: What happens when γ > 1?
A: For γ > 1 (more rows than columns), there are additional zero eigenvalues not described by this density.

Q5: How accurate is this for finite matrices?
A: The distribution becomes exact as m,n → ∞ with m/n → γ, but works well for moderately large matrices (m,n > 50).