Appendix B: Matrix Multiplication in Optics - Exploring Optics: From Fundamentals to Advanced Applications

“In mathematics you don’t understand things. You just get used to them.”

John von Neumann

Matrix multiplication provides one of the most powerful and systematic approaches to analyzing optical systems. Far from being merely computational shortcuts, matrices encode the fundamental transformations that light undergoes, revealing deep connections between seemingly different optical phenomena. This appendix will guide you through matrix operations with a focus on their physical meaning in optics, building from basic concepts to sophisticated applications in polarization and ray optics.

1What Are Matrices and Why Do We Need Them in Optics?¶

1.1The Power of Linear Transformations¶

Many optical operations in a specified linear regime can be described as linear transformations of suitable state vectors. Examples include:

A polarizer acts on a fully polarized Jones vector (unpolarized light needs a different description)
A wave plate changes the polarization state of light
A lens transforms a parallel beam into a converging beam
A sequence of paraxial interfaces changes a ray’s height and angle

Each of these transformations can be represented mathematically as a matrix operation, where the input state is multiplied by a transformation matrix to yield the output state.

1.2Matrix Basics: Structure and Notation¶

A matrix is a rectangular array of numbers arranged in rows and columns. For optics, we primarily work with 2×2 matrices for both polarization and ray optics:

\mathbf{M} = \begin{pmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{pmatrix}

(1)

1.3Vectors: Representing Optical States¶

In matrix optics, we represent physical states as column vectors:

Polarization States (Jones vectors):

\vec{E} = \begin{pmatrix} E_x \\ E_y \end{pmatrix}

(2)

Ray States (ray vectors):

\vec{r} = \begin{pmatrix} y \\ \theta \end{pmatrix}

(3)

where $y$ is the ray height and $\theta$ is the ray angle.

This appendix uses the common $(y,\theta)^T$ ray-vector order. Chapter 11 instead uses $(n\alpha,y)^T$ , so its matrix entries occupy different positions. Both conventions describe the same paraxial rays once the coordinates and refractive-index factors are converted. In this $(y,\theta)^T$ convention, two transverse planes are conjugate when $B=0$ ; in Chapter 11’s $(n\alpha,y)^T$ convention, the corresponding condition is $C=0$ .

1.4Why Matrix Multiplication Works for Optics¶

The reason matrices are so powerful in optics is that optical systems are compositional: if you have two optical elements in series, the combined effect is the product of their individual effects. Mathematically:

If element A transforms state $\vec{s}_1$ to $\vec{s}_2$ : $\vec{s}_2 = \mathbf{A}\vec{s}_1$ And element B transforms state $\vec{s}_2$ to $\vec{s}_3$ : $\vec{s}_3 = \mathbf{B}\vec{s}_2$

Then the combined transformation is: $\vec{s}_3 = \mathbf{B}(\mathbf{A}\vec{s}_1) = (\mathbf{B}\mathbf{A})\vec{s}_1$

The combined system has matrix $\mathbf{BA}$ —the product of the individual matrices.

2Matrix Multiplication: Rules and Mechanics¶

2.1The Fundamental Rule¶

Matrix multiplication follows a specific pattern. For matrices $\mathbf{A}$ and $\mathbf{B}$ :

(\mathbf{AB})_{ij} = \sum_{k} A_{ik}B_{kj}

(4)

In plain English: to get element $(i,j)$ of the product, take the dot product of row $i$ from the first matrix with column $j$ from the second matrix.

2.2Step-by-Step Process for 2×2 Matrices¶

For two 2×2 matrices:

\mathbf{A} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}

(5)

The product $\mathbf{C} = \mathbf{AB}$ has elements:

\mathbf{C} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix}

(6)

2.3Matrix-Vector Multiplication¶

When multiplying a matrix by a vector:

\mathbf{A}\vec{v} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} a_{11}v_1 + a_{12}v_2 \\ a_{21}v_1 + a_{22}v_2 \end{pmatrix}

(9)

This is how we transform optical states—the matrix represents the optical element, and the vector represents the light state.

