Why vectors and matrices?

• To understand methods like MDS, PCA and Factor Modelling it helps to understand vectors and matrices.
• In this lecture we cover some basic rules of matrices
• Later on we will cover the geometric intuition behind matrices.

Matrices

• A matrix is an array of numbers. If a matrix has $r$ rows and $c$ columns then we say it is a $r\times c$ matrix.
• A vector is just a special case of a matrix with either a single column $r\times 1$ or a single row $1\times c$.
• These are called a column vector and row vector respectively
• A matrix with a single column and a single row is called a scalar.

Adding Matrices

• To add two matrices simply add the corresponding elements

$$\left(\begin{array}{cc}1& 2\\ 3& 3\\ 4& 5\end{array}\right)+\left(\begin{array}{cc}6& 2\\ 4& 2\\ 5& 6\end{array}\right)=\left(\begin{array}{cc}7& 4\\ 7& 5\\ 9& 11\end{array}\right)$$

• Only matrices with the same dimensions can be added together

Scalar Multiplication

• To multiply a matrix and a scalar, simply multiply each element of the matrix by the scalar

$$3\times \left(\begin{array}{cc}1& 2\\ 3& 3\\ 4& 5\end{array}\right)=\left(\begin{array}{cc}3& 6\\ 9& 9\\ 12& 15\end{array}\right)$$

Matrix Transpose

• Another important matrix operation is the transpose.
• Every row becomes a column
• Every column becomes a row.

$${\left(\begin{array}{cc}1& 2\\ 3& 3\\ 4& 5\end{array}\right)}^{\prime }=\left(\begin{array}{ccc}1& 3& 4\\ 2& 3& 5\end{array}\right)$$

Matrix multiplication

• A more confusing operation is matrix multiplication.
• It does NOT involve simply multiplying corresponding elements.
• The intuition behind matrix multiplication will become clearer when we look at the geometry.

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=$$

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=1\times 6+...$$

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=6+...$$

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=6+3\times 4+...$$

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=6+12+...$$

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=6+12+4\times 5$$

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=6+12+20$$

Row times column

• First learn to multiply a row by a column

$$\left(\begin{array}{ccc}1& 3& 4\end{array}\right)\times \left(\begin{array}{c}6\\ 4\\ 5\end{array}\right)=38$$

Things to note

• We multiplied a $1\times 3$ vector by a $3\times 1$ vector
• The result is a single number (scalar)
• Let $z=\left(\begin{array}{c}{z}_1\\ z_2\\ ⋮\\ z_n\end{array}\right)$
• What is $z^{\prime }z$?

Sum of squares

$$\begin{array}{cc}{z}^{\prime }z& =\left(\begin{array}{cccc}{z}_1& z_2& \cdots & z_n\end{array}\right)\left(\begin{array}{c}{z}_1\\ z_2\\ ⋮\\ z_n\end{array}\right)\\ & =z_1^2+z_2^2+\dots z_n^2\\ & =\sum _{i=1}^n{z}_i^2\end{array}$$

Matrix Multiplication

• To multiply two matrices
• Multiply every row from the first matrix by every column in the second matrix
• Now an example

$$\left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right)\times \left(\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right)$$

Matrix Multiplication

$$\begin{array}{cc}& \left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right)\times \left(\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right)\\ & =\left(\begin{array}{cc}ag+bi+ck& ...\\ ...& ...\end{array}\right)\end{array}$$

Matrix Multiplication

$$\begin{array}{cc}& \left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right)\times \left(\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right)\\ & =\left(\begin{array}{cc}ag+bi+ck& ah+bj+cl\\ ...& ...\end{array}\right)\end{array}$$

Matrix Multiplication

$$\begin{array}{cc}& \left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right)\times \left(\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right)\\ & =\left(\begin{array}{cc}ag+bi+ck& ah+bj+cl\\ dg+ei+fk& ...\end{array}\right)\end{array}$$

Matrix Multiplication

$$\begin{array}{cc}& \left(\begin{array}{ccc}a& b& c\\ d& e& f\end{array}\right)\times \left(\begin{array}{cc}g& h\\ i& j\\ k& l\end{array}\right)\\ & =\left(\begin{array}{cc}ag+bi+ck& ah+bj+cl\\ dg+ei+fk& dh+ej+fl\end{array}\right)\end{array}$$

Multiplication checklist

When multiplying matrices check two things

$$\underset{(a\times b)}{X}\underset{(b\times c)}{Y}=\underset{\left(a\times c\right)}{Z}$$

1. First check dimensions on the ‘inside’ (blue). If equal the multiplication is conformable
2. Then check the dimensions on the ‘outside’ (red). These give the dimensions of the result (they do not have to be equal).

Be careful!

• Not all matrices can be multplied!
• Often two matrices can be added but NOT multiplied.
  • Consider the very first example of addition in these slides.
• Sometimes $XY$ is possible but not $YX$.
• Even when both $XY$ and $YX$ are both conformable they may still not be equal, i.e $XY\ne YX$.

