Mathematics Prerequisites

1 Mathematics

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Math is not just a way of calculating numerical answers; it is a way of thinking, using clear definitions for concepts and rigorous logic to organize our thoughts and back up our assertions.

Cheng (2025)

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These lecture notes use:

• algebra
• precalculus
• univariate calculus
• linear algebra
• vector calculus

Some key results are listed here.

1.1 Elementary Algebra

Mastery of Elementary Algebra (a.k.a. “College Algebra”) is a prerequisite for calculus, which is a prerequisite for Epi 202 and Epi 203, which are prerequisites for this course (Epi 204). Nevertheless, each year, some Epi 204 students are still uncomfortable with algebraic manipulations of mathematical formulas. Therefore, I include this section as a quick reference.

1.1.1 Equalities

Theorem 1 (Equalities are transitive) If \(a=b\) and \(b=c\), then \(a=c\)

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Theorem 2 (Substituting equivalent expressions) If \(a = b\), then for any function \(f(x)\), \(f(a) = f(b)\)

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1.1.2 Inequalities

Theorem 3 If \(a<b\), then \(a+c < b+c\)

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Theorem 4 (negating both sides of an inequality) If \(a < b\), then: \(-a > -b\)

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Theorem 5 If \(a < b\) and \(c \geq 0\), then \(ca < cb\).

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Theorem 6 \[-a = (-1)*a\]

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1.1.3 Sums

Theorem 7 (adding zero changes nothing) \[a+0=a\]

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Theorem 8 (Sums are symmetric) \[a+b = b+a\]

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Theorem 9 (Sums are associative)  

When summing three or more terms, the order in which you sum them does not matter:

\[(a + b) + c = a + (b + c)\]

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1.1.4 Products

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Theorem 10 (Multiplying by 1 changes nothing) \[a \times 1 = a\]

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Theorem 11 (Products are symmetric) \[a \times b = b \times a\]

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Theorem 12 (Products are associative) \[(a \times b) \times c = a \times (b \times c)\]

1.1.5 Division

Theorem 13 (Division can be written as a product) \[\frac {a}{b} = a \times \frac{1}{b}\]

1.1.6 Sums and products together

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Theorem 14 (Multiplication is distributive) \[a(b+c) = ab + ac\]

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1.1.7 Quotients

Definition 1 (Quotients, fractions, rates)  

A quotient, fraction, or rate is a division of one quantity by another:

\[\frac{a}{b}\]

In epidemiology, rates typically have a quantity involving time or population in the denominator.

c.f. https://en.wikipedia.org/wiki/Rate_(mathematics)

Definition 2 (Ratios) A ratio is a quotient in which the numerator and denominator are measured using the same unit scales.

c.f. https://en.wikipedia.org/wiki/Ratio

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Definition 3 (Proportion) In statistics, a “proportion” typically means a ratio where the numerator represents a subset of the denominator.

See https://en.wikipedia.org/wiki/Population_proportion.

See also https://en.wikipedia.org/wiki/Proportion_(mathematics) for other meanings.

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Definition 4 (Proportional) Two functions \(f(x)\) and \(g(x)\) are proportional if their ratio \(\frac{f(x)}{g(x)}\) does not depend on \(x\). (c.f. https://en.wikipedia.org/wiki/Proportionality_(mathematics))

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Additional reference for elementary algebra: https://en.wikipedia.org/wiki/Population_proportion#Mathematical_definition

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1.2 Exponentials and Logarithms

Theorem 15 (logarithm of a product is the sum of the logs of the factors) \[ \log{a\cdot b} = \log{a} + \log{b} \]

Corollary 1 (logarithm of a quotient)  

The logarithm of a quotient is equal to the difference of the logs of the factors:

\[\log{\frac{a}{b}} = \log{a} - \log{b}\]

Theorem 16 (logarithm of an exponential function) \[ \text{log}{\left\{a^b\right\}} = b \cdot\text{log}{\left\{a\right\}} \]

Theorem 17 (exponential of a sum)  

The exponential of a sum is equal to the product of the exponentials of the addends:

\[\text{exp}{\left\{a+b\right\}} = \text{exp}{\left\{a\right\}} \cdot\text{exp}{\left\{b\right\}}\]

Corollary 2 (exponential of a difference)  

The exponential of a difference is equal to the quotient of the exponentials of the addends:

\[\text{exp}{\left\{a-b\right\}} = \frac{\text{exp}{\left\{a\right\}}}{\text{exp}{\left\{b\right\}}}\]

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Theorem 18 (exponential of a product) \[a^{bc} = {\left(a^b\right)}^c = {\left(a^c\right)}^b\]

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Corollary 3 (natural exponential of a product) \[\text{exp}{\left\{ab\right\}} = (\text{exp}{\left\{a\right\}})^b = (\text{exp}{\left\{b\right\}})^a\]

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Exercise 1 For \(a \ge 0,~b,c \in \mathbb{R}\), When does \((a^b)^c = a^{(b^c)}\)?