2.4Important Properties of Matrix Multiplication¶

Table 1:Matrix Multiplication Properties

Property	Mathematical Statement	Physical Meaning
Associative	$(\mathbf{AB})\mathbf{C} = \mathbf{A}(\mathbf{BC})$	Order of grouping doesn’t matter
Not Commutative	$\mathbf{AB} \neq \mathbf{BA}$ (usually)	Order of optical elements matters!
Identity Element	$\mathbf{AI} = \mathbf{IA} = \mathbf{A}$	Identity represents “do nothing”
Distributive	$\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC}$	Superposition principle

3Special Matrices in Optics¶

3.1The Identity Matrix¶

The identity matrix represents “no change”:

\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

(10)

For any matrix $\mathbf{A}$ or vector $\vec{v}$ :

$\mathbf{AI} = \mathbf{IA} = \mathbf{A}$
$\mathbf{I}\vec{v} = \vec{v}$

Physical meaning: No change between two coincident reference planes. Propagation over a finite distance changes ray height and has its own matrix.

3.2Rotation Matrices¶

A rotation matrix rotates vectors by angle $\theta$ :

\mathbf{R}(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

(11)

3.3Inverse Matrices¶

The inverse of matrix $\mathbf{A}$ is denoted $\mathbf{A}^{-1}$ and satisfies:

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

(14)

For a 2×2 matrix:

\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \begin{pmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{pmatrix}

(15)

where $\det(\mathbf{A}) = a_{11}a_{22} - a_{12}a_{21}$ is the determinant, provided $\det(\mathbf A)\ne0$ .

Physical meaning: For an invertible transformation, the inverse maps an output state back to its input. An ideal polarizer is singular and cannot be inverted.

4Jones Matrices: Polarization Optics¶

4.1Representing Polarized Light¶

In Jones calculus, we represent polarized light as a complex 2D vector:

\mathbf E = \begin{pmatrix} \mathcal E_x \\ \mathcal E_y \end{pmatrix} = \begin{pmatrix} A_x e^{i\phi_x} \\ A_y e^{i\phi_y} \end{pmatrix},\qquad A_x,A_y\ge0.

(16)

The physical field is $\operatorname{Re}[\mathbf E e^{-i\omega t}]$ . Jones vectors describe fully polarized, coherent fields.

Table 2:Common Polarization States

Polarization Type	Jones Vector	Physical Description
Horizontal linear	$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$	Electric field oscillates in x-direction
Vertical linear	$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$	Electric field oscillates in y-direction
45° linear	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$	Equal x and y components, in phase
Right circular	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}$	Quadrature components; clockwise rotation when viewed toward the source
Left circular	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix}$	Quadrature components; counterclockwise rotation when viewed toward the source

4.2Jones Matrices for Common Optical Elements¶

Linear Polarizer (transmission axis at angle $\theta$ ):

\mathbf{P}(\theta) = \begin{pmatrix} \cos^2\theta & \cos\theta\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta \end{pmatrix}

(17)

Special cases:

Horizontal polarizer ( $\theta = 0$ ): $\mathbf{P}_H = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
Vertical polarizer ( $\theta = 90°$ ): $\mathbf{P}_V = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$

Quarter-Wave Plate (fast axis at angle $\theta$ ):

\mathbf{Q}(\theta) = \mathbf{R}(\theta) \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \mathbf{R}(-\theta)

(18)

Half-Wave Plate (fast axis at angle $\theta$ ):

\mathbf{H}(\theta) = \mathbf{R}(\theta) \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \mathbf{R}(-\theta)

(19)

These forms follow from the active rotation matrix $\mathbf R$ defined above: rotate into the element’s axes with $\mathbf R(-\theta)$ , apply its principal-axis matrix, then rotate back.

4.3Worked Example: Polarization Analysis¶

Example: Light Through Crossed Polarizers

Analyze what happens when horizontally polarized light passes through:

A 45° polarizer
A vertical polarizer

Solution:

Initial state: $\vec{E}_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ (horizontal)

Step 1: 45° polarizer

\mathbf{P}_{45°} = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}

(20)

\vec{E}_1 = \mathbf{P}_{45°}\vec{E}_0 = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1/2 \\ 1/2 \end{pmatrix}

(21)

Step 2: Vertical polarizer

\mathbf{P}_V = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}

(22)

\vec{E}_2 = \mathbf{P}_V\vec{E}_1 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1/2 \\ 1/2 \end{pmatrix} = \begin{pmatrix} 0 \\ 1/2 \end{pmatrix}

(23)

Combined effect: $$\mathbf P_V\mathbf P_{45°} = \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}

\begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}

(24)

= \begin{pmatrix} 0 & 0 \ 1/2 & 1/2 \end{pmatrix},\qquad \vec E_2=(\mathbf P_V\mathbf P_{45°})\vec E_0 =\begin{pmatrix}0\1/2\end{pmatrix}.$$

Intensity: $I = |\vec{E}_2|^2 = |1/2|^2 = 1/4$ of the original intensity.

