🎓 Matrix Learning Lab

Interactive visualizations and explorations for linear algebra — from first principles to matrix decompositions

🎯 What Are Matrices?

Welcome to your interactive journey into linear algebra! Let's start with the absolute basics and build up from there.

Part 1: A Matrix is Just a Grid of Numbers

Think of it like a table or spreadsheet:

│  1   2  │
│  3   4  │

This is a 2×2 matrix (2 rows, 2 columns).
We read it as "2 by 2".

How to Read Matrix Notation

Matrix A = │  a   b  │
           │  c   d  │

Entry notation:
• A₁₁ = a (row 1, col 1)
• A₁₂ = b (row 1, col 2)
• A₂₁ = c (row 2, col 1)
• A₂₂ = d (row 2, col 2)

Here's the magic: Matrices transform vectors into new vectors. Imagine a matrix as a machine:

🎯 Input Vector → Matrix Machine → 🎯 Output Vector

The matrix "transforms" the input into something new!

📝 Simple Example:

Let's see how a matrix transforms a vector (point):

Matrix    │ 2  0 │    Input      │ 3 │
          │ 0  3 │  × Vector   │ 2 │

Result = (2×3, 0×3+3×2) = │ 6 │  ← Output Vector
                          │ 6 │

What happened?
The point (3, 2) was transformed to (6, 6):
• x-coordinate: 2 × 3 + 0 × 2 = 6 (doubled)
• y-coordinate: 0 × 3 + 3 × 2 = 6 (tripled)
The matrix stretched the point along both axes!

Real-World Applications

🎮 Computer Graphics

Rotate, scale, and move 3D game objects. Every rotation in a video game uses matrices!

📱 Image Filters

Instagram filters? Matrices! They transform pixel colors to create effects like blur, sharpen, and edge detection.

🤖 Machine Learning

Neural networks are built from matrices. Every AI model you use (ChatGPT, image recognition) runs on matrix math!

📊 Data Analysis

Organize and analyze large datasets. Matrices help find patterns in millions of data points (like Netflix recommendations).

⚡ Engineering

Model electrical circuits, structural forces, and quantum mechanics. Engineers use matrices to predict how systems behave.

🔐 Cryptography

Encrypt secret messages! Some encryption methods multiply your message by a secret matrix to scramble it.

🎯 Lesson 2: Matrix Arithmetic

Learning Objectives:
✓ Master matrix addition and subtraction
✓ Understand matrix multiplication (it's different!)
✓ Learn transpose, determinant, and inverse operations

Part 1: Addition & Subtraction (The Easy Ones!)

Good news: Adding matrices is like adding regular numbers — just do it element-by-element!

Example: Add these 2×2 matrices

│ 1  2 │
│ 3  4 │Matrix A
+
│ 5  6 │
│ 7  8 │Matrix B
=
│  6   8 │
│ 10  12 │Result

Each element adds: (1+5=6, 2+6=8, 3+7=10, 4+8=12)

⚠️ Important Rule: You can only add/subtract matrices with the SAME dimensions! A 2×3 matrix + 2×3 matrix = ✓ Valid. A 2×2 matrix + 3×3 matrix = ✗ Error!

Part 2: Matrix Multiplication (The Tricky One!)

Warning: Matrix multiplication is NOT element-by-element! It's more like "mix and combine".

The Rule: To multiply matrices A × B:

Take a row from A
Take a column from B
Multiply corresponding elements and add them up
That's one entry in the result!

Example: Calculate entry (1,1) of A × B

Row 1 of A = [1, 2]
Column 1 of B = [5, 7]

Result = (1×5) + (2×7) = 5 + 14 = 19

💡 Dimension Rule: You can only multiply A × B if columns_of_A = rows_of_B. A (2×3) × B (3×4) = ✓ Valid (result is 2×4). A (2×3) × B (4×2) = ✗ Invalid!

Part 3: Transpose (The Flipper!)

The transpose of a matrix flips rows into columns (and columns into rows).