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SolutionSolution

Solution 1. Short answer: rarely (that’s all you need to know for this course).

Long answer:

If \((a^b)^c = a^{(b^c)}\), then since \((a^b)^c = a^{bc}\), we have: \[a^{bc} = a^{(b^c)}\] \[\text{log}{\left\{a^{bc}\right\}} = \text{log}{\left\{a^{(b^c)}\right\}}\] \[bc \cdot \text{log}{\left\{a\right\}} = b^c\cdot \text{log}{\left\{a\right\}} \tag{1}\]

Equation 1 holds in each of the following cases:

1. \(bc = b^c\) (see Exercise 2).
2. \(a=1\) (i.e., \(\text{log}{\left\{a\right\}} = 0\)).
3. \(a=0\) (i.e., \(\text{log}{\left\{a\right\}}= -\infty\)) and \(\text{sign}{\left\{bc\right\}}=\text{sign}{\left\{b^c\right\}}\).

In particular, when \(a=0\) and \(c=0\), \(bc = 0\) and \(b^c = 1\) (for any \(b \in \mathbb{R}\)), so \(\text{sign}{\left\{bc\right\}}\neq \text{sign}{\left\{b^c\right\}}\), and \((a^b)^c \neq a^{(b^c)}\):

\[ \begin{aligned} (a^b)^c &= (0^b)^0 \\ &= 1 \end{aligned} \]

\[ \begin{aligned} a^{(b^c)} &= 0^{(b^0)} \\ &= 0^1 \\ &= 0 \end{aligned} \]

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Exercise 2 For \(b,c \in \mathbb{R}\), when does \(b^c = bc\)?

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SolutionSolution

Solution 2. \(bc = b^c\) in each of the following cases:

1. \(c = 1\).
2. \(b=0\) and \(c > 0\).
3. \(b = \text{exp}{\left\{\frac{\log{c}}{c-1}\right\}}\) (for \(c \ge 0\)).

See the red contours in Figure 2 for a visualization.

`b*c_f` <- function(b, c) b*c
`b^c_f` <- function(b, c) b^c
values_b <- seq(0, 5, by = .01)
values_c <- seq(-.5, 3, by = .01)

`b*c` <- outer(values_b, values_c, `b*c_f`)
`b^c` <- outer(values_b, values_c, `b^c_f`)
`b^c`[is.infinite(`b^c`)] = NA

opacity <- .3
z_min <- min(`b*c`, `b^c`, na.rm = TRUE)
z_max <- 5
plotly::plot_ly(
  x = ~values_b,
  y = ~values_c
) |>
  plotly::add_surface(
    z = ~ t(`b*c`),
    contours = list(
      z = list(
        show = TRUE,
        start = -1,
        end = 1,
        size = .1
      )
    ),
    name = "b*c",
    showscale = FALSE,
    opacity = opacity,
    colorscale = list(c(0, 1), c("green", "green"))
  ) |>
  plotly::add_surface(
    opacity = opacity,
    colorscale = list(c(0, 1), c("red", "red")),
    z = ~ t(`b^c`),
    contours = list(
      z = list(
        show = TRUE,
        start = z_min,
        end = z_max,
        size = .2
      )
    ),
    showscale = FALSE,
    name = "b^c"
  ) |>
  plotly::layout(
    scene = list(
      xaxis = list(
        # type = "log",
        title = "b"
      ),
      yaxis = list(
        # type = "log",
        title = "c"
      ),
      zaxis = list(
        # type = "log",
        range = c(z_min, z_max),
        title = "outcome"
      ),
      camera = list(eye = list(x = -1.25, y = -1.25, z = 0.5)),
      aspectratio = list(x = .9, y = .8, z = 0.7)
    )
  )

Figure 1: Graph of \(b*c\) and \(b^c\)

`b^c - b*c_f` <- function(b, c) `b^c_f`(b,c) - `b*c_f`(b,c)

mat1 <- outer(values_b, values_c, `b^c - b*c_f`)
mat1[is.infinite(mat1)] = NA

opacity <- .3
plotly::plot_ly(
  x = ~values_b,
  y = ~values_c
) |>
  plotly::add_surface(
    z = ~ t(mat1),
    contours = list(
      z = list(
        show = TRUE,
        start = 0,
        end = 1,
        size = 1,
        color = "red"
      )
    ),
    name = "b^c - b*c",
    showscale = TRUE,
    opacity = opacity
  ) |>
  plotly::layout(
    scene = list(
      xaxis = list(
        # type = "log",
        title = "b"
      ),
      yaxis = list(
        # type = "log",
        title = "c"
      ),
      zaxis = list(
        title = "outcome"
      ),
      camera = list(eye = list(x = -1.25, y = -1.25, z = 0.5)),
      aspectratio = list(x = .9, y = .8, z = 0.7)
    )
  )

Figure 2: Graph of \(b^c - b*c\). Red contour lines show where \(b^c = b*c\).

