Econometric Theory/Matrix Algebra

Matrices

A matrix is an array of numbers arranged into rows and columns. Some examples of matrices are,

$A=\left[\begin{array}{ccc}2& 4& 6\\ 0& -5& 7.105\\ 1& -3& 2\end{array}\right],B=\left[\begin{array}{cc}-3& -4\\ \pi & \sqrt{2}\end{array}\right],\text{ and }C=\left[\begin{array}{cccc}-3& -5& 0& 0\\ -1& 4.56& 3.28& 19\end{array}\right].$

When describing matrices we indicate the number of rows first, then the number of columns. For example, the matrix $C$ with two rows and four columns is said to be a $2\times 4$ matrix.

It is standard notation to name matrices with capital letters and to use lower case letters with subscripts to identify particular entries in a matrix.

For example, to identify the entry in row 1 and column 3 of matrix $A$ we would write $a_{13}$. To indicate that this entry is a six we would write the equation $a_{13}=6$.

Two matrices are considered to be equal only if they are the same size and every pair of corresponding elements are equal.

A column matrix is a matrix with only one column. Similarly, a row matrix has only one row.

Vectors

A vector is an object often defined by a long list of properties. However, for now we will avoid the more complicated definition, and just say that a vector is an ordered list of numbers. Later we will see that vectors can really be much more.

An ordered pair, $(x,y)$, that is used to identify a point in the plane can be considered to be a vector.

Similarly, an ordered triple, $(x,y,z)$ is a vector.

Obviously, row and column matrices can also be considered to be a vector.

It is common to name vectors using variables with arrows above.

For example, we might write $\overrightarrow{v}=(2,3,5,-4),\text{ or }\overrightarrow{w}=\left[\begin{array}{c}4\\ 5\\ 0\end{array}\right]$.

For the most part, it will convenient to think of vectors as column matrices.