#### QR Decomposition with Householder Reflections

The more common approach to QR decomposition is employing Householder reflections rather than utilizing Gram-Schmidt. In practice, the Gram-Schmidt procedure is not recommended as it can lead to cancellation that...

#### QR Decomposition with the Gram-Schmidt Algorithm

QR decomposition is another technique for decomposing a matrix into a form that is easier to work with in further applications. The QR decomposition technique decomposes a square or rectangular...

#### Image Compression with Singular Value Decomposition

As mentioned in a previous post, image compression with singular value decomposition is a frequently occurring application of the method. The image is treated as a matrix of pixels with...

#### Singular Value Decomposition in R

Following from a previous post on the Cholesky decomposition of a matrix, I wanted to explore another often used decomposition method known as Singular Value Decomposition, also called SVD. SVD...

#### How to Calculate Eigenvalues and Eigenvectors Manually and with R

Eigenvalues and eigenvectors prominently appear in many statistical and other computational fields that require transformations of linear systems or are interested in the evolution of systems from an initial point....

#### The Matrix Trace in R and Some Properties of the Trace

Although comparatively straightforward in nature, the matrix trace has many properties related to other matrix operations and often appears in statistical methods such as maximum likelihood estimation of the covariance...

#### Cholesky Decomposition of a Positive-Definite Matrix

Cholesky decomposition, also known as Cholesky factorization, is a method of decomposing a positive-definite matrix. A positive-definite matrix is defined as a symmetric matrix where for all possible vectors $x$,...

#### How to Calculate the Inverse Matrix for 2×2 and 3×3 Matrices

Inverses of Numbers and Matrices The inverse of a number is its reciprocal. For example, the inverse of 8 is $\frac{1}{8}$, the inverse of 20 is $\frac{1}{20}$ and so on. Therefore,...