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Theorem 19 (\(\text{exp}{\left\{\right\}}\) and \(\text{log}{\left\{\right\}}\) are mutual inverses) \[\text{exp}{\left\{\text{log}{\left\{a\right\}}\right\}} = \text{log}{\left\{\text{exp}{\left\{a\right\}}\right\}} = a\]

2 Derivatives

Theorem 20 (Constant rule) \[\frac{\partial}{\partial x}c = 0\]

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Theorem 21 (Power rule) If \(a\) is constant with respect to \(x\), then: \[\frac{\partial}{\partial x}ay = a \frac{\partial x}{\partial y}\]

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Theorem 22 (Power rule) \[\frac{\partial}{\partial x}x^q = qx^{q-1}\]

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Theorem 23 (Derivative of natural logarithm) \[\text{log}'{\left\{x\right\}} = \frac{1}{x} = x^{-1}\]

Theorem 24 (derivative of exponential) \[\text{exp}'{\left\{x\right\}} = \text{exp}{\left\{x\right\}}\]

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Theorem 25 (Product rule) \[(ab)' = ab' + ba'\]

Theorem 26 (Quotient rule) \[(a/b)' = a'/b - (a/b^2)b'\]

Theorem 27 (Chain rule) \[\begin{aligned} \frac{\partial a}{\partial c} &= \frac{\partial a}{\partial b} \frac{\partial b}{\partial c} \\ &= \frac{\partial b}{\partial c} \frac{\partial a}{\partial b} \end{aligned} \]

or in Euler/Lagrange notation:

\[(f(g(x)))' = g'(x) f'(g(x))\]

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Corollary 4 (Chain rule for logarithms) \[ \frac{\partial}{\partial x}\log{f(x)} = \frac{f'(x)}{f(x)} \]

ProofProof

Proof. Apply Theorem 27 and Theorem 23.

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3 Linear Algebra

3.1 Vectors

Definition 5 (Column vector) A column vector of length \(p\) is an ordered list of \(p\) numbers, written vertically:

\[ \tilde{x}= \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{p} \end{bmatrix} \]

Column vectors are the default convention in these notes and in most statistics textbooks. They are also called \(p \times 1\) matrices.

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Definition 6 (Transpose) The transpose of a column vector \(\tilde{x}\) is the row vector with the same sequence of entries, written horizontally:

\[ {\tilde{x}}^{\top} \equiv \tilde{x}' \equiv [x_1,\; x_2,\; \ldots,\; x_p] \]

The transpose operation converts a column vector to a row vector, or more generally, swaps the rows and columns of a matrix (Definition 14).

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3.1.1 Special vectors

Definition 7 (Zero vector) The zero vector \(\tilde{0}\) of length \(p\) has all entries equal to zero:

\[ \tilde{0}= \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \]

The zero vector is the additive identity for vector addition: \(\tilde{x}+ \tilde{0}= \tilde{x}\) for any vector \(\tilde{x}\) of the same length.

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Definition 8 (Ones vector) The ones vector \(\tilde{1}\) of length \(p\) has all entries equal to one:

\[ \tilde{1} = \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix} \]

The dot product \({\tilde{1}}^{\top}\tilde{x}= \tilde{1} \cdot \tilde{x}= \sum_{i=1}^p x_i\) is the sum of all entries of \(\tilde{x}\).

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Definition 9 (Indicator vector / standard basis vector) The \(j\)-th indicator vector (or standard basis vector) \(\tilde{e}_j\) of length \(p\) has a \(1\) in position \(j\) and \(0\)s elsewhere:

\[ (\tilde{e}_j)_i = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} \qquad \tilde{e}_j = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \leftarrow \text{position } j \]

They are also called unit vectors or standard basis vectors.

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Theorem 28 (Indicator vectors select entries) For any vector \(\tilde{x}\) of length \(p\) and any \(j \in \{1, \ldots, p\}\):

\[{\tilde{e}_j}^{\top}\tilde{x}= x_j\]

ProofProof

Proof. Writing the product componentwise:

\[ \begin{aligned} {\tilde{e}_j}^{\top}\tilde{x} &= \sum_{i=1}^{p} (\tilde{e}_j)_i\, x_i \\&= \sum_{i=1}^{p} \begin{cases} 1 \cdot x_i & \text{if } i = j \\ 0 \cdot x_i & \text{if } i \neq j \end{cases} \\&= x_j \end{aligned} \]

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Definition 10 (Dot product/linear combination/inner product) For any two real-valued vectors \(\tilde{x}= (x_1, \ldots, x_n)\) and \(\tilde{y}= (y_1, \ldots, y_n)\), the dot-product, linear combination, or inner product of \(\tilde{x}\) and \(\tilde{y}\) is:

\[\tilde{x}\cdot \tilde{y}= \tilde{x}^{\top} \tilde{y}\stackrel{\text{def}}{=}\sum_{i=1}^nx_i y_i\]

Note

See also the definitions in

• Dobson and Barnett (2018), §1.3 (equation 1.1, page 7)

• Kaplan (2022), here.

