# Matrix Review

EC655

Wilfrid Laurier University

Fall 2022

# Introduction

## Introduction

• Undergrad metrics normally uses scalar notation

• More accessible for students without advanced math background
• At the graduate level, it is often taught using matrix algebra

• Some advantages to matrix notation

• More compact

• Easier to express some estimators

• In this section, we review matrix algebra essentials for econometrics

• Not a comprehensive review
• We will switch between scalar and matrix notation in the course

• Depending on which is clearer in each context

# Matrices and Vectors

## Matrix

• A matrix is a rectangular array of numbers organized in rows and columns

• For example, matrix $\mathbf{A}$ with 2 rows and 3 columns could be

$\mathbf{A} = \begin{bmatrix} 1 & 2 & 3 \\ 4 &5 & 6 \end{bmatrix}$

• More generally, matrix $\mathbf{A}$ with m rows and n columns is

$\mathbf{A}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mn} \end{bmatrix}$

## Vectors

• A vector is a matrix with one column or one row

• A row vector $\mathbf{a}$ with n elements is

$\mathbf{a}= \begin{bmatrix} a_{1}& a_{2} &\cdots & a_{n} \end{bmatrix}$

• A .red[column vector] $\mathbf{a}$ with m elements is

$\mathbf{a}= \begin{bmatrix} a_{1}\\ a_{2}\\ \vdots \\ a_{m} \end{bmatrix}$

## Special Matrices

• A Square Matrix has the same number of rows and columns

$\mathbf{A}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1m} \\ a_{21}& a_{22} &\cdots & a_{2m} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mm} \end{bmatrix}$

• A Diagonal Matrix is a square matrix with zeroes for all off-diagonal elements

$\mathbf{A}= \begin{bmatrix} a_{11}& 0&\cdots & 0 \\ 0& a_{22} &\cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ 0& 0&\cdots & a_{mm} \end{bmatrix}$

## Special Matrices

• The Identity Matrix is a square matrix with ones on the diagonal and zeroes on the off-diagonals

$\mathbf{I}= \begin{bmatrix} 1& 0&\cdots & 0 \\ 0& 1 &\cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ 0& 0&\cdots & 1 \end{bmatrix}$

• The Zero Matrix is a matrix with zeroes for all elements

$\mathbf{0}= \begin{bmatrix} 0& 0&\cdots & 0 \\ 0& 0 &\cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ 0& 0&\cdots & 0 \end{bmatrix}$

• You can add and subtract matrices with the same dimensions

• Matrices with different dimensions are not conformable for addition or subtraction
• The sum of matrices $\mathbf{A}$ and $\mathbf{B}$ with dimension $m \times n$ is

$\mathbf{A} + \mathbf{B}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mn} \end{bmatrix} + \begin{bmatrix} b_{11}& b_{12} &\cdots & b_{1n} \\ b_{21}& b_{22} &\cdots & b_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ b_{m1}& b_{m2} &\cdots & b_{mn} \end{bmatrix}$

$= \begin{bmatrix} a_{11} + b_{11}& a_{12} + b_{12} &\cdots & a_{1n}+ b_{1n} \\ a_{21} + b_{21}& a_{22} + b_{22} &\cdots & a_{2n}+ b_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1} + b_{m1}& a_{m2} +b_{m2} &\cdots & a_{mn}+ b_{mn} \\ \end{bmatrix}$

## Matrix Subtraction

• Similarly, the difference between matrices $\mathbf{A}$ and $\mathbf{B}$ is

$\mathbf{A} - \mathbf{B}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mn} \end{bmatrix} - \begin{bmatrix} b_{11}& b_{12} &\cdots & b_{1n} \\ b_{21}& b_{22} &\cdots & b_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ b_{m1}& b_{m2} &\cdots & b_{mn} \end{bmatrix}$

$= \begin{bmatrix} a_{11} - b_{11}& a_{12} - b_{12} &\cdots & a_{1n}- b_{1n} \\ a_{21} - b_{21}& a_{22} - b_{22} &\cdots & a_{2n}- b_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1} - b_{m1}& a_{m2} -b_{m2} &\cdots & a_{mn}- b_{mn} \\ \end{bmatrix}$