Without the intermediate polarizer, crossed polarizers would transmit zero intensity (Malus’s law). The 45° polarizer allows 25% transmission.

4.4Complex Polarization States¶

Real optical systems often involve complex Jones matrices with both real and imaginary elements:

5Ray Transfer Matrices: Geometrical Optics¶

5.1The ABCD Matrix Formalism¶

In geometrical optics, we can represent ray propagation using 2×2 matrices. A ray is characterized by:

Height $y$ above the optical axis
Angle $\theta$ with respect to the optical axis

\vec{r} = \begin{pmatrix} y \\ \theta \end{pmatrix}

(26)

An optical element transforms the ray according to:

\vec{r}_{out} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\vec{r}_{in} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\begin{pmatrix} y_{in} \\ \theta_{in} \end{pmatrix}

(27)

5.2ABCD Matrices for Common Elements¶

Table 3:Ray Transfer Matrices

Optical Element	ABCD Matrix	Physical Effect
Free space (distance d)	$\begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}$	Ray height changes, angle unchanged
Thin lens (focal length f)	$\begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix}$	Height unchanged, angle changes
Curved mirror ( $R>0$ for a focusing concave mirror; unfolded path)	$\begin{pmatrix} 1 & 0 \\ -2/R & 1 \end{pmatrix}$	Reflection and focusing with the optical axis unfolded after reflection
Flat interface (n₁ to n₂)	$\begin{pmatrix} 1 & 0 \\ 0 & n_1/n_2 \end{pmatrix}$	Refraction changes angle

5.3System Analysis Using Matrix Products¶

For multiple elements in series, multiply their matrices in reverse order:

\mathbf{M}_{total} = \mathbf{M}_N \mathbf{M}_{N-1} \cdots \mathbf{M}_2 \mathbf{M}_1

(28)

Example: Simple Telescope

Analyze a telescope consisting of:

Objective lens (focal length $f_1 = 100$ mm)
Distance $d = 120$ mm
Eyepiece lens (focal length $f_2 = 20$ mm)

Solution:

Individual matrices:

Objective: $\mathbf{L}_1 = \begin{pmatrix} 1 & 0 \\ -1/100 & 1 \end{pmatrix}$
Free space: $\mathbf{D} = \begin{pmatrix} 1 & 120 \\ 0 & 1 \end{pmatrix}$
Eyepiece: $\mathbf{L}_2 = \begin{pmatrix} 1 & 0 \\ -1/20 & 1 \end{pmatrix}$

System matrix:

\mathbf{M} = \mathbf{L}_2 \mathbf{D} \mathbf{L}_1

(29)

Step 1: $\mathbf{D}\mathbf{L}_1 = \begin{pmatrix} 1 & 120 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -1/100 & 1 \end{pmatrix} = \begin{pmatrix} -0.2 & 120 \\ -0.01 & 1 \end{pmatrix}$

Step 2: $\mathbf{M} = \begin{pmatrix} 1 & 0 \\ -1/20 & 1 \end{pmatrix}\begin{pmatrix} -0.2 & 120 \\ -0.01 & 1 \end{pmatrix} = \begin{pmatrix} -0.2 & 120 \\ 0 & -5 \end{pmatrix}$

Analysis:

$C=0$ , so an on-axis parallel input bundle leaves parallel: this telescope is afocal.
The output beam-height ratio for a parallel input bundle is $A=-0.2$ ; its magnitude is one fifth of the input beam height.
The signed angular ratio is $D=-5$ for a ray entering on the axis, so the angular magnification has magnitude 5 and the view is inverted. A finite-plane lateral image magnification is not defined for this afocal setup.

5.4Physical Meaning of ABCD Elements¶

Each element of the ABCD matrix has physical significance:

\begin{pmatrix} y_{out} \\ \theta_{out} \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}\begin{pmatrix} y_{in} \\ \theta_{in} \end{pmatrix}

(30)

A: Height magnification ( $y_{out}/y_{in}$ when $\theta_{in} = 0$ )
B: Height displacement per unit input angle (length per radian)
C: Angle change per unit input height (radians per length)
D: Angle magnification ( $\theta_{out}/\theta_{in}$ when $y_{in} = 0$ )

6Practical Applications and Examples¶

6.1Polarization State Analysis¶

Example: Complete Polarization System

Design a system to convert right-handed circular polarization to 30° linear polarization.