• wikipedia

“Linear combination” can also refer to weighted sums of vectors, or in other words matrix-vector multiplication.

The dot-product has a different generalization for two matrices; see wikipedia for more.

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Theorem 29 (Dot product is symmetric) The dot product is symmetric:

\[\tilde{x}\cdot \tilde{y}= \tilde{y}\cdot \tilde{x}\]

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ProofProof

Proof. Apply:

• Definition 10
• symmetry of scalar multiplication
• Definition 10 again

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Example 1 (Dot product as matrix multiplication) The dot product of two column vectors \(\tilde{x}\) and \(\tilde{\beta}\) can be written as a matrix product of the row vector \({\tilde{x}}^{\top}\) with the column vector \(\tilde{\beta}\):

\[ \begin{aligned} \tilde{x}\cdot \tilde{\beta} &= {\tilde{x}}^{\top}\, \tilde{\beta} \\ &= [x_1,\; x_2,\; \ldots,\; x_p] \begin{bmatrix} \beta_{1} \\ \beta_{2} \\ \vdots \\ \beta_{p} \end{bmatrix} \\ &= x_1\beta_1 + x_2\beta_2 + \cdots + x_p \beta_p \end{aligned} \]

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3.1.2 Orthogonality

Definition 11 (Orthogonal vectors) Two vectors \(\tilde{x}\) and \(\tilde{y}\) of the same length are orthogonal (written \(\tilde{x}\perp \tilde{y}\)) if their dot product is zero:

\[\tilde{x}\perp \tilde{y}\iff {\tilde{x}}^{\top}\tilde{y}= 0\]

Orthogonality generalizes the geometric notion of perpendicularity to arbitrary dimensions.

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Definition 12 (Orthonormal vectors) A set of vectors \(\{\tilde{x}_1, \tilde{x}_2, \ldots, \tilde{x}_k\}\) is orthonormal if the vectors are mutually orthogonal and each has unit length:

\[{\tilde{x}_i}^{\top}\tilde{x}_j = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}\]

The indicator vectors \(\tilde{e}_1, \tilde{e}_2, \ldots, \tilde{e}_p\) (Definition 9) form an orthonormal set.

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3.2 Matrices

Definition 13 (Matrix) A matrix of dimensions \(m \times n\) is a rectangular array of \(m \cdot n\) numbers, arranged in \(m\) rows and \(n\) columns:

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

The entry in row \(i\) and column \(j\) is denoted \(a_{ij}\) or \((\mathbf{A})_{ij}\). A column vector of length \(p\) is a special case: a \(p \times 1\) matrix. A row vector of length \(p\) is a \(1 \times p\) matrix.

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3.2.1 Matrix transpose

Definition 14 (Matrix transpose) The transpose of an \(m \times n\) matrix \(\mathbf{A}\) is the \(n \times m\) matrix \({\mathbf{A}}^{\top}\) obtained by swapping the rows and columns of \(\mathbf{A}\):

\[({\mathbf{A}}^{\top})_{ij} = a_{ji}\]

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Theorem 30 (Transpose of a sum) \[{(\mathbf{A} + \mathbf{B})}^{\top} = {\mathbf{A}}^{\top} + {\mathbf{B}}^{\top}\]

In particular, for column vectors \(\tilde{x}\) and \(\tilde{y}\):

\[{(\tilde{x}+ \tilde{y})}^{\top} = {\tilde{x}}^{\top} + {\tilde{y}}^{\top}\]

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Theorem 31 (Transpose of a product) For compatible matrices \(\mathbf{A}\) and \(\mathbf{B}\):

\[{(\mathbf{A}\mathbf{B})}^{\top} = {\mathbf{B}}^{\top}\,{\mathbf{A}}^{\top}\]

The order of the factors reverses when transposing a product.

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3.2.2 Matrix addition

Definition 15 (Zero matrix) The \(m \times n\) zero matrix \(\mathbf{0}_{m \times n}\) (or \(\mathbf{0}\) when dimensions are clear from context) has all entries equal to zero:

\[ \mathbf{0}_{m \times n} = \begin{bmatrix} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix} \]

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Definition 16 (Matrix addition) Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) of the same dimensions \(m \times n\) can be added element-wise:

\[(\mathbf{A} + \mathbf{B})_{ij} = a_{ij} + b_{ij}\]

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Theorem 32 (Matrix addition is commutative) \[\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}\]

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Theorem 33 (Matrix addition is associative) \[(\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})\]

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Theorem 34 (Zero matrix is the additive identity) \[\mathbf{A} + \mathbf{0} = \mathbf{A}\]

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Theorem 35 (Additive inverse) For any matrix \(\mathbf{A}\), the matrix \(-\mathbf{A}\) (defined by \((-\mathbf{A})_{ij} = -a_{ij}\)) satisfies:

\[\mathbf{A} + (-\mathbf{A}) = \mathbf{0}\]

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3.2.3 Scalar multiplication

Definition 17 (Scalar multiplication) A matrix \(\mathbf{A}\) can be multiplied by a scalar \(c\):

\[(c\mathbf{A})_{ij} = c \cdot a_{ij}\]

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3.2.4 Matrix multiplication

Definition 18 (Matrix multiplication) The product of an \(m \times k\) matrix \(\mathbf{A}\) and a \(k \times n\) matrix \(\mathbf{B}\) is the \(m \times n\) matrix \(\mathbf{C} = \mathbf{A}\mathbf{B}\) with entries:

\[c_{ij} = \sum_{s=1}^{k} a_{is}\, b_{sj}\]

Matrix multiplication is only defined when the number of columns in \(\mathbf{A}\) equals the number of rows in \(\mathbf{B}\).