## Rules for Addition and Subtraction

• The following rules apply to matrix addition and subtraction

• Commutativity $\mathbf{A + B = B + A}$

• Associativity $\mathbf{A + (B + C) = (A+B) + C}$

• Effectively, both rules mean order does not matter

• Similar to scalar math
• For subtraction, replace plus sign with minus sign and same rules apply

## Matrix Multiplication

• To multiply matrix $\mathbf{A}$ and $\mathbf{B}$, the number of columns in $\mathbf{A}$ must equal the number of rows in $\mathbf{B}$

• Suppose matrix $\mathbf{A}$ is $m \times n$ and matrix $\mathbf{B}$ is $n \times p$

• Define product as $\mathbf{C}$= $\mathbf{AB}$

• The $ij$ element of $\mathbf{C}$ is the sum of the product of the corresponding elements along the $i$th row of $\mathbf{A}$ and $j$th column of $\mathbf{B}$
$c_{ij} = \sum_{k} a_{ik}b_{kj}$

• The product matrix $\mathbf{C}$ will have dimension $m \times p$

• The number of rows of $\textbf{A}$ and number of columns of $\textbf{B}$

## Matrix Multiplication

• The product $\mathbf{AB}$ is

$\mathbf{AB}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mn} \end{bmatrix} \times \begin{bmatrix} b_{11}& b_{12} &\cdots & b_{1p} \\ b_{21}& b_{22} &\cdots & b_{2p} \\ \vdots & \vdots &\ddots & \vdots \\ b_{n1}& b_{n2} &\cdots & b_{np} \end{bmatrix}$

$= \begin{bmatrix} a_{11} b_{11} + a_{12} b_{21} + \cdots + a_{1n} b_{n1} &a_{11} b_{12} + a_{12} b_{22} + \cdots + a_{1n} b_{n2} &\cdots&a_{11} b_{1p} + a_{12} b_{2p} + \cdots + a_{1n} b_{np}\\ a_{21} b_{11} + a_{22} b_{21} + \cdots + a_{2n} b_{n1} &a_{21} b_{12} + a_{22} b_{22} + \cdots + a_{2n} b_{n2} &\cdots&a_{21} b_{1p} + a_{22} b_{2p} + \cdots + a_{2n} b_{np}\\ \vdots &\ddots & \vdots \\ a_{m1} b_{11} + a_{m2} b_{21} + \cdots + a_{mn} b_{n1} &a_{m1} b_{12} + a_{m2} b_{22} + \cdots + a_{mn} b_{n2} &\cdots&a_{m1} b_{1p} + a_{m2} b_{2p} + \cdots + a_{mn} b_{np}\\ \end{bmatrix}$

## Matrix Multiplication

• As an illustration suppose we have the following matrices $\mathbf{A}= \begin{bmatrix} 1& 2\\ 3& 4 \\ \end{bmatrix} \mathbf{B}= \begin{bmatrix} 5&6&7 \\ 8&9 &10 \end{bmatrix}$

• We can multiply $\mathbf{AB}$ because $\mathbf{A}$ has 2 columns, and $\mathbf{B}$ has 2 rows

• The product $\mathbf{C}$ = $\mathbf{AB}$ is

$\mathbf{C}= \begin{bmatrix} 1& 2\\ 3& 4 \\ \end{bmatrix} \times \begin{bmatrix} 5&6&7 \\ 8&9 &10 \end{bmatrix} = \begin{bmatrix} 1 \times 5 + 2\times 8&1 \times 6 + 2 \times 9 & 1 \times 7 + 2 \times 10 \\ 3 \times 5 + 4\times 8&3 \times 6 + 4 \times 9 & 3 \times 7 + 4 \times 10 \end{bmatrix}$

$= \begin{bmatrix} 21& 24& 27 \\ 47&54& 61 \end{bmatrix}$

## Scalar Multiplication

• A scalar is a single real number

• You can also multiply a scalar by a matrix

• If $\gamma$ is a scalar, and $\mathbf{A}$ is a matrix, then

$\mathbf{\gamma A}= \gamma \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mn} \end{bmatrix} = \begin{bmatrix} \gamma a_{11}&\gamma a_{12} &\cdots & \gamma a_{1n} \\ \gamma a_{21}& \gamma a_{22} &\cdots & \gamma a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ \gamma a_{m1}& \gamma a_{m2} &\cdots & \gamma a_{mn} \end{bmatrix}$