│ 1  2  3 │
│ 4  5  6 │

Original (2×3)

→ Aᵀ →

│ 1  4 │
│ 2  5 │
│ 3  6 │

Transposed (3×2)

Notice: Rows became columns! The first row [1,2,3] became the first column.

Part 4: Determinant (The "Size" of a Matrix)

The determinant tells you if a matrix is "invertible" and measures scaling. For 2×2 matrices:

│  a   b  │
│  c   d  │

det(A) = ad - bc

Example:

Matrix: │  2   3  │
        │  1   4  │

det = (2×4) - (3×1) = 8 - 3 = 5

🔍 What does determinant mean?
• det = 0 → Matrix is "singular" (not invertible, collapses space)
• det > 0 → Matrix preserves orientation
• det < 0 → Matrix flips orientation (mirrors)
• |det| = scaling factor (how much the matrix stretches area/volume)

Exercise 3: Determinant

Find det(A) for:

          A =       
              │ 3  -2│
              │ 1   4│

Show Solution

det(A) = (3×4) - (-2×1) = 12 - (-2) = 12 + 2 = 14

💻 Interactive Matrix Calculator

Now try these operations yourself! Load presets or enter your own matrices.

Quick Load Presets:

1. Choose Operation:

2. Matrix Size:

Count:

3. Actions:

🎯 Lesson 3: Meet the Matrix Celebrities

Learning Objectives:
✓ Recognize identity, diagonal, rotation, and symmetric matrices
✓ Understand what makes each matrix type special
✓ Know when to use each type in practice

Just like numbers have special cases (0, 1, π), matrices have "celebrities" — special types that show up everywhere!

🟦 Identity Matrix (I) — The "Number 1" of Matrices

What is it? Ones on the diagonal, zeros everywhere else.

│ 1  0  0 │
│ 0  1  0 │
│ 0  0  1 │

Superpower: Multiplying by I does nothing!
A × I = A (just like 5 × 1 = 5)

Why it matters: Starting point for transformations, used in algorithms, essential for matrix inverses

🔄 Rotation Matrix — The Spinner

What is it? Rotates vectors by angle θ without stretching.

                  
│ cos(θ)  -sin(θ) │
│ sin(θ)   cos(θ) │

Superpower: Pure rotation, preserves lengths!

Example: θ = 45° →
cos(45°) ≈ 0.707, sin(45°) ≈ 0.707

Why it matters: Computer graphics, robotics, animations, GPS systems

📏 Diagonal Matrix — The Scaler

What is it? Only diagonal entries are non-zero.

│ 2  0  0 │
│ 0  5  0 │
│ 0  0  3 │

Superpower: Scales each axis independently!
(x,y,z) → (2x, 5y, 3z)

Why it matters: Easy to compute powers (D² = just square each entry!), eigenvalue decompositions

🔁 Symmetric Matrix — The Mirror

What is it? Equals its own transpose: A = Aᵀ (mirror across diagonal).

         
│ 4  1  2│
│ 1  3  5│  ← Notice symmetry
│ 2  5  6│     across diagonal!

Superpower: Always has real eigenvalues!

Check: A₁₂ = A₂₁ = 1 ✓
A₁₃ = A₃₁ = 2 ✓
A₂₃ = A₃₂ = 5 ✓

Why it matters: Physics (stress tensors), optimization, covariance matrices in statistics

🪞 Orthogonal Matrix — The Preserver

What is it? Preserves lengths and angles. QᵀQ = I

               
│ 0.8  -0.6  0│
│ 0.6   0.8  0│
│ 0     0    1│

Superpower: Rotations + reflections, no distortion!

Distance between points = unchanged
Angles = unchanged
det(Q) = ±1

Why it matters: Computer vision, QR decomposition, numerical stability

⚪ Zero Matrix — The Destroyer

What is it? All entries are zero (the "0" of matrices).