Matrix multiplication is not commutative in general: \(\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}\).

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Theorem 36 (Matrix multiplication is associative) \[(\mathbf{A}\mathbf{B})\mathbf{C} = \mathbf{A}(\mathbf{B}\mathbf{C})\]

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Theorem 37 (Matrix multiplication is distributive over addition) \[\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}\]

\[(\mathbf{A} + \mathbf{B})\mathbf{C} = \mathbf{A}\mathbf{C} + \mathbf{B}\mathbf{C}\]

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3.2.5 Matrix-vector multiplication

Definition 19 (Matrix-vector multiplication) The product of an \(m \times p\) matrix \(\mathbf{A}\) and a \(p \times 1\) column vector \(\tilde{x}\) is the \(m \times 1\) column vector \(\mathbf{A}\tilde{x}\) with entries:

\[(\mathbf{A}\tilde{x})_i = \sum_{j=1}^{p} a_{ij}\, x_j\]

Matrix-vector multiplication is a generalization of the dot product. Each entry of the result is a dot product of a row of \(\mathbf{A}\) with the vector \(\tilde{x}\).

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3.3 Special Matrices

See also Definition 15 for the zero matrix.

Definition 20 (Square matrix) A matrix is square if it has the same number of rows as columns. The number of rows (= columns) is the order of the matrix.

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Definition 21 (Matrix power) For a square matrix \(\mathbf{A}\) of order \(p\) and a positive integer \(k\), the \(k\)-th power of \(\mathbf{A}\) is:

\[\mathbf{A}^k = \underbrace{\mathbf{A}\,\mathbf{A}\cdots\mathbf{A}}_{k \text{ copies}}\]

In particular, \(\mathbf{A}^2 = \mathbf{A}\mathbf{A}\).

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Definition 22 (Identity matrix) The \(p \times p\) identity matrix \(\mathbf{I}_p\) (or \(\mathbf{I}\) when the size is clear from context) has ones on the main diagonal and zeros elsewhere:

\[ (\mathbf{I}_p)_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} \qquad \mathbf{I}_p = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} \]

Theorem 38 (Identity matrix is a multiplicative identity) For any \(m \times p\) matrix \(\mathbf{A}\):

\[\mathbf{A}\,\mathbf{I}_p = \mathbf{A}\]

\[\mathbf{I}_m\,\mathbf{A} = \mathbf{A}\]

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Definition 23 (Symmetric matrix) A square matrix \(\mathbf{A}\) is symmetric if \({\mathbf{A}}^{\top} = \mathbf{A}\), i.e., \(a_{ij} = a_{ji}\) for all \(i\) and \(j\).

Covariance matrices and information matrices are symmetric.

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Definition 24 (Diagonal matrix) A square matrix \(\mathbf{D}\) is a diagonal matrix if all off-diagonal entries are zero: \(d_{ij} = 0\) whenever \(i \neq j\):

\[ \mathbf{D} = \begin{bmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_p \end{bmatrix} \]

Diagonal matrices are denoted \(\mathbf{D} = \text{diag}(d_1, d_2, \ldots, d_p)\), where \(d_1, \ldots, d_p\) are the diagonal entries.

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Definition 25 (Matrix inverse) For a square \(p \times p\) matrix \(\mathbf{A}\), the inverse \(\mathbf{A}^{-1}\) (if it exists) is the unique matrix satisfying:

\[\mathbf{A}\,\mathbf{A}^{-1} = \mathbf{A}^{-1}\,\mathbf{A} = \mathbf{I}_p\]

A matrix that has an inverse is called invertible or non-singular.

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Theorem 39 (Inverse of a product) For invertible matrices \(\mathbf{A}\) and \(\mathbf{B}\):

\[(\mathbf{A}\mathbf{B})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\]

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Definition 26 (Idempotent matrix) A square matrix \(\mathbf{A}\) is idempotent if

\[\mathbf{A}^2 = \mathbf{A}\]

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Definition 27 (Projection matrix) A square matrix \(\mathbf{P}\) is a projection matrix (also called an orthogonal projector) if it is both symmetric and idempotent:

\[{\mathbf{P}}^{\top} = \mathbf{P} \qquad \text{and} \qquad \mathbf{P}^2 = \mathbf{P}\]

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Theorem 40 (Complement of a projection matrix) If \(\mathbf{P}\) is a projection matrix, then \(\mathbf{I} - \mathbf{P}\) is also a projection matrix.