• You multiply the scalar by each element of the matrix

## Transpose

• The transpose of a matrix is one where the rows and columns are switched

• Suppose matrix $\mathbf{A}$ is

$\mathbf{A}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mn} \end{bmatrix}$

• Then its transpose $\mathbf{A'}$ is

$\mathbf{A'}= \begin{bmatrix} a_{11}& a_{21} &\cdots & a_{m1} \\ a_{12}& a_{22} &\cdots & a_{m2} \\ \vdots & \vdots &\ddots & \vdots \\ a_{1n}& a_{2n} &\cdots & a_{mn} \end{bmatrix}$

## Transpose

• The transpose has the following properties

$\mathbf{(A')' = A }$ $\mathbf{(\alpha A)' = \alpha A' }$ $\mathbf{(A + B)' = A' + B' }$ $\mathbf{(AB)' = B'A' }$

• There are additional rules for different types of matrices that we will cover below

## Partitioned Matrix Multiplication

• You may sometimes want to break matrices into vectors before you multiply

• Multiplication works the same way, but notation can be cleaner and more intuitive

• Suppose we have the following matrices $\mathbf{A}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{m1}& a_{m2} &\cdots & a_{mn} \end{bmatrix} \mathbf{B}= \begin{bmatrix} b_{11}& b_{12} &\cdots & b_{1p} \\ b_{21}& b_{22} &\cdots & b_{2p} \\ \vdots & \vdots &\ddots & \vdots \\ b_{n1}& b_{n2} &\cdots & b_{np} \end{bmatrix}$

• We are interested in the product $\mathbf{AB}$

## Partitioned Matrix Multiplication

• Break these matrices into vectors conformable for multiplication

$\mathbf{A}= \begin{bmatrix} \mathbf{a_{1}}&\mathbf{a_{2}} & \cdots & \mathbf{a_{n}} \end{bmatrix} \mathbf{B}= \begin{bmatrix} \mathbf{b_{1}}\\ \mathbf{b_{2} }\\ \vdots \\ \mathbf{b_{n}} \end{bmatrix}$

• Where

$\mathbf{a_{1}}= \begin{bmatrix} a_{11}\\ a_{21}\\ \cdots\\ a_{m1} \end{bmatrix} \mathbf{b_{1}}= \begin{bmatrix} b_{11}&b_{12} & \cdots & b_{1p} \end{bmatrix}$

## Partitioned Matrix Multiplication

• Multiply the vectors to get

$\mathbf{AB} = \sum_{i=1}^{n} \mathbf{a_{i}b_{i}}$

• This breaks the product $\mathbf{AB}$ into the sum of $n$ sub-matrices

• Each sub-matrix is product of corresponding vectors

• Also each sub-matrix will have dimension $m \times p$

• This will be useful for some econometric estimators we derive

• Makes notation simpler and more intuitive
• Again, note that you get the same answer as doing straight matrix multiplication

## Rules for Matrix Multiplication

• There are several useful properties for matrix (and scalar) multiplication

$(\alpha + \beta)\mathbf{A} = \alpha \mathbf{A} + \beta\mathbf{A}$ $\alpha (\mathbf{A} +\mathbf{B}) =\alpha \mathbf{A} +\alpha\mathbf{B}$ $(\alpha\beta) \mathbf{A} =\alpha(\beta \mathbf{A})$ $\alpha (\mathbf{A}\mathbf{B}) =(\alpha \mathbf{A}) \mathbf{B}$ $(\mathbf{A}\mathbf{B} )\mathbf{C} =\mathbf{A}(\mathbf{B} \mathbf{C})$ $\mathbf{A}(\mathbf{B} +\mathbf{C}) =\mathbf{A}\mathbf{B} +\mathbf{A} \mathbf{C}$ $(\mathbf{A}+\mathbf{B} )\mathbf{C} =\mathbf{A}\mathbf{C} +\mathbf{B} \mathbf{C}$ $\mathbf{A}\mathbf{I} =\mathbf{I}\mathbf{A} = \mathbf{A}$ $\mathbf{A}\mathbf{0} =\mathbf{0}\mathbf{A} = \mathbf{0}$ $\mathbf{A}\mathbf{B} \neq\mathbf{B}\mathbf{A}$