         
│ 0  0  0│
│ 0  0  0│

Superpower: Collapses everything to zero!
A × 0 = 0

Why it matters: Represents "no transformation", used as initializer, theoretical importance

🧪 Try It Yourself!

Use the presets below to explore each special matrix type.

🎯 Lesson 4: Matrices in Motion

Learning Objectives:
✓ Visualize how matrices transform space
✓ Understand rotation, scaling, shearing, and reflection
✓ Connect determinant to geometric area changes

The Big Idea: Matrices Transform Space

A 2×2 matrix takes every point (x, y) in the plane and moves it to a new location. Think of it like molding clay!

How It Works:

Matrix        times vector      equals new vector         
       │ a b│              │ x │                  │ ax + by│
       │ c d│              │ y │                  │ cx + dy│

The matrix "mixes" x and y coordinates to create the new position!

Common Transformations

🔄 Rotation (by θ degrees)

                  
│ cos(θ)  -sin(θ) │
│ sin(θ)   cos(θ) │

What it does: Spins vectors around the origin
Determinant: Always 1 (no area change!)
Example: θ=90° → Rotate left 90 degrees

📏 Scaling (by factors sx, sy)

         
│ sx   0 │
│  0  sy │

What it does: Stretches/squashes along axes
Determinant: sx × sy (area multiplier)
Example: [2, 0; 0, 3] → doubles x, triples y

↔️ Shear (slant by k)

       
│ 1  k │  ← Horizontal shear
│ 0  1 │

What it does: "Slants" space (like pushing cards)
Determinant: Always 1 (area preserved!)
Example: k=1 → skews parallelogram

🪞 Reflection (over x-axis)

        
│  1   0│
│  0  -1│  ← Flip y-coordinate

What it does: Mirrors across an axis
Determinant: -1 (flips orientation!)
Note: Negative det = mirror/flip

🔍 Determinant Tells The Story:
• |det| = how much area gets scaled (2 = double area, 0.5 = half area)
• det > 0 = preserves orientation (no flip)
• det < 0 = flips orientation (mirror)
• det = 0 = collapses to a line (singular matrix!)

💻 Interactive Transformer

Try transforming the unit square! Enter matrix values or use presets.

Enter Matrix Values

Determinant: 1.00

Area scaling factor
(negative = orientation flip)

Click plot to set vector

Rotate

Scale

Vector

Click plot to set vector

Interactive Controls: ● Original Vector (blue) | ● Transformed Vector (green)
Grid: coordinate system Click on plot to change vector position

🎯 Lesson 5: Solving Systems with Matrices

Learning Objectives:
✓ Convert equation systems to matrix form (Ax = b)
✓ Solve systems using matrix operations
✓ Understand when systems have unique, infinite, or no solutions

Why Matrices for Equations?

Remember solving systems by hand? Painful! Matrices make it systematic and scalable to 100s of variables.

Step 1: Convert to Matrix Form

System of Equations:

2x + 3y = 8
x - y = 1

→

Matrix Form (Ax = b):

                  
│ 2  3│ · │ x│ = │ 8│
│ 1 -1│   │ y│   │ 1│
                  
   A        x       b

A = coefficients, x = unknowns, b = right-hand sides

Step 2: Solve using Matrix Inverse

If A is invertible (det ≠ 0), the solution is: x = A⁻¹b

Example: Solve the system above

Step 1: Find det(A) = (2)(-1) - (3)(1) = -2 - 3 = -5 ✓ (non-zero!)

Step 2: Find A⁻¹ (use formula for 2×2):
A⁻¹ = (1/det) ×         =  (1/-5) ×         =           
                 │ d -b│              │-1 -3│     │ 0.2  0.6│
                 │-c  a│              │-1  2│     │ 0.2 -0.4│
                                                         

Step 3: Multiply x = A⁻¹b:
                                                
│ x│ = │ 0.2  0.6│ · │ 8│ = │ (0.2×8)+(0.6×1) │ = │2.2│
│ y│   │ 0.2 -0.4│   │ 1│   │ (0.2×8)+(-0.4×1)│   │1.2│
                                                

Answer: x = 2.2, y = 1.2

✓ Check: 2(2.2) + 3(1.2) = 4.4 + 3.6 = 8.0 ✓ | 2.2 - 1.2 = 1.0 ✓

The Three Cases: How Many Solutions?