ProofProof

Proof. We verify symmetry and idempotency.

Symmetry: \[{(\mathbf{I} - \mathbf{P})}^{\top} = {\mathbf{I}}^{\top} - {\mathbf{P}}^{\top} = \mathbf{I} - \mathbf{P}\]

Idempotency: \[\begin{aligned} (\mathbf{I} - \mathbf{P})^2 &= (\mathbf{I} - \mathbf{P})(\mathbf{I} - \mathbf{P}) \\ &= \mathbf{I} - \mathbf{P} - \mathbf{P} + \mathbf{P}^2 \\ &= \mathbf{I} - \mathbf{P} - \mathbf{P} + \mathbf{P} \\ &= \mathbf{I} - \mathbf{P} \end{aligned}\]

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Theorem 41 (Hat matrix is a projection matrix) In a linear regression model with full-rank design matrix \(\mathbf{X}\), the hat matrix

\[\mathbf{H} = \mathbf{X}({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top}\]

is a projection matrix.

ProofProof

Proof. We verify symmetry and idempotency.

Symmetry: \[\begin{aligned} {\mathbf{H}}^{\top} &= {\left(\mathbf{X}({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top}\right)}^{\top} \\ &= {({\mathbf{X}}^{\top})}^{\top} \cdot {\left(({\mathbf{X}}^{\top}\mathbf{X})^{-1}\right)}^{\top} \cdot {\mathbf{X}}^{\top} \\ &= \mathbf{X}\cdot ({\mathbf{X}}^{\top}\mathbf{X})^{-1} \cdot {\mathbf{X}}^{\top} \\ &= \mathbf{H} \end{aligned}\]

where the third line uses \({({\mathbf{X}}^{\top})}^{\top} = \mathbf{X}\) and the fact that \({\mathbf{X}}^{\top}\mathbf{X}\) is symmetric, so its inverse is also symmetric (\({\left(({\mathbf{X}}^{\top}\mathbf{X})^{-1}\right)}^{\top} = ({\mathbf{X}}^{\top}\mathbf{X})^{-1}\)).

Idempotency: \[\begin{aligned} \mathbf{H}^2 &= \mathbf{X}({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top} \cdot \mathbf{X}({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top} \\ &= \mathbf{X}({\mathbf{X}}^{\top}\mathbf{X})^{-1}({\mathbf{X}}^{\top}\mathbf{X})({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top} \\ &= \mathbf{X}({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top} \\ &= \mathbf{H} \end{aligned}\]

The hat matrix appears in the formula for fitted values in linear regression: \(\hat{\tilde{y}} = \mathbf{X}\hat{\tilde{\beta}} = \mathbf{X}({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top}\tilde{y}= \mathbf{H}\tilde{y}\). It “puts a hat” on \(\tilde{y}\) — hence the name.

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Theorem 42 (Projection matrices produce orthogonal decompositions) If \(\mathbf{P}\) is a projection matrix and \(\tilde{v}\) is any vector of compatible dimension, then the two components of the decomposition

\[\tilde{v} = \underbrace{\mathbf{P}\tilde{v}}_{\text{projected}} + \underbrace{(\mathbf{I} - \mathbf{P})\tilde{v}}_{\text{residual}}\]

are orthogonal:

\[\mathbf{P}\tilde{v} \;\perp\; (\mathbf{I} - \mathbf{P})\tilde{v}\]

ProofProof

Proof. \[\begin{aligned} {(\mathbf{P}\tilde{v})}^{\top}\,(\mathbf{I} - \mathbf{P})\tilde{v} &= {\tilde{v}}^{\top}\,{\mathbf{P}}^{\top}\,(\mathbf{I} - \mathbf{P})\tilde{v} \\ &= {\tilde{v}}^{\top}\,\mathbf{P}\,(\mathbf{I} - \mathbf{P})\tilde{v} \\ &= {\tilde{v}}^{\top}\,(\mathbf{P} - \mathbf{P}^2)\tilde{v} \\ &= {\tilde{v}}^{\top}\,(\mathbf{P} - \mathbf{P})\tilde{v} \\ &= {\tilde{v}}^{\top}\,\mathbf{0}\,\tilde{v} \\ &= 0 \end{aligned}\]

where the second line uses symmetry (\({\mathbf{P}}^{\top} = \mathbf{P}\)) and the fourth line uses idempotency (\(\mathbf{P}^2 = \mathbf{P}\)).

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3.4 Quadratic Forms

Definition 28 (Quadratic form) A quadratic form is a mathematical expression of the structure

\[{\tilde{x}}^{\top}\, \mathbf{S}\, \tilde{x}\]

where \(\tilde{x}\) is a \(p \times 1\) vector and \(\mathbf{S}\) is a \(p \times p\) matrix.

