THE GRAM-SCHMIDT PROCESS
The Gram-Schmidt Process produces an orthogonal basis for a nonzero subspace of . It is based on the following theorem. Theorem Let be a basis for a nonzero subspace W of and: . Then is an orthogonal basis forRead More
DIAGONALIZATION OF SYMMETRIC MATRICES
What is so special about symmetric matrices? While not every square matrix is diagonalizable, every symmetric matrix can be diagonalized. Diagonal matrices are easier to work with and have many fascinating properties. In addition, every symmetric matrix can be representedRead More
SIMILAR MATRICES
As discussed in the article Matrices of Linear Transformations, the matrix representing a linear transformation depends of the bases chosen for the vector spaces associated with the transformation. For example, let T be the linear transformation from to . SomeRead More
MATRICES OF LINEAR TRANSFORMATIONS
The central proposition in this article is that for every linear transformation from a vector space to another there exists a matrix that can be viewed as a representation of the transformation and the entries of the matrix depend onRead More
COORDINATES AND CHANGE OF BASIS
One may get confused when asked to specify the coordinates of T as depicted below. Figure 1 However, they may think it easy to determine the coordinates of T if instead of Figure 1, they are given the following figure.Read More
DIAGONALIZATION OF A SQUARE MATRIX
Just as the coordinate matrix of a vector depends on the basis chosen for the associated vector space, the matrix of a linear operator on a vector space depends on the basis for the vector space. It is often favorableRead More
FIRST ENCOUNTER WITH EIGENVALUES AND EIGENVECTORS
Consider the matrix , which can be viewed as a matrix representation of a linear operator on , and the vector . It is easy to check that . In this example, the output of the matrix transformation is aRead More
SOME SOLVED PROBLEMS ON PRINCIPAL COMPONENT ANALYSIS
Problem 1 Find the population principal components Y1 and Y2 for the covariance matrix . Then, calculate the proportion of the total population variance explained by the first principal component. Answer Find the eigenvalues and the corresponding eigenvectors. (λ-5)(λ-2) –Read More
FIRST ENCOUNTER WITH ARITHMETIC SEQUENCES (2)
Part 1 of this article introduces the concept of arithmetic sequence. Like sequences in general, arithmetic sequences can generate what we call arithmetic series. In brief, for every term of an arithmetic sequence there corresponds a partial sum; the sequenceRead More
FIRST ENCOUNTER WITH ARITHMETIC SEQUENCES
Consider the following sequences of real numbers. Sequence 1: 2, 5, 8, 11, 14, … Sequence 2: 11, 16, 21, 26, 31, … Sequence 3: 15, 13, 11, 9, 7, … They are examples of what weRead More