## Trace

• The trace of a square matrix is the sum of the diagonal elements

• If square matrix $\mathbf{A}$ is

$\mathbf{A}= \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{n1}& a_{n2} &\cdots & a_{nn} \end{bmatrix}$

• Then its trace is

$tr(\mathbf{A})= \sum_{i=1}^{n} a_{ii}$

## Trace

• Important properties of the trace are

$tr(\mathbf{I_{n}})= n$ $tr(\mathbf{A}')=tr(\mathbf{A})$ $tr(\mathbf{A +B})=tr(\mathbf{A}) + tr(\mathbf{B})$ $tr(\alpha \mathbf{A})=\alpha tr(\mathbf{A})$ $tr(\mathbf{AB})=tr(\mathbf{BA})$

## Marix Determinant

• The determinant is a scalar value associated with a square matrix

• Helpful concept for several things in matrix algebra

• For econometrics, most useful for solving systems of equations and finding inverse of a matrix

• For $2 \times 2$ matrix $\mathbf{A}$ $\mathbf{A}= \begin{bmatrix} a_{11}& a_{12} \\ a_{21}& a_{22} \\ \end{bmatrix}$

• The determinant is

$|\mathbf{A}|=a_{11}a_{22} - a_{12}a_{21}$

## Marix Determinant

• For $3 \times 3$ matrix $\mathbf{A}$

$\mathbf{A}= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \\ \end{bmatrix}$

• The determinant is

$|\mathbf{A}|=a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} +a_{13}a_{21}a_{32}$ $-(a_{12}a_{21}a_{33} + a_{11}a_{23}a_{32} +a_{13}a_{22}a_{31})$

$=a_{11}(a_{22}a_{33} - a_{23}a_{32}) + a_{12}(a_{23}a_{31} -a_{21}a_{33} ) +a_{13}(a_{21}a_{32} - a_{22}a_{31} )$

## Marix Determinant

• For $n \times n$ matrix $\mathbf{A}$ the determinant is

$|\mathbf{A}|=a_{i1}c_{i1} + a_{i2}c_{i2} + \cdots + a_{in}c_{in} \text{ for choice of any row i}$

• Where

• $a_{ij}$ is the $ij$ element of matrix $\mathbf{A}$

• $c_{ij}$ is the $ij$ cofactor of matrix $\mathbf{A}$ defined as $c_{ij} = (-1)^{i+j}|\mathbf{A}_{ij}|$

• $|\mathbf{A}_{ij}|$ is the minor of matrix $\mathbf{A}$

• Determinant of the sub-matrix formed by deleting the $i$th row and $j$th column of $\mathbf{A}$
• Process is long and tedious for large matrices

## Marix Determinant

• Example of $3 \times 3$ matrix

$\mathbf{A}= \begin{bmatrix} 1& 2 & 3 \\ 4& 5&6 \\ 7& 8 &9 \end{bmatrix}$

• Choose any row to find cofactors and compute determinant

• Does not matter which
• Let us expand along row 1

$|\mathbf{A}|=1(-1)^{1+1} \begin{vmatrix} 5&6 \\ 8 &9 \end{vmatrix} +2(-1)^{1+2} \begin{vmatrix} 4&6 \\ 7 &9 \end{vmatrix} +3(-1)^{1+3} \begin{vmatrix} 4&5 \\ 7 &8 \end{vmatrix}$

$|\mathbf{A}|= -3 +12 -9 = 0$

## Matrix Inverse

• The inverse of a square matrix $\mathbf{A}$ is defined such that

$\mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I}$

• It is roughly the equivalent of taking the reciprocal in scalar math

• But it is not generally the reciprocal of the elements of a matrix
• The formula for the inverse is

