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Eigen Values

The world of mathematics has many unusual words, and eigenvalues are one of them. Although the term may seem complicated at first, it plays an important role in understanding how things work, such as moving objects, sound vibrations, or even face recognition software.

In this blog, we will explain what an eigenvalue is, what it tells us, how it connects to eigenvectors, and how to calculate it all in a straightforward way.

Table Of Contents

What Is an Eigen Value?
Why Are Eigen Values Important?
What Do Eigen Values Tell Us?
What Is an Eigen Vector?
How to Calculate Eigen Values
Example of an Eigen Value
What Is the Relationship Between Eigen Values and Eigen Vectors?
Applications of Eigen Values and Eigen Vectors
Fun Facts About Eigen Values
Common Mistakes When Learning Eigen Values
Tips to Remember
Conclusion
FAQs on Eigen Values

What Is an Eigen Value?

An eigen value is a special number related to a matrix. It tells us how a vector changes when a certain transformation is applied to it using that matrix.

Let’s say you stretch or rotate something using a matrix. If there’s a direction where the object doesn’t change direction but only gets stretched or squished, the amount of stretch is called the eigen value. The direction itself is the eigen vector.

In simple words:

Eigen values = amount of stretch or compression
Eigen vectors = the direction that doesn’t change

So, eigen values eigen vectors help us understand what happens to things when we apply certain transformations.

Why Are Eigen Values Important?

You might wonder, “Why do we find eigen values?” The answer is: they show up in many areas of real life and science.

Eigen values are useful in:

Physics - to study vibrations and waves
Engineering - to check if buildings or bridges will shake
Computer Science - for face recognition and data compression
Mathematics - to understand matrices and transformations
Finance - to analyse risk and trends

Understanding eigen value eigen vector relationships helps in breaking down complex systems into simpler parts.

What Do Eigen Values Tell Us?

Eigen values tell us how a system behaves when something is applied to it like a force, change, or movement.

For example:

If an eigen value is 1, it means the vector stays the same size.
If the value is 2, the vector doubles in size.
If the value is 0, the vector gets flattened (disappears).
A negative eigen value means the vector flips direction.

This way, eigen values eigen vectors describe both how much change happens and in which direction.

What Is an Eigen Vector?

Let’s quickly understand eigen vectors because they go hand in hand with eigen values.

An eigen vector is a special kind of vector. When a transformation (like stretching, turning, etc.) is applied to it through a matrix, it does not change direction. It might get longer or shorter, but the line it lies on stays the same.

Together, eigen value eigen vector describe how a matrix acts like a machine that stretches or squashes space in specific ways.

How to Calculate Eigen Values

Now let’s see how to calculate an eigen value step by step.

Step 1: Start with a square matrix

Example:

A =

| 2 1 |

| 1 2 |

Step 2: Subtract λ (lambda) from the diagonals

Make a new matrix:

| 2-λ 1 |

| 1 2-λ |

Step 3: Find the determinant and set it to 0

Determinant of a 2×2 matrix:

(2-λ)(2-λ) - (1×1) = 0

(2-λ)² - 1 = 0

λ² - 4λ + 3 = 0

Step 4: Solve for λ

Using the quadratic formula:

λ = 1 and 3

These are your eigen values of matrix A.

Example of an Eigen Value

Let’s look at a simple example:

Matrix A =

| 4 2 |

| 1 3 |

Find the eigen values.

Step 1: Subtract λ from the diagonal:

| 4-λ 2 |

| 1 3-λ |

Step 2: Find the determinant:

(4-λ)(3-λ) - 2×1 = 0

(4-λ)(3-λ) - 2 = 0

λ² - 7λ + 10 = 0

Step 3: Solve:

λ = 2 and 5

So, 2 and 5 are the eigen values of this matrix.

This shows how eigen value is calculated and gives us insight into how the transformation behaves.

What Is the Relationship Between Eigen Values and Eigen Vectors?

Each eigen value has at least one corresponding eigen vector. They work together like a pair.

When a matrix A acts on its eigen vector v, the result is just a scaled version of v:

A⋅v=λ⋅vA \cdot v = λ \cdot vA⋅v=λ⋅v

This equation says:

Take the matrix A
Multiply it by vector v (the eigen vector)
The result is λ (the eigen value) times vector v

That’s why the pair eigen value eigen vector is always discussed together.

Applications of Eigen Values and Eigen Vectors

Let’s explore some real-world places where eigen values eigen vectors appear:

In Google’s Search Algorithm

Google uses eigen values to rank websites using something called “PageRank.”

In Face Recognition Software

Cameras and apps use eigen faces based on eigen vectors to recognize your face.

In Earthquake Engineering

Engineers calculate eigen values to see how much a building will shake in an earthquake.

In Vibration Analysis

Cars and planes are tested for vibrations using eigen values and eigen vectors.

In Quantum Physics

Eigen values describe energy levels of electrons in atoms.

These examples show why it’s important to learn about eigen values even if they sound hard at first.

Fun Facts About Eigen Values

The word “eigen” comes from German and means “own” or “self.”
An eigen value can be positive, negative, zero, or even complex (with i).
Eigen values are used in machine learning, AI, and data science.

Even though they’re abstract, they help solve practical problems.

Common Mistakes When Learning Eigen Values

Confusing matrix multiplication rules
Forgetting to set the determinant to 0
Using the wrong formula for determinant
Mixing up eigen value and eigen vector
Thinking eigen values only apply to square matrices they do!

Stay calm and follow each step carefully when solving.

Tips to Remember

Practice with small 2×2 matrices before moving to bigger ones
Learn how to find determinants first
Write the eigen value equation clearly:
A⋅v=λ⋅vA \cdot v = λ \cdot vA⋅v=λ⋅v
Use the quadratic formula if needed
Check your final answer by plugging back into the equation

The more you practice, the easier eigen values eigen vectors become.

Conclusion

Eigen values might sound like a big topic, but they’re just numbers that show how things change when you apply a transformation. When combined with eigen vectors, they provide useful insights into systems ranging from simple math to physics and even AI.

Understanding the relationship between eigen values and eigen vectors makes it easier to solve problems, break down matrices, and recognize patterns in real life.

So next time you hear about eigen values, just remember they aren't scary; they're just special numbers that help us understand the world a little better.

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Frequently Asked Questions on Eigen Values

1. What do you mean by eigen value?

Ans: An eigen value is a number that shows how much a vector stretches or shrinks when a transformation (matrix) is applied to it.

2. What eigenvalues tell us?

Ans: Eigen values tell us how a system changes. They show if a shape or vector is stretched, squashed, flipped, or stays the same during transformation.

3. What is an example of an eigenvalue?

Ans: For matrix A =