Quadratic forms are the matrix generalizations of the scalar expression \(c x^2\). They occur frequently in statistics:

• The residual sum of squares in linear regression
  1. is a quadratic form.
• The variance of a linear combination of estimates
  2. is a quadratic form: \(\text{Var}{\left({\tilde{x}}^{\top}\hat{\tilde{\beta}}\right)} = {\tilde{x}}^{\top}\,\text{Var}{\left(\hat{\tilde{\beta}}\right)}\,\tilde{x}\).

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Theorem 43 (Symmetric part of a quadratic form) If \(\mathbf{S}\) is a square matrix, then

\[ {\tilde{x}}^{\top}\mathbf{S}\tilde{x} = {\tilde{x}}^{\top}\left(\frac{1}{2}(\mathbf{S}+{\mathbf{S}}^{\top})\right)\tilde{x}. \]

So the value of a quadratic form depends only on the symmetric part of \(\mathbf{S}\).

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3.5 Design Matrix

Definition 29 (Design matrix) In a regression model with \(n\) observations and \(p\) predictors, the design matrix (or model matrix) \(\mathbf{X}\) is the \(n \times p\) matrix whose \(i\)-th row is the covariate vector \({\tilde{x}_i}^{\top}\) for observation \(i\):

\[ \mathbf{X}= \begin{bmatrix} {\tilde{x}_1}^{\top} \\ {\tilde{x}_2}^{\top} \\ \vdots \\ {\tilde{x}_n}^{\top} \end{bmatrix} = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1p} \\ x_{21} & x_{22} & \cdots & x_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & \cdots & x_{np} \end{bmatrix} \]

The product \(\mathbf{X}\tilde{\beta}\) collects all the linear predictors \({\tilde{x}_i}^{\top}\tilde{\beta}\) into a single \(n \times 1\) vector:

\[ \mathbf{X}\tilde{\beta}= \begin{bmatrix} {\tilde{x}_1}^{\top}\tilde{\beta}\\ \vdots \\ {\tilde{x}_n}^{\top}\tilde{\beta} \end{bmatrix} \]

The matrix \({\mathbf{X}}^{\top}\mathbf{X}\) is a \(p \times p\) symmetric matrix that appears in the OLS estimator \(\hat{\tilde{\beta}} = ({\mathbf{X}}^{\top}\mathbf{X})^{-1}{\mathbf{X}}^{\top}\tilde{y}\).

4 Vector Calculus

(adapted from Fieller (2016), §7.2)

This section covers derivatives of functions of vectors and matrices. Linear algebra prerequisites — including vectors, matrices, transpose, dot product, and quadratic forms — are covered in Section 3.

Let \(\tilde{x}\) and \(\tilde{\beta}\) be column vectors of length \(p\) (see Definition 5 and Definition 10).

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Definition 30 (Vector derivative) If \(f(\tilde{\beta})\) is a function that takes a vector \(\tilde{\beta}\) as input, such as \(f(\tilde{\beta}) = x'\tilde{\beta}\), then:

\[ \frac{\partial}{\partial \tilde{\beta}} f(\tilde{\beta}) = \begin{bmatrix} \frac{\partial}{\partial \beta_1}f(\tilde{\beta}) \\ \frac{\partial}{\partial \beta_2}f(\tilde{\beta}) \\ \vdots \\ \frac{\partial}{\partial \beta_p}f(\tilde{\beta}) \end{bmatrix} \]

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Definition 31 (Row-vector derivative) If \(f(\tilde{\beta})\) is a function that takes a vector \(\tilde{\beta}\) as input, such as \(f(\tilde{\beta}) = x'\tilde{\beta}\), then:

\[ \frac{\partial}{\partial \tilde{\beta}^{\top}} f(\tilde{\beta}) = \begin{bmatrix} \frac{\partial}{\partial \beta_1}f(\tilde{\beta}) & \frac{\partial}{\partial \beta_2}f(\tilde{\beta}) & \cdots & \frac{\partial}{\partial \beta_p}f(\tilde{\beta}) \end{bmatrix} \]

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Theorem 44 (Row and column derivatives are transposes) \[\frac{\partial}{\partial \tilde{\beta}^{\top}} f(\tilde{\beta}) = {\left(\frac{\partial}{\partial \tilde{\beta}} f(\tilde{\beta})\right)}^{\top}\]

\[\frac{\partial}{\partial \tilde{\beta}} f(\tilde{\beta}) = {\left(\frac{\partial}{\partial \tilde{\beta}^{\top}} f(\tilde{\beta})\right)}^{\top}\]

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Theorem 45 (Derivative of a dot product) \[ \frac{\partial}{\partial \tilde{\beta}} \tilde{x}\cdot \tilde{\beta}= \frac{\partial}{\partial \tilde{\beta}} \tilde{\beta}\cdot \tilde{x}= \tilde{x} \]

This looks a lot like non-vector calculus, except that you have to transpose the coefficient.

