Matrices and Vectors

This chapter covers the following topics:

• Matrices and their operations
• Vectors
• Gradient, divergence, and curl
• Identities
• Newton’s method for root extraction
• Trapezoidal rule for numerical integration
• Simpson’s rule for numerical integration

Matrices and their operations

A matrix is a rectangular array of numbers arranged in rows and columns. We usually write an $m\times n$ matrix (with $m$ rows and $n$ columns) as:

$$A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]$$

Addition and subtraction

You can add or subtract two matrices only if they have the same dimensions. The operation is done element by element.

Addition:

$$C=A+B=\left[\begin{array}{cccc}{a}_{11}+b_{11}& a_{12}+b_{12}& \dots & a_{1n}+b_{1n}\\ a_{21}+b_{21}& a_{22}+b_{22}& \dots & a_{2n}+b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}+b_{m1}& a_{m2}+b_{m2}& \dots & a_{mn}+b_{mn}\end{array}\right]$$

Subtraction:

$$C=A-B=\left[\begin{array}{cccc}{a}_{11}-b_{11}& a_{12}-b_{12}& \dots & a_{1n}-b_{1n}\\ a_{21}-b_{21}& a_{22}-b_{22}& \dots & a_{2n}-b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}-b_{m1}& a_{m2}-b_{m2}& \dots & a_{mn}-b_{mn}\end{array}\right]$$

Matrix multiplication

Matrix multiplication is not done element by element. Instead, each entry of the product comes from a row-by-column dot product.

$$C=AB$$

If $A$ is an $m\times p$ matrix and $B$ is a $p\times n$ matrix, then $C$ is an $m\times n$ matrix, and each entry is

$$c_{ij}=\sum _{k=1}^p{a}_{ik}{b}_{kj}$$

Matrix multiplication is defined only when the number of columns of $A$ matches the number of rows of $B$.

Determinant of a matrix

The determinant is defined for square matrices. It produces a single number that is used in several places, including checking whether a matrix is invertible.

For a $2\times 2$ matrix:

$$\det (A)=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$$

For a $3\times 3$ matrix:

$$\det (A)=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|=a_{11}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{12}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|+a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$$

Adjoint of a matrix

The adjoint (also called the adjugate) of a square matrix $A$ is the transpose of its cofactor matrix:

$$\text{adj}(A)=\text{Cof}(A{)}^T$$

Inverse of a matrix

A square matrix $A$ has an inverse only if its determinant is nonzero (that is, $\det (A)\ne 0$). When the inverse exists, it is given by:

$$A^{-1}=\frac{1}{\det (A)}\text{adj}(A)$$

Identity matrix

An identity matrix is a square matrix with ones on the diagonal and zeros everywhere else:

$$I_n=\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 1& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 1\end{array}\right]$$

It acts like the number 1 in matrix multiplication:

$$AI=IA=A$$

Properties of matrices

1. Associative property: $(AB)C=A(BC)$
2. Distributive property: $A(B+C)=AB+AC$
3. Commutative property (addition): $A+B=B+A$
4. Transpose of a product: $(AB{)}^T=B^T{A}^T$
5. Inverse property: $A{A}^{-1}=A^{-1}A=I$

This section has outlined the main matrix operations and the key definitions used with square matrices.

Vectors

A vector is a quantity that has both magnitude and direction. In three dimensions, a vector is often written in component form as:

$$\vec{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$$

Where:

• $a_1,a_2,a_3$ are the components of the vector in the x, y, and z directions.
• $\hat{i},\hat{j},\hat{k}$ are unit vectors in the x, y, and z directions, respectively.

Vector addition and subtraction

Addition

Vector addition combines components in the same direction:

$$\vec{A}+\vec{B}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}$$

Subtraction

Vector subtraction subtracts components in the same direction:

$$\vec{A}-\vec{B}=(a_1-b_1)\hat{i}+(a_2-b_2)\hat{j}+(a_3-b_3)\hat{k}$$

Dot product (scalar product)

The dot product of two vectors produces a scalar. It can be defined using magnitudes and the angle $\theta$ between the vectors:

$$\vec{A}\cdot \vec{B}=|\vec{A}||\vec{B}|\cos \theta$$

Or computed directly from components:

$$\vec{A}\cdot \vec{B}=a_1{b}_1+a_2{b}_2+a_3{b}_3$$

• Result: Scalar
• Measures projection of one vector onto another.

Cross product (vector product)

The cross product of two vectors produces a vector perpendicular to both. Its magnitude depends on the angle $\theta$ between the vectors:

$$\vec{A}\times \vec{B}=|\vec{A}||\vec{B}|\sin \theta \hat{n}$$

Where $\hat{n}$ is the unit vector perpendicular to both $\vec{A}$ and $\vec{B}$.

Using a determinant form:

$$\vec{A}\times \vec{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$$

• Result: Vector
• Direction follows the right-hand rule.

Angle between two vectors

You can find the angle between two nonzero vectors using the dot product:

$$\cos \theta =\frac{\vec{A}\cdot \vec{B}}{|\vec{A}||\vec{B}|}$$

Projection of a vector

Scalar projection of $\vec{A}$ on $\vec{B}$:

The scalar projection gives the signed length of $\vec{A}$ in the direction of $\vec{B}$:

$${\text{proj}}_{\vec{B}}\vec{A}=\frac{\vec{A}\cdot \vec{B}}{|\vec{B}|}$$

Vector projection of $\vec{A}$ on $\vec{B}$:

The vector projection gives the actual vector component of $\vec{A}$ parallel to $\vec{B}$:

$${\vec{A}}_{\parallel }=\left(\frac{\vec{A}\cdot \vec{B}}{|\vec{B}{|}^2}\right)\vec{B}$$

Gradient, divergence, and curl

Gradient (∇f)

The gradient takes a scalar field $f(x,y,z)$ and produces a vector field. The gradient points in the direction where $f$ increases most rapidly.

$$\nabla f=\frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial y}\hat{j}+\frac{\partial f}{\partial z}\hat{k}$$

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Divergence (∇·F)

The divergence takes a vector field $\vec{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$ and produces a scalar field:

$$\nabla \cdot \vec{F}=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$$

• Measures the “outflowing-ness” of a vector field at a point.

