If is an matrix, consider the product T If vector is an column vector, then T is a row vector. The product is of dimension or a scalar. symmetric matrix with the property that T for all is...

If

is an

matrix, consider the product

^T

If vector

is an

column vector, then

^T
is a

row vector. The product is of dimension

or a scalar.

symmetric matrix with the property that

^T

for all



is said to be positive definite. Positive definite matrices play a role in many fields of engineering and science. We will study these matrices in Chapter 13 and subsequent chapters.

Show that the matrix is positive definite by showing that

positive definite matrix can be uniquely factored into the product

^T

where

is an upper-triangular matrix.

The MATLAB command gallery produces many different kinds of matrices to use for testing purposes. Enter the command

It generates a

positive-definite matrix. The MATLAB command chol
computes the matrix

Use it to find the factorization

^T

of
.