$\mathbf{A}^{-1}= \frac{1}{|\mathbf{A}|} \begin{bmatrix} c_{11}& c_{12} &\cdots & c_{1n} \\ c_{21}& c_{22} &\cdots & c_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ c_{n1}& c_{n2} &\cdots & c_{nn} \end{bmatrix}$

• where $c_{ij}$ are the cofactors defined above

## Matrix Inverse

• The inverse exists only when $|\mathbf{A}| \neq 0$

• This is why it is important to know the determinant

• In example above, inverse does not exist

• We will see later that it is because the columns are linearly dependent
• A matrix that cannot be inverted is singular

• A matrix that has an inverse is nonsingular

• Inverse matrices have the following properties

$\mathbf{(\alpha A)^{-1} = \frac{1}{\alpha} A^{-1} }$ $\mathbf{(A')^{-1}} = \mathbf{(A^{-1})' }$ $\mathbf{(A^{-1})^{-1}} = \mathbf{A}$ $\mathbf{(AB)^{-1}= B^{-1}A^{-1} }$

# Linear Independence and Rank of a Matrix

## Summary

• Now that we can manipulate matrices, we can move to more advanced topics

• Matrix algebra is useful for expressing and solving systems of equations

• This is how we will use it in econometrics
• We will learn you can solve for the OLS estimator when regressors are linearly independent

• They are not linear functions of one another
• To check linear independence, we use the concept of rank

• The rank of a matrix is the maximum number of independent rows or columns

• For non-square matrices, the maximum rank is the lesser of the number or rows or columns

## Linear Independence

• A set of vectors are linearly independent if you cannot express any of them as linear functions the others

• Mathematically, suppose that $\mathbf{A}=\begin{bmatrix} \mathbf{a}_{1}& \mathbf{a}_{2} &\cdots & \mathbf{a}_{m} \end{bmatrix}$

• where $\mathbf{a}_{1}, \mathbf{a}_{2}, \cdots,\mathbf{a}_{m}$ are $n \times 1$ vectors
• The vectors are independent if the only solution to

$\alpha_{1}\mathbf{a}_{1}+ \alpha_{2}\mathbf{a}_{2}+ \cdots+\alpha_{m}\mathbf{a}_{m}= 0$

• is

$\alpha_{1} = \alpha_{2}= \cdots=\alpha_{m}= 0$ - If at least one $\alpha_{i} \neq 0$, then the vectors are linearly dependent

## Rank of a Matrix

• The rank of a matrix is the maximum number of linearly independent rows or columns

• The rank of the rows will always equal the rank of the columns

• If the number of rows is less than columns, the highest rank is the number of rows

• Vice versa if the number of columns is less than the number of rows

• A matrix has full rank if rank equals the minimum of the number of rows/columns

• In econometrics, we mostly deal with matrices with more rows than columns

• So the matrix will be full rank if the rank equals the number of columns
• We will see later we need our matrix of regressors to have full rank

• None of the regressors can be linear functions of each other (no multicollinearity)

## Rank of a Matrix

• Some useful properties of the rank of a matrix

• The rank of a matrix and transpose are the same $rank(\mathbf{A'}) = rank(\mathbf{A})$

• If $\mathbf{A}$ is $n \times m$ then $rank(\mathbf{A}) \le min(n,m)$

• If $\mathbf{A}$ is $n \times n$ and $rank(\mathbf{A}) =n$ then $\mathbf{A}$ is nonsingular (invertible)

# Quadratic Forms and Positive Definite Matrices

• If $\mathbf{A}$ is $n \times n$ and symmetric, and $\mathbf{x}$ is $n \times 1$, the quadratic form for $\mathbf{A}$ is

$\mathbf{x'Ax}= \begin{bmatrix} x_{1}& x_{2} &\cdots & x_{n} \end{bmatrix} \begin{bmatrix} a_{11}& a_{12} &\cdots & a_{1n} \\ a_{21}& a_{22} &\cdots & a_{2n} \\ \vdots & \vdots &\ddots & \vdots \\ a_{n1}& a_{n2} &\cdots & a_{nn} \end{bmatrix} \begin{bmatrix} x_{1}\\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix}$