Solution:

Input: $\vec{E}_{in} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}$ (RCP)

Target: $\vec{E}_{target} = \begin{pmatrix} \cos 30° \\ \sin 30° \end{pmatrix} = \begin{pmatrix} \sqrt{3}/2 \\ 1/2 \end{pmatrix}$ (30° linear)

Step 1: A quarter-wave plate with fast axis at $-45°$ gives

\mathbf Q_{-45°} =\frac12\begin{pmatrix}1+i&i-1\\i-1&1+i\end{pmatrix}, \qquad \mathbf E_1=\mathbf Q_{-45°}\mathbf E_{\rm in} =\frac{1+i}{\sqrt2}\begin{pmatrix}1\\0\end{pmatrix}.

(31)

The output is horizontal linear polarization, up to an overall phase.

Step 2: A half-wave plate with fast axis at $15°$ rotates horizontal linear polarization to $30°$ :

\mathbf H_{15°}=\begin{pmatrix}\cos30°&\sin30°\\ \sin30°&-\cos30°\end{pmatrix},\qquad \mathbf E_{\rm out}=\frac{1+i}{\sqrt2} \begin{pmatrix}\cos30°\\\sin30°\end{pmatrix}.

(32)

The prefactor has unit modulus, so ideal plates preserve intensity.

6.2Optical System Design¶

Example: Beam Expander Design

Design a Galilean beam expander with a threefold increase in the width of a collimated beam, using lenses with focal lengths -50 mm and +150 mm.

Solution:

System layout: Negative lens → distance $d$ → positive lens

Matrices:

$\mathbf{L}_1 = \begin{pmatrix} 1 & 0 \\ 1/50 & 1 \end{pmatrix}$ ( $f_1=-50$ mm)
$\mathbf{D} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}$ (distance d)
$\mathbf{L}_2 = \begin{pmatrix} 1 & 0 \\ -1/150 & 1 \end{pmatrix}$ ( $f_2=150$ mm)

For an afocal threefold expansion: $C=0$ and $A=3$ for parallel incident rays. These conditions give $d=f_1+f_2=100$ mm.

System matrix: $\mathbf{M} = \mathbf{L}_2\mathbf{D}\mathbf{L}_1$

Multiplication gives $\mathbf M=\begin{pmatrix}3&100\ {\rm mm}\\0&1/3\end{pmatrix}$ , so a collimated input beam exits three times wider with one third the angular spread.

7Common Pitfalls and How to Avoid Them¶

7.1Order of Operations¶

7.2Sign Conventions¶

7.3Physical Interpretation¶

7.4Complex vs. Real Matrices¶

8Matrix Multiplication Techniques and Shortcuts¶

8.1Special Matrix Products¶

Rotation matrices:

\mathbf{R}(\alpha)\mathbf{R}(\beta) = \mathbf{R}(\alpha + \beta)

(33)

Translation matrices:

\begin{pmatrix} 1 & d_1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & d_2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & d_1 + d_2 \\ 0 & 1 \end{pmatrix}

(34)

Thin lens combinations: Two thin lenses in contact: $\mathbf{L}_{total} = \mathbf{L}_2\mathbf{L}_1$

\begin{pmatrix} 1 & 0 \\ -1/f_2 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -1/f_1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -1/f_{eff} & 1 \end{pmatrix}

(35)

where $\frac{1}{f_{eff}} = \frac{1}{f_1} + \frac{1}{f_2}$ is the power-addition rule for thin lenses in contact.

8.2Computational Tips¶

For hand calculations:

Look for patterns (identity elements, zeros)
Factor common terms
Use special properties (orthogonal, symmetric matrices)
Check dimensions and units at each step

For computer calculations:

import numpy as np

# Define matrices
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

# Matrix multiplication
C = A @ B  # or np.dot(A, B)

9Summary and Physical Insights¶

9.1Why Matrix Methods Work So Well¶

Matrix multiplication succeeds in optics because:

Table 4:Fundamental Reasons for Matrix Success

Mathematical Property	Physical Principle	Optical Example
Linearity	Superposition of waves	Interference, diffraction
Composition	Sequential operations	Multi-element systems
Invertible subset	Invertibility of lossless ray propagation	Reverse tracing through lenses
Representation theory	Symmetries	Crystal optics, polarization

9.2Connections to Other Physics¶

Quantum Mechanics: Matrix methods in optics directly parallel quantum mechanical operators acting on state vectors.

