in solving large-scale linear systems, focusing on computational efficiency, numerical stability, and versatility

across different problem domains. At its core, matrix decomposition is a crucial tool in numerical linear algebra,

allowing us to break down complex systems into more manageable forms.

The study delves into both the theoretical underpinnings and practical performance of these methods,

highlighting their respective strengths and weaknesses. SVD stands out for its ability to handle ill-conditioned

matrices and is widely used in dimensionality reduction and data analysis, whereas Eigenvalue Decomposition

is commonly applied to structured problems and spectral analysis.

To validate the performance claims, computational experiments were conducted using randomly generated

matrices of sizes 100×100, 1000×1000, and 5000×5000. The results show that both methods perform efficiently

for small matrices, but as matrix size increases, SVD demonstrates better computational performance on the

Matrix decomposition, also known as matrix factorization, is a fundamental tool in linear algebra and numerical

analysis that simplifies complex matrix operations by breaking them into the product of simpler, structured

matrices. Historically, the motivation for decomposition arose from the need to solve large systems of linear

equations, which form the backbone of scientific computation, engineering models, and data-driven research.

Early techniques such as Gaussian elimination laid the foundation for systematic approaches to matrix reduction,

but it was not until the mid-20th century with the advent of digital computers that decomposition methods gained

prominence (Trefethen, L.N and Bau, D. 1997) . Householder’s Principles of Numerical Analysis (1954)

emphasized the role of partitioned matrices and block-based LU decomposition as an efficient strategy for high

speed computation, marking a significant departure from purely manual methods (Householder 1954); (Lu, J.

For instance, single value decomposition and Eigenvalue decomposition, are widely employed in solving

systems of equations, least squares problems, eigenvalue computations, and optimization challenges (Higham,

2002 & Ejikeme et al, 2015). In computer science and machine learning, matrix decomposition plays a critical

role in the development of algorithms for dimensionality reduction, collaborative filtering, and feature extraction

(Meyer 2000). For example, SVD is foundational in Principal Component Analysis (PCA), which is used for

pattern recognition and data compression. Similarly, factorization techniques underpin neural network training

via back propagation, reinforcing the link between numerical linear algebra and artificial intelligence (Lu, 2021;

Duan, Jiang, and Jain, 2022).

Despite their ubiquity, decomposition methods encounter limitations when applied to modern large-scale

datasets and computational systems. Sparse and structured matrices, common in graph theory and scientific

systems in mathematics now power modern innovations in artificial intelligence, business analytic, and decision

support systems. Such applications reaffirm that matrix decomposition is not merely a mathematical tool but a

foundational bridge between theory and practice (Benzi 2002 and Demnel 2007).

Matrix decomposition methods, although powerful, face significant challenges when applied to large-scale linear

systems(Golu, G.H., 2009) and (Peng, R.and Vempala, S. 2021).

While several classical decomposition methods such as LU and QR exist in the literature, they are not the focus

of this study. Instead, this research concentrates on evaluating the performance of Singular Value Decomposition

(SVD) and Eigenvalues Decomposition in large scale settings.

These methods are particularly suitable for modern applications involving high dimension and structured data,

where numerical stability and robustness are critical.

datasets and randomly generated matrices. Consequently, the scope of this research does not extend to

implementations in specialized software packages such as LAPACK, ARPACK, or Suite-sparse.

Singular Value Decomposition

In linear algebra, few tools are as powerful and versatile as the Singular Value Decomposition (SVD). While LU

decomposition targets square matrices for solving linear systems, SVD extends its utility far beyond,

encompassing even non-square matrices, whether tall or wide (Ryan 2008). From numerical stability to

dimensionality reduction, SVD is a cornerstone in applied mathematics, machine learning, signal processing,

and statistics.

Given any m × n real matrix A, SVD enables us to express A as the product of three matrices:

the result. These insights are critical when analyzing or compressing data, computing pseudoinverses, or solving

ill-posed problems.

such that:

,

where:

admits a singular value

decomposition.

. Both are positive semi-definite and hence have orthonormal eigenvectors

with non-negative eigenvalues. Let:

where Σ† is the pseudoinverse of Σ. It can be shown that U is orthogonal, and thus:

Remark:

(i). The singular values of A are the square roots of the eigenvalues of ATA.

