Econometric Theory/Matrix Algebra

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Matrices[edit]

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[edit]

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.