Econometric Theory/Matrix Algebra

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

$$A= \begin{bmatrix} 2 & 4 & 6 \\ 0 & -5 & 7.105\\ 1 & -3 & 2 \end{bmatrix} ,\quad B= \begin{bmatrix} -3 & -4 \\ \pi & \sqrt{2} \end{bmatrix} ,\mbox{ and }\,\,\, C= \begin{bmatrix} -3 & -5 & 0 & 0 \\ -1 & 4.56 & 3.28 & 19 \end{bmatrix} .$$

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 $$\vec{v}=(2, 3, 5, -4),\mbox{ or } \vec{w}= \begin{bmatrix} 4 \\ 5 \\ 0 \end{bmatrix}$$.

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