Understanding matrix operations

I will start with basic matrix definitions, including an orthogonal matrix and determinant. They will help you to understand a transformation matrix and eigenvalues and eigenvectors.

An orthogonal matrix

Let me start with a few matrix definitions. A vector is called “normal” if it has a length of one. Two vectors are “orthogonal” if they are at right angles to each other. A matrix is called an “orthonormal matrix” if it is a real square matrix whose columns and rows are orthonormal vectors. Figure 4.1 gives some examples:

Figure 4.1 – An orthogonal matrix

Why are we interested in an orthogonal matrix? The product of two orthogonal matrices will also be an orthogonal matrix. The transpose of the orthogonal matrix will also be an orthogonal matrix. These properties will be used to derive SVD.

The determinant of a matrix

The determinant of a matrix is the scalar value...