Curl (∇×F)

The curl takes a vector field $\vec{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$ and produces another vector field:

$$\nabla \times \vec{F}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ F_x& F_y& F_z\end{array}\right|$$

• Measures the rotation or swirling strength of the field.

Identities

Vector algebra identities

Vector dot product properties

1. Commutativity:

   $$\mathbf{A}\cdot \mathbf{B}=\mathbf{B}\cdot \mathbf{A}$$

2. Distributive property:

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

3. Dot product with itself (magnitude squared):

   $$\mathbf{A}\cdot \mathbf{A}=|\mathbf{A}{|}^2$$

4. Unit vector dot products:

   $$\mathbf{i}\cdot \mathbf{i}=\mathbf{j}\cdot \mathbf{j}=\mathbf{k}\cdot \mathbf{k}=1$$

5. Perpendicular unit vectors:

   $$\mathbf{i}\cdot \mathbf{j}=\mathbf{j}\cdot \mathbf{k}=\mathbf{k}\cdot \mathbf{i}=0$$

6. Zero dot product condition:

   If $\mathbf{A}\cdot \mathbf{B}=0$, then either:

   • $\mathbf{A}=0$,
   • $\mathbf{B}=0$,
   • or $\mathbf{A}$ is perpendicular to $\mathbf{B}$.

Vector cross product properties

1. Anti-commutativity:

   $$\mathbf{A}\times \mathbf{B}=-\mathbf{B}\times \mathbf{A}$$

2. Distributive property:

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

3. Alternative distributive property:

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

4. Unit vector cross products:

   $$\mathbf{i}\times \mathbf{i}=\mathbf{j}\times \mathbf{j}=\mathbf{k}\times \mathbf{k}=0$$

5. Cyclic permutation:

   $$\mathbf{i}\times \mathbf{j}=\mathbf{k},\mathbf{j}\times \mathbf{k}=\mathbf{i},\mathbf{k}\times \mathbf{i}=\mathbf{j}$$

6. Anti-symmetry property:

   $$\mathbf{j}\times \mathbf{i}=-\mathbf{k},\mathbf{k}\times \mathbf{j}=-\mathbf{i},\mathbf{i}\times \mathbf{k}=-\mathbf{j}$$

7. Zero cross product condition:

   If $\mathbf{A}\times \mathbf{B}=0$, then either:

   • $\mathbf{A}=0$,
   • $\mathbf{B}=0$,
   • or $\mathbf{A}$ is parallel to $\mathbf{B}$.

Vector calculus identities

1. Laplacian operator:

   $${\nabla }^2\varphi =\nabla \cdot (\nabla \varphi )=(\nabla \cdot \nabla )\varphi$$

2. Curl of a gradient:

   $$\nabla \times \nabla \varphi =0$$

3. Divergence of a curl:

   $$\nabla \cdot (\nabla \times \mathbf{A})=0$$

4. Vector Laplacian identity:

   $$\nabla \times (\nabla \times \mathbf{A})=\nabla (\nabla \cdot \mathbf{A})-{\nabla }^2\mathbf{A}$$

Newton’s method for root extraction

Newton’s method (also called the Newton-Raphson method) is an iterative numerical technique for finding roots of a real-valued function $f(x)$ (that is, values of $x$ where $f(x)=0$).

The iteration formula is:

$$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$$

1. Choose an initial guess $x_0$
2. Compute successive approximations using the formula
3. Stop when the difference $|x_{n+1}-x_n|$ is below a chosen tolerance

Example

To find the square root of 2, define a function whose root is $\sqrt{2}$:

• $f(x)=x^2-2$
• $f^{\prime }(x)=2x$

Substitute into the iteration formula:

$$x_{n+1}=x_n-\frac{x_n^2-2}{2x_n}$$

Trapezoidal rule for numerical integration

The trapezoidal rule estimates a definite integral by approximating the graph of $f(x)$ with straight-line segments, forming trapezoids.

Single application

For $f(x)$ over $[a,b]$:

$${\int }_a^{b}f(x)dx\approx \frac{b-a}{2}\left[f(a)+f(b)\right]$$

Composite trapezoidal rule

Divide the interval into $n$ equal subintervals of width $h=\frac{b-a}{n}$, then:

$${\int }_a^{b}f(x)dx\approx \frac{h}{2}\left[f(x_0)+2\sum _{i=1}^{n-1}f(x_i)+f(x_n)\right]$$

Simpson’s rule for numerical integration

Simpson’s rule estimates a definite integral by fitting parabolas through points on the curve.

Single application

For $f(x)$ over $[a,b]$, with midpoint $m=\frac{a+b}{2}$:

$${\int }_a^{b}f(x)dx\approx \frac{b-a}{6}\left[f(a)+4f(m)+f(b)\right]$$

Composite simpson’s rule

Let $n$ be even and $h=\frac{b-a}{n}$:

$${\int }_a^{b}f(x)dx\approx \frac{h}{3}\left[f(x_0)+4\sum _{i=1,3,5,\dots }^{n-1}f(x_i)+2\sum _{i=2,4,6,\dots }^{n-2}f(x_i)+f(x_n)\right]$$