Classical Mechanics: ABCD matrices are analogous to transfer matrices in mechanical vibrations and electrical circuits.

Signal Processing: Jones matrices operate on complex signals just like digital filters operate on electronic signals.

9.3Practical Benefits¶

10Practice Problems¶

10.1Basic Matrix Operations¶

10.2Polarization Analysis¶

10.3Ray Optics System¶

10.4System Design¶

11Solutions to Practice Problems¶

Solution to Problem 1

a) $\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 4 & 9 \\ 6 & 3 \end{pmatrix}$

b) $\begin{pmatrix} 1 & 0 \\ -1/50 & 1 \end{pmatrix}\begin{pmatrix} 1 & 100 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 100 \\ -1/50 & -1 \end{pmatrix}$

c) $\begin{pmatrix} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{pmatrix}$ (rotation by $2\theta$ )

Solution to Problem 2

For unit input amplitude, $\mathbf E_{\rm in}=(1,1)^T/\sqrt2$ . The quarter-wave plate gives $\mathbf E_1=(1,i)^T/\sqrt2$ . The half-wave plate has matrix

\mathbf H_{22.5°}=\frac1{\sqrt2} \begin{pmatrix}1&1\\1&-1\end{pmatrix},\qquad \mathbf E_2=\frac12\begin{pmatrix}1+i\\1-i\end{pmatrix}.

(36)

The final polarizer transmits along $\mathbf p=(1/2,\sqrt3/2)^T$ , so

\mathbf E_{\rm out} =\mathbf p(\mathbf p^T\mathbf E_2) =\frac{(1+\sqrt3)+i(1-\sqrt3)}4 \begin{pmatrix}1/2\\\sqrt3/2\end{pmatrix}.

(37)

The transmitted intensity fraction is $|\mathbf p^T\mathbf E_2|^2=1/2$ .

Solution to Problem 3

The interface matrix in $(y,\theta)^T$ coordinates is $\begin{pmatrix}1&0\\-(n_2-n_1)/(n_2R)&n_1/n_2\end{pmatrix}$ . Thus

\mathbf M =\underbrace{\begin{pmatrix}1&0\\-P_2&n\end{pmatrix}}_{\rm back} \underbrace{\begin{pmatrix}1&t\\0&1\end{pmatrix}}_{\rm glass} \underbrace{\begin{pmatrix}1&0\\-P_1/n&1/n\end{pmatrix}}_{\rm front} =\begin{pmatrix} 1-tP_1/n&t/n\\ -P_1-P_2+tP_1P_2/n&1-tP_2/n \end{pmatrix}.

(38)

The determinant is one because the exterior medium is air on both sides.

Solution to Problem 4

a) The object plane is 200 mm before lens 1, the lenses are 300 mm apart, and the image plane is 100 mm after lens 2.

b) With $\mathbf D(d)=\begin{pmatrix}1&d\\0&1\end{pmatrix}$ and $\mathbf L(f)=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}$ ,

\mathbf M=\mathbf D(100)\mathbf L(100)\mathbf D(300) \mathbf L(200)\mathbf D(200) =\begin{pmatrix}-1/2&0\\0&-2\end{pmatrix}.

(39)

c) Since $B=0$ in this $(y,\theta)^T$ convention, the two planes are conjugate and the lateral magnification is $A=-f_2/f_1=-0.5$ .

d) The angular magnification is $D=-f_1/f_2=-2$ .

12Final Thoughts: The Elegance of Linear Algebra in Physics¶

Matrix multiplication reveals deep structural relationships in optics that would be nearly impossible to see otherwise. The mathematical framework doesn’t just make calculations easier—it reveals the underlying symmetries and conservation laws that govern how light behaves.

“Mathematics is the language with which God has written the universe.”

Galileo Galilei

As you continue studying optics, you’ll discover that matrix methods appear everywhere: from the simplest polarizer to the most sophisticated laser system. The investment in understanding these mathematical tools pays dividends throughout your career in optics and photonics.

Matrix multiplication in optics is more than a computational tool—it’s a window into the mathematical structure of the electromagnetic world. Master these methods, and you’ll have one of the most powerful techniques in all of physics at your command.