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ProofProof

Proof. \[ \begin{aligned} \frac{\partial}{\partial \beta} (x^{\top}\beta) &= \begin{bmatrix} \frac{\partial}{\partial \beta_1}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) \\ \frac{\partial}{\partial \beta_2}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) \\ \vdots \\ \frac{\partial}{\partial \beta_p}(x_1\beta_1+x_2\beta_2 +...+x_p \beta_p ) \end{bmatrix} \\ &= \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{p} \end{bmatrix} \\ &= \tilde{x} \end{aligned} \]

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Theorem 46 (Derivative of a quadratic form) For a quadratic form (Definition 28), if \(S\) is a \(p\times p\) matrix that is constant with respect to \(\beta\), then:

\[ \frac{\partial}{\partial \beta} \beta'S\beta = 2S\beta \]

This is like taking the derivative of \(cx^2\) with respect to \(x\) in non-vector calculus.

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Corollary 5 (Derivative of a simple quadratic form) \[ \frac{\partial}{\partial \tilde{\beta}} \tilde{\beta}'\tilde{\beta}= 2\tilde{\beta} \]

This is like taking the derivative of \(x^2\).

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Theorem 47 (Vector chain rule) \[\frac{\partial z}{\partial \tilde{x}} = \frac{\partial y}{\partial \tilde{x}} \frac{\partial z}{\partial y}\]

or in Euler/Lagrange notation:

\[(f(g(\tilde{x})))' = \tilde{g}'(\tilde{x}) f'(g(\tilde{x}))\]

See https://quickfem.com/finite-element-analysis/, specifically https://quickfem.com/wp-content/uploads/IFEM.AppF_.pdf

See also https://en.wikipedia.org/wiki/Gradient#Relationship_with_Fr%C3%A9chet_derivative

This chain rule is like the univariate chain rule (Theorem 27), but the order matters now. The version presented here is for the gradient (column vector); the total derivative (row vector) would be the transpose of the gradient.

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Corollary 6 (Vector chain rule for quadratic forms) \[\frac{\partial}{\partial \tilde{\beta}}{{\left(\tilde{\varepsilon}(\tilde{\beta})\cdot \tilde{\varepsilon}(\tilde{\beta})\right)}} = {\left(\frac{\partial}{\partial \tilde{\beta}}\tilde{\varepsilon}(\tilde{\beta})\right)} {\left(2 \tilde{\varepsilon}(\tilde{\beta})\right)}\]

5 Additional resources

5.1 Calculus

• Kaplan (2022)
• Khuri (2003)
• Banner (2007)
• Miller (2016)
  • http://www.youtube.com/watch?v=xYzQL0TUtBA
  • http://www.youtube.com/watch?v=Ps2SBo_WjoE

5.2 Linear Algebra and Vector Calculus

• Fieller (2016)
• Banerjee and Roy (2014)
• Searle and Khuri (2017)

5.3 Numerical Analysis

• Hua Zhou’s lecture notes for “UCLA Biostat 216 - Mathematical Methods for Biostatistics” (2023 Fall)

5.4 Real Analysis

• Grinberg (2017)

References

Banerjee, Sudipto, and Anindya Roy. 2014. Linear Algebra and Matrix Analysis for Statistics. Vol. 181. Crc Press Boca Raton. https://www.routledge.com/Linear-Algebra-and-Matrix-Analysis-for-Statistics/Banerjee-Roy/p/book/9781420095388.

Banner, Adrian D. 2007. The Calculus Lifesaver : All the Tools You Need to Excel at Calculus. A Princeton Lifesaver Study Guide. Princeton University Press. https://press.princeton.edu/books/paperback/9780691130880/the-calculus-lifesaver.

Cheng, Eugenia. 2025. “Opinion | How Math Turned Me from a D.E.I. Skeptic to a Supporter.” The New York Times. https://www.nytimes.com/2025/09/05/opinion/math-dei.html.

Dobson, Annette J, and Adrian G Barnett. 2018. An Introduction to Generalized Linear Models. 4th ed. CRC press. https://doi.org/10.1201/9781315182780.

Fieller, Nick. 2016. Basics of Matrix Algebra for Statistics with R. Chapman; Hall/CRC. https://doi.org/10.1201/9781315370200.

Grinberg, Raffi. 2017. The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs. 1st ed. Princeton Lifesaver Study Guides. Princeton University Press. https://press.princeton.edu/books/paperback/9780691172934/the-real-analysis-lifesaver.

Kaplan, Daniel. 2022. MOSAIC Calculus. Www.mosaic-web.org. www.mosaic-web.org.

Khuri, André I. 2003. Advanced Calculus with Applications in Statistics. John Wiley & Sons. https://www.wiley.com/en-us/Advanced+Calculus+with+Applications+in+Statistics%2C+2nd+Edition-p-9780471391043.

Miller, Steven J. 2016. The Probability Lifesaver: Calculus Review Problems. https://web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/index.htm#:~:text=http%3A//web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/supplementalchap_calcreview.pdf.

Searle, Shayle R, and Andre I Khuri. 2017. Matrix Algebra Useful for Statistics. John Wiley & Sons.