Question

Answer the following questions.
(a) What is the implication of a correlation matric not being positive-semidefinite?
(b) Why are the diagonal elements of a correlation matrix always 1?
(c) Making small changes to a positive-semidefinite matrix with 100 variables will have no effect on the matrix. Explain this statement

Answer 1

Suppose you have data set comprising various variables, say n variables. Then correlation matrix, C, is a matrix giving correlation of one variable to another. It is a table showing correlation coefficients between sets of variables. Each random variable in the table is correlated with each of the other values in the table . The correlation matrix computes the correlation coefficients of the columns of a matrix. That is, row i and column j of the correlation matrix is the correlation between column i and column j of the original matrix.

1. Roundoff error may cause one or more eigenvalues to be slightly negative. Sampling error causes an eigenvalue to become negative
2. . The diagonal of the table is always a set of ones, because the correlation between a variable and itself is always 1.

The correlation matrix is also symmetric since the correlation of column i with column j is the same as the correlation of column j with column i.

3. Making small changes to a positive-semidefinite matrix with 100 variables will have no effect on the matrix. It means that a small perturbation added to a matrix will not change the solution and the eigen values will not change significantly.