$=\sum_{i=1}^n a_{ii}x_{i}^2 + 2\sum_{i=1}^n \sum_{j>i}a_{ij}x_{i}x_{j}$

• A matrix is positive definite if for all $\mathbf{x} \neq 0$

$\mathbf{x'Ax} > 0$

## Positive Definite Matrices

• A matrix is positive semidefinite if for all $\mathbf{x} \neq 0$

$\mathbf{x'Ax} \ge 0$ - Positive definite matrices have diagonal elements that are strictly positive

• Positive semidefinite matrices have diagonal elements that are nonnegative

• Some other useful properties of positive definite/semidefinite matrices

• If $\mathbf{A}$ is positive definite, then $\mathbf{A}^{-1}$ exists and is also positive definite

• If $\mathbf{A}$ is $n \times m$, then $\mathbf{A'A}$ and $\mathbf{AA'}$ are positive definite

• If $\mathbf{A}$ is $n \times m$ and $rank(\mathbf{A}) = m$ then $\mathbf{A'A}$ is positive definite

• These concepts are used mostly for variance-covariance matrices in econometrics

## Idempotent Matrices

• An idempotent matrix is one that does not change when multiplied by itself

• Mathematically, $\mathbf{A}$ is idempotent when

$\mathbf{AA} = \mathbf{A}$

• When we discuss OLS, we will work with the following idempotent matrices

• Suppose $\mathbf{X}$ is $n \times k$ with full rank. Define

$\mathbf{P} = \mathbf{X(X'X)^{-1}X'}$ $\mathbf{M} =\mathbf{I_{n}} - \mathbf{X(X'X)^{-1}X'}$

• You can verify they are idempotent my multiplying each by itself

• Some important properties of idempotent matrices are

• $rank(\mathbf{A}) = tr(\mathbf{A})$

• $\mathbf{A}$ is positive semidefinite

# Moments of Random Vectors

## Expected Value

• The expected value of a random matrix is the matrix of expected values

• If $\mathbf{X}$ is an $n \times m$ matrix, then

$\mathbf{E}(\mathbf{X})= \begin{bmatrix} \mathbf{E}(x_{11}) & \mathbf{E}(x_{12}) & \cdots & \mathbf{E}(x_{1m})\\ \mathbf{E}(x_{21}) & \mathbf{E}(x_{22}) & \cdots &\mathbf{E}(x_{2m})\\ \vdots & \vdots &\ddots & \vdots \\ \mathbf{E}(x_{n1}) & \mathbf{E}(x_{n2}) & \cdots &\mathbf{E}(x_{nm})\\ \end{bmatrix}$

• Properties of expected values are similar to those in scalar math

• If $\mathbf{x}$ is a random vector, $\mathbf{b}$ is a nonrandom vector, and $\mathbf{A}$ is a nonrandom matrix, then $\mathbf{E}(\mathbf{Ax+b}) = \mathbf{A}\mathbf{E}(\mathbf{x})+\mathbf{b}$

• If $\mathbf{X}$ is a random matrix, and $\mathbf{B}$ and $\mathbf{A}$ are nonrandom matrices, then $\mathbf{E}(\mathbf{AXB}) = \mathbf{A}\mathbf{E}(\mathbf{X})\mathbf{B}$

## Variance-Covariance Matrix

• The variance-covariance matrix of random vector $\mathbf{y}$ has variances on the diagonal, covariances in the off-diagonal

• If $\mathbf{y}$ is an $n \times 1$ random vector, then

$var(\mathbf{y})= \mathbf{\sigma_{y}} = \mathbf{E[(y-E[y])(y-E[y])']}$ $= \begin{bmatrix} \text{var}(y_{1}) & \text{cov}(y_{1},y_{2}) & \cdots &\text{cov}(y_{1},y_{n}) \\ \text{cov}(y_{2},y_{1}) & \text{var}(y_{2}) & \cdots &\text{cov}(y_{2},y_{n}) \\ \vdots & \vdots &\ddots & \vdots \\ \text{cov}(y_{n},y_{1}) & \text{cov}(y_{n},y_{2}) & \cdots &\text{var}(y_{n})\\ \end{bmatrix}$