✅ Case 1: Unique Solution (det ≠ 0)

When: Matrix A is invertible (full rank)
Means: Exactly ONE solution exists
Solve: x = A⁻¹b

Example:        det = 2  →  Unique solution!
         │ 2 3│
         │ 0 1│

⚠️ Case 2: Infinite Solutions (det = 0, consistent)

When: Equations are dependent (one is a multiple of another)
Means: Infinite solutions (a line/plane of solutions)
Example: 2x + 4y = 6 and x + 2y = 3 (same line!)

Matrix:        det = 0  →  Infinite solutions
        │ 2 4│
        │ 1 2│

❌ Case 3: No Solution (det = 0, inconsistent)

When: Equations are contradictory
Means: No solution exists (parallel lines never meet!)
Example: 2x + 4y = 6 and 2x + 4y = 10 (impossible!)

Same matrix but different b → No solution

🔍 Quick Check: Calculate det(A) first!
• det ≠ 0 → Unique solution guaranteed ✓
• det = 0 → Check if consistent (infinite) or inconsistent (no solution)

💻 Interactive System Solver

Solve your own linear systems! Enter the number of equations and let the matrix solver work its magic.

Number of equations:

🧪 Practice Problems

Exercise 1: Convert to Matrix Form

Write this system as Ax = b:

3x + 2y = 7
x - 4y = -5

Show Solution

A =           x =        b =     
    │ 3  2│        │ x│        │  7│
    │ 1 -4│        │ y│        │ -5│

Exercise 2: Solve by Hand

Solve using matrix inverse:

                 
│ 1 2│ · │ x│ = │ 5│
│ 3 4│   │ y│   │11│

Show Solution

det(A) = (1×4) - (2×3) = 4 - 6 = -2 ✓

A⁻¹ = (1/-2) ×         =         
               │  4 -2│     │ -2  1│
               │ -3  1│     │1.5 -0.5│
                                 

x = A⁻¹b =         ×     =                 =    
           │ -2  1│   │ 5│   │(-2×5)+(1×11)│   │ 1│
           │1.5 -0.5│   │11│   │(1.5×5)+(-0.5×11)│   │ 2│
                                                

Answer: x = 1, y = 2

Exercise 3: Determine Solution Type

How many solutions does each system have?

System A:              System B:      
          │ 2 6│                  │ 1 2│
          │ 1 3│                  │ 3 5│

Show Solution

System A: det = (2×3) - (6×1) = 6 - 6 = 0
→ Either infinite solutions or no solution (need to check consistency)

System B: det = (1×5) - (2×3) = 5 - 6 = -1 ≠ 0
→ Unique solution exists! ✓

� Lesson 6: Eigenvalues — The Magic Directions

Learning Objectives:
✓ Understand what eigenvalues and eigenvectors represent geometrically
✓ Learn how to find eigenvalues (det(A - λI) = 0)
✓ Recognize why eigenvalues matter in applications

The Mind-Blowing Idea

Most matrices rotate AND stretch vectors. But there are special directions where the matrix ONLY stretches!

The Eigenvalue Equation

Av = λv

A
The matrix

v
Eigenvector
(special direction)

λ
Eigenvalue
(stretch factor)

"Matrix A just scales vector v by λ — no rotation!"