(ii). The columns of V are the eigenvectors of ATA; those of U are the eigenvectors of AAT .

(iii). The number of non-zero singular values equals the rank of A.

4. Image processing: SVD compresses grayscale or RGB images by truncating smaller singular values

Singular Value Decomposition: Step-by-Step Method

Let A be an m × n real matrix.

.

Then

So,

Example 3: Low-Rank Matrix

Let

Then

Eigenvectors of

So,

of the matrix, identifies its rank, and provides the optimal low-rank approximation. Its numerical stability and

theoretical depth make it indispensable in applied linear algebra.

Eigenvalue Decomposition:

In linear algebra, understanding the internal structure of matrices plays a central role in both

theoretical and applied mathematics. One such structural insight is offered by eigenvalue

decomposition, also known as spectral decomposition. This decomposition provides a

way of analyzing a square matrix by breaking it down into a set of eigenvectors and eigenvalues quantities that

capture the matrix’s most essential properties.

Eigenvalue decomposition applies to a special class of matrices specifically, square matrices

that are diagonalizable. For such matrices, the decomposition expresses the matrix as the

where D is a diagonal matrix whose entries are the eigenvalues of A, and P is a matrix whose columns are the

corresponding eigenvectors.

The utility of eigenvalue decomposition is vast. It underlies the solution of differential equations, stability

analysis, quantum mechanics, and machine learning algorithms such as Principal Component Analysis (PCA).

Diagonal matrices are computationally simple to manipulate raising them to powers, computing exponentials,

and applying functions become straight forward. Thus, diagonalizing a matrix simplifies these operations

substantially.

However, not all matrices are diagonalizable. A matrix may fail to have enough linearly independent

eigenvectors, making P non-invertible. In such cases, the Jordan canonical form or Schur decomposition is

employed instead. Nevertheless, for symmetric (or Hermitian in the complex case) matrices, diagonalizability is

where v is a non-zero vector (the eigenvector) and λ is a scalar (the eigenvalue). This equation states that the

action of matrix A on vector v merely stretches (or compresses or flips) the vector without changing its direction.

eigenvectors, then A is diagonalizable, and there exists an invertible matrix P and a diagonal matrix D such that:

where:

 The columns of P are the eigenvectors of A,

 The diagonal entries of D are the corresponding eigenvalues.

Summary

if it has n linearly independent eigenvectors.

Principal Component Analysis (PCA)

Eigenvalue decomposition is at the core of PCA, which is used for dimensionality reduction in statistics and

machine learning.

Stability Analysis

In dynamical systems, the eigenvalues of the system matrix determine stability. The system is stable if all

eigenvalues have negative real parts.

on their numerical stability, computational efficiency, and suitability for different types of problems.

Eigenvalue Decomposition proved particularly useful for theoretical analysis, offering insights into matrix

structure, stability, and diagonalization. It's also a crucial component in spectral analysis and applications like

Principal Component Analysis (PCA). However, our experiments showed that its computational performance

was outpaced by SVD in solving large-scale systems.

SVD, on the other hand, demonstrated strong performance across the board, showcasing better computational

efficiency and superior numerical stability - especially when dealing with large, ill-conditioned matrices. This

makes SVD a reliable choice for practical applications, including machine learning, recommender systems, and

dimensionality reduction. Although SVD is theoretically computationally intensive, our results suggest that it

can outperform Eigenvalue Decomposition in practice, depending on the system configuration and

1. Use SVD for solving ill-conditioned or rank-deficient problems where accuracy is critical, such as in signal

processing, machine learning, or inverse problems.

2. Employ Eigenvalue decomposition mainly for spectral analysis, stability studies, or system where matrix

diagonalization is required.

There is no one size solution when it comes to solving large scale systems of equations using matrix

decomposition. Each method has its advantages depending on the nature of the matrix and the requirements of

the problem. Eigenvalue decomposition serves well in theoretical and analytical contexts while SVD offer

greater stability for ill-conditioned or over determined systems. The best approach is to first analyze the structure

and condition of the matrix. Based on this, select the decomposition method that offers the best trade off between

(2nd ed.) Academic Press.