In order to describe and solve the basic linear algebra problems of this chapter,
we need some notation.
We will refer frequently to * matrices*, * vectors* and
* scalars*. A matrix will be denoted by an upper case letter like **A**,
and its -th element by . Occasionally in detailed algorithmic
descriptions we will instead write . The submatrix of **A** occupying
rows **i** though **j** and columns **k** through **l** will be denoted .
A lower case letter like **x** will denote a vector, and its **i**-th element will
be written .
Vectors will almost always be column vectors, which are the same as
matrices with one column. Lower case Greek letters (and occasionally lower
case letters) will denote scalars. will denote the set of real numbers,
the set of **n**-dimensional real vectors,
and the set of **m**-by-**n** real matrices.
will denote the * transpose* of the
matrix **A**: . A matrix is called * symmetric* if .
A matrix **A** is * positive definite* if for all nonzero
vectors **x**. Other notation will be introduced as needed.