Example: See It In Action

Matrix A =          Eigenvector v =        Eigenvalue λ = 4
           │ 3 1│                    │ 1│
           │ 0 4│                    │ 0│
                                       

Let's verify Av = λv:

Av =       ×     =         =    
     │ 3 1│   │ 1│   │ 3×1+1×0│   │ 3│
     │ 0 4│   │ 0│   │ 0×1+4×0│   │ 0│
                                   ← This should equal 4v

λv = 4 ×     =       ← They DON'T match! So v=[1,0] is NOT an eigenvector.
         │ 1│   │ 4│
         │ 0│   │ 0│
                  

Try v =    :
        │ 1│
        │ 1│
           

Av =       ×     =        and    4v =       ✓ Match! This IS an eigenvector!
     │ 3 1│   │ 1│   │ 4│                  │ 4│
     │ 0 4│   │ 1│   │ 4│                  │ 4│

How to Find Eigenvalues (The Algorithm)

Step 1: Set up the characteristic equation

det(A - λI) = 0

Why? We need Av = λv, which rearranges to (A - λI)v = 0.
This only has non-trivial solutions when det(A - λI) = 0.

Step 2: Calculate the determinant (gives a polynomial)

Example: A =      
             │ 4 2│
             │ 1 3│
                  

A - λI =          
         │ 4-λ   2│
         │  1  3-λ│
                  

det(A - λI) = (4-λ)(3-λ) - (2)(1)
            = 12 - 4λ - 3λ + λ² - 2
            = λ² - 7λ + 10

Step 3: Solve for λ (find roots)

λ² - 7λ + 10 = 0
(λ - 5)(λ - 2) = 0

Eigenvalues: λ₁ = 5, λ₂ = 2

Step 4: Find eigenvectors (plug λ back in)

For λ = 5:  (A - 5I)v = 0  →      v = 0  →  Eigenvector v₁ =    
                                   │-1  2│                       │ 2│
                                   │ 1 -2│                       │ 1│

Why Eigenvalues Matter

🎬 Google PageRank
The web is a giant matrix! Google finds the "importance" eigenvector of this matrix.

🔬 Physics & Engineering
Vibration modes, quantum mechanics, stability analysis — all use eigenvalues!

📊 Principal Component Analysis (PCA)
Data science's favorite tool! Uses eigenvectors to find important patterns in data.

🎮 Computer Graphics
Eigenvectors define "principal axes" for 3D object rotations.

🔍 Quick Facts:
• An n×n matrix has n eigenvalues (counting multiplicities)
• Symmetric matrices → always real eigenvalues
• Trace(A) = sum of eigenvalues
• det(A) = product of eigenvalues
• Eigenvalues of diagonal matrices = the diagonal entries!

💻 Compute & Visualize

Generate a random square matrix to explore its eigenvalues and eigenvectors.

Matrix Size (N×N):

📊 Visualization

💡 How to read this plot:
• Arrows (Real Eigenvalues): Show the eigenvectors ($v$) and how they get scaled ($\lambda v$).
• Points (Complex Eigenvalues): Show the eigenvalues on the complex plane (Rotation + Scaling).

🎯 Lesson 7: SVD & PCA — Data Science Superpowers

Learning Objectives:
✓ Understand Singular Value Decomposition (A = UΣVᵀ)
✓ Learn how SVD enables data compression
✓ Master Principal Component Analysis (PCA) for dimensionality reduction

SVD: The Ultimate Matrix Factorization

Every matrix can be broken down into three simple pieces!

A = U Σ Vᵀ

A
Original
Matrix

U
Rotate
(left)

Σ
Stretch
(diagonal)

Vᵀ
Rotate
(right)

Any transformation = Rotate → Stretch → Rotate!

The Three Pieces Explained

U (Left Singular Vectors)
• Orthogonal matrix (columns are perpendicular)
• Defines output space directions
• Size: m × m

Σ (Singular Values)
• Diagonal matrix with σ₁ ≥ σ₂ ≥ ... ≥ 0
• Shows "importance" of each direction
• Large σ = important, small σ = can be dropped!

Vᵀ (Right Singular Vectors)
• Orthogonal matrix (rows are perpendicular)
• Defines input space directions
• Size: n × n

🗜️ The Magic: Data Compression!

Keep only the TOP k singular values → throw away the rest → get close approximation with less data!