Regression

In a regression model

$$\begin{array}{cc}{y}_1& ={\beta }_1{x}_{11}+{\beta }_2{x}_{12}+\dots +{\beta }_k{x}_{1k}+{ϵ}_1\\ y_2& ={\beta }_1{x}_{21}+{\beta }_2{x}_{22}+\dots +{\beta }_k{x}_{2k}+{ϵ}_2\\ ⋮& =⋮+⋮+\dots +⋮+⋮\\ y_n& ={\beta }_1{x}_{n1}+{\beta }_2{x}_{n2}+\dots +{\beta }_k{x}_{nk}+{ϵ}_n\end{array}$$

Regression Model

• All these equations can be written as

$$y=X\beta +ϵ$$

• The vector $y$ is $n\times 1$
  • $y_i$ is the $i^{th}$ observation of the dependent variable
• The matrix $X$ is $n\times k$
  • $x_{ij}$ is the $i^{th}$ observation of the $j^{th}$ dependent variable.
  • For an intercept set $x_{i1}=1$ for all $i$

Regression Model

• The vector $\beta$ is $k\times 1$
  • ${\beta }_j$ is the $j^{th}$ observation of the dependent variable
• The vector $ϵ$ is $n\times 1$
  • ${ϵ}_i$ is the $i^{th}$ error term.
• Check for yourselves that all multiplication is conformable.

Variance Covariance matrix

• Another important matrix to understand is the variance covariance matrix.
• To start we will revise variance for scalar valued y which is

$${\sigma }^2=E\left[\right(y-\mu {)}^2]$$

• Or the sample equivalent

$$s^2=\frac{1}{n-1}\sum _{i=1}^n\left[\right(y_i-\overline{y}{)}^2]$$

Variance Covariance matrix

• What about when $y=\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)$ where $y_1$ and $y_2$ are different variables?
• The equivalent formulas are

$$\Sigma =E\left[\right(y-\mu \left)\right(y-\mu {)}^{\prime }]$$ and

$$S=\frac{1}{n-1}\sum _{i=1}^n\left[\right(y_i-\overline{y}\left)\right(y-\overline{y}{)}^{\prime }]$$

Variance Covariance matrix

• To unpack this, keep things simple by looking at the expected variance and assume $\mu =0$
• What are the dimensions of $y{y}^{\prime }$?
• Since $y$ is a $2\times 1$ matrix $y{y}^{\prime }$ will be a $2\times 1$ vector multiplied by a $1\times 2$ vector.
• This is a $2\times 2$ matrix.
• What is in this $2\times 2$?

Variance Covariance matrix

$$\begin{array}{cc}E\left[y{y}^{\prime }\right]& =E\left[\left(\begin{array}{c}{y}_1\\ y_2\end{array}\right)\left(\begin{array}{cc}{y}_1& y_2\end{array}\right)\right]\\ & =E\left[\left(\begin{array}{cc}{y}_1^2& y_1{y}_2\\ y_2{y}_1& y_2^2\end{array}\right)\right]\\ & =\left(\begin{array}{cc}E\left[y_1^2\right]& E\left[y_1{y}_2\right]\\ E\left[y_2{y}_1\right]& E\left[y_2^2\right]\end{array}\right)\end{array}$$

Variance Covariance matrix

$$\Sigma =\left(\begin{array}{cc}E\left[y_1^2\right]& E\left[y_1{y}_2\right]\\ E\left[y_2{y}_1\right]& E\left[y_2^2\right]\end{array}\right)$$

• The diagonal elements are variances.
• The off-diagonal elements are covariances.

In general

• For a $p$-dimensional vector, the variance covariance matrix is a $p\times p$ matrix.
• The element in the row $i$ and column $j$ for $i\ne j$ is the covariance between $y_i$ and $y_j$.
• The element in the row $i$ and column $i$ is a diagonal element and is the variance of $y_i$.

Can we divide by matrices?

• We can add and multiply matrices. Can we divide them?
• Strictly speaking we cannot.
• However it is possible to multiply bu a matrix inverse.
• What is a matrix inverse?

Inverses for scalars

• In scalar algebra dividing $a$ by $b$ is the same as multiplying $a$ by $1/b$
• The inverse has the property that multiplying a number by its inverse yields an answer of 1.
• In scalar algebra this means $b\times 1/b=1$
• What is the equivalent of 1 in matrix algebra?

Identity Matrix

The Identity Matrix is

$$\left(\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \cdots & 0& 1\end{array}\right)$$

Multiplying any matrix by the identity matrix gives the same matrix. I.e. $AI=IA=A$.

Matrix Inverse

• The matrix inverse of $X$ denoted $X^{-1}$ is defined so that $$X^{-1}X={XX}^{-1}=I$$
• This is only possible when $X$ is a square matrix
• A square matrix has the same number of rows as columns.

Rotation

• A very special matrix is a rotation matrix.
• If $R$ is a rotation matrix then it has the following property

$$R^{-1}=R^{\prime }$$

• Another way of stating this is that $R^{\prime }R=I$
• The reason we call this a rotation will be covered later on.

Conclusions

• In factor modelling matrices are used heavily.
• Make sure you understand how multiplication works.
• In particular understand when multiplication is possible and when it is not possible.