## Variance-Covariance Matrix

• Useful properties of variance-covariance matrices are

• If $\mathbf{a}$ is a nonrandom vector, then $\text{var}(\mathbf{a'y}) =\mathbf{a'}\text{var}\mathbf{(y)a}$

• If $\text{var}(\mathbf{a'y})>0$ for all $\mathbf{a>0}$, $\text{var}(\mathbf{y})$ is positive definite

• If $\mathbf{A}$ is a nonrandom matrix, $\mathbf{b}$ is a nonrandom vector, then $\text{var}(\mathbf{Ay + b}) =\mathbf{A'}\text{var}\mathbf{(y)A}$

• If $\text{var}(y_{j})=\sigma^{2}$ for all $j=1,2,...,n$, and the elements of $\textbf{y}$ are uncorrelated, then $\text{var}(\mathbf{y})=\sigma^{2}\mathbf{I_{n}}$

# Matrix Differentiation

## Scalar Functions

• A scalar function of a vector is a single function with respect to several variables

• A vector function is a set of one or more scalar functions, each with respect to several variables

• We will not cover these

• Consider the scalar function $y = f(\mathbf{x}) =f(x_{1}, x_{2},...,x_{n})$

• The function takes the vector $\mathbf{x}$ and returns a scalar

• This is just another way to write a multivariate function

• The derivative of this function is

$\frac{\partial f(\mathbf{x})}{\mathbf{x}}= \begin{bmatrix} \frac{\partial f(\mathbf{x})}{x_{1}} & \frac{\partial f(\mathbf{x})}{x_{2}} & \cdots & \frac{\partial f(\mathbf{x})}{x_{n}} \end{bmatrix}$

## Derivative of Scalar Function

• We simply collect the derivative with respect to each element of $\mathbf{x}$ in a vector

• Ex: linear function of $\mathbf{x}$

• Suppose $\mathbf{a}$ is an $n \times 1$ vector and $y = f(\mathbf{x}) = \mathbf{a'x} = \sum_{i=1}^{n} a_{i}x_{i}$

• The derivative is

$\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}}=\frac{\partial \mathbf{a'x} }{\partial \mathbf{x}}= \mathbf{a'} = \begin{bmatrix} a_{1}& a_{2}& \cdots & a_{n} \end{bmatrix}$

## Derivative of Scalar Function

• Ex: Quadratic form of $\mathbf{x}$

• Suppose $\mathbf{A}$ is an $n \times n$ symmetric matrix. The quadratic form is $y = f(\mathbf{x}) = \mathbf{x'Ax} =\sum_{i=1}^n a_{ii}x_{i}^2 + 2\sum_{i=1}^n \sum_{j>i}a_{ij}x_{i}x_{j}$

• The derivative is $\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}}=\frac{\partial \mathbf{x'Ax} }{\partial \mathbf{x}}= \mathbf{2x'A}$

# Linear Regression Model in Matrix Notation

## Population Regression Model

• In undergraduate textbooks, the population linear regression model is written as

$y= \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \cdots + \beta_{k}x_{k} + u$

• $y$ and $x_{1},...,x_{k}$ are observable random variables

• $u$ is an unobservable random variable

• We can write more compactly in vector form as

$y= \mathbf{x}\boldsymbol{\beta} + u$

• $\mathbf{x}$ is a $1 \times (k+1)$ vector of independent variables

• There are $k$ independent variables, plus an intercept
• $\boldsymbol{\beta}$ is a $(k+1) \times 1$ vector of slope parameters

## Population Regression Model

• Now suppose we take a random sample of $n$ people from the population

• The population model holds for each member of the sample

$y_{i}= \mathbf{x_{i}}\boldsymbol{\beta} + u_{i}, \forall i=1,...,n$

• We can express this more compactly with full matrix notation

$\mathbf{y}= \mathbf{X}\boldsymbol{\beta} + \mathbf{u}$

• $\mathbf{X}$ is an $n \times (k+1)$ matrix of observations on each regressor

• $\boldsymbol{\beta}$ is still a $(k+1) \times 1$ vector of slope parameters

• $\mathbf{y}$ is an $n \times 1$ vector of observations on the dependent variable

• $\mathbf{u}$ is an $n \times 1$ vector of error terms