Example: Compress an Image

Original 1000×1000 image matrix = 1,000,000 numbers

After SVD: Keep top 50 singular values
→ Storage needed: 50 × (1000 + 1000 + 1) ≈ 100,000 numbers
→ Compression ratio: 10x smaller! Only 10% of original size

Result: Image looks almost identical but file is 90% smaller! 🎉

This is exactly what JPEG compression does! (with a twist using DCT, a cousin of SVD)

📊 PCA: Finding Patterns in Data

Principal Component Analysis uses SVD to find the "most important directions" in your data.

The PCA Recipe (4 Steps)

Center the data: Subtract mean from each feature
Compute covariance matrix: Shows how features vary together
Find eigenvalues & eigenvectors: Get principal components (eigenvectors of covariance matrix)
Project onto top components: Reduce from 100D → 2D!

Real Example: Netflix Recommendations
• You have 10,000 movies (10,000 dimensions!)
• PCA finds ~50 "hidden factors" (action, romance, comedy, etc.)
• Each user's taste = combination of these 50 factors
• 10,000D → 50D = 200x compression! 🚀

💡 Key Insight: Not all directions matter equally! SVD/PCA find the directions with the MOST variation. Drop the boring directions, keep the interesting ones.

💻 Interactive SVD Decomposition

Generate a random matrix or enter one to see its Singular Value Decomposition.

Rows:

Cols:

🎯 Interactive PCA Analysis

Generate a random dataset (rows=samples, cols=features) and find its principal components.

Samples (Rows):

Features (Cols):

🎓 Matrix Learning Lab

🎯 What Are Matrices?

Part 1: A Matrix is Just a Grid of Numbers

How to Read Matrix Notation

Real-World Applications

🎯 Lesson 2: Matrix Arithmetic

Part 1: Addition & Subtraction (The Easy Ones!)

Part 2: Matrix Multiplication (The Tricky One!)

Part 3: Transpose (The Flipper!)

Part 4: Determinant (The "Size" of a Matrix)

💻 Interactive Matrix Calculator

Quick Load Presets:

Enter Matrix (semicolon-separated rows)

🎯 Lesson 3: Meet the Matrix Celebrities

🟦 Identity Matrix (I) — The "Number 1" of Matrices

🔄 Rotation Matrix — The Spinner

📏 Diagonal Matrix — The Scaler

🔁 Symmetric Matrix — The Mirror

🪞 Orthogonal Matrix — The Preserver

⚪ Zero Matrix — The Destroyer

🧪 Try It Yourself!

🎯 Lesson 4: Matrices in Motion

The Big Idea: Matrices Transform Space

Common Transformations

🔄 Rotation (by θ degrees)

📏 Scaling (by factors sx, sy)

↔️ Shear (slant by k)

🪞 Reflection (over x-axis)

💻 Interactive Transformer

Enter Matrix Values

🎯 Lesson 5: Solving Systems with Matrices

Why Matrices for Equations?

Step 2: Solve using Matrix Inverse

The Three Cases: How Many Solutions?

✅ Case 1: Unique Solution (det ≠ 0)

⚠️ Case 2: Infinite Solutions (det = 0, consistent)

❌ Case 3: No Solution (det = 0, inconsistent)

💻 Interactive System Solver

🧪 Practice Problems

� Lesson 6: Eigenvalues — The Magic Directions

The Mind-Blowing Idea

Example: See It In Action

How to Find Eigenvalues (The Algorithm)

Why Eigenvalues Matter

💻 Compute & Visualize

📊 Visualization

🎯 Lesson 7: SVD & PCA — Data Science Superpowers

SVD: The Ultimate Matrix Factorization

The Three Pieces Explained

🗜️ The Magic: Data Compression!

📊 PCA: Finding Patterns in Data

💻 Interactive SVD Decomposition

🎯 Interactive PCA Analysis

Explained Variance (Scree Plot)

2D Projection (PC1 vs PC2)