What Are Eigenvalues?

Steps to Determine Eigenvalues

Key Points to Remember

Applications of Eigenvalues

Eigenvalues Of A Matrix FAQs

How To Determine The Eigenvalues Of A Matrix

Eigenvalues are special numbers associated with a matrix that provide important information about its properties and behavior in various mathematical and engineering applications. If you're studying linear algebra or working with matrices, understanding how to determine eigenvalues is essential. Here's a step-by-step guide to finding the eigenvalues of a matrix, explained in simple terms.

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What Are Eigenvalues?

In simple terms, eigenvalues are numbers that satisfy the equation:

A \cdot v = \lambda \cdot v

$A$ $\cdot$ $v$ $=$ $λ$ $\cdot$ $v$

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Here:

$A$ $A$ is the matrix,
$v$ $v$ is a non-zero vector (called an eigenvector),
$\lambda$ $λ$ is the eigenvalue.

The eigenvalues tell us how the matrix transforms the eigenvector $v$ $v$ .

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Steps to Determine Eigenvalues

Follow these steps to calculate the eigenvalues of a matrix:

1. Understand the Characteristic Equation

To find eigenvalues, solve the characteristic equation:

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\text{det}(A - \lambda I) = 0

$det$ $($ $A$ $-$ $λI$ $)$ $=$ $0$

$A$ $A$ is the given matrix,
$\lambda$ $λ$ is the eigenvalue (unknown to be solved),
$I$ $I$ is the identity matrix of the same size as $A$ $A$ ,
$\text{det}$ $det$ stands for the determinant.

This equation ensures that $A - \lambda I$ $A$ $-$ $λI$ has no inverse, which is a key property for eigenvalues.

2. Set Up the Matrix $A - \lambda I$ $A$ $-$ $λI$

Subtract $\lambda I$ $λI$ (a matrix with $\lambda$ $λ$ on the diagonal) from the matrix $A$ $A$ . For example:

If $A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}$ $A$ $=$ $[$ $41$ $$ $23$ $$ $]$ , then:

A - \lambda I = \begin{bmatrix} 4-\lambda & 2 \\ 1 & 3-\lambda \end{bmatrix}

$A$ $-$ $λI$ $=$ $[$ $4$ $-$ $λ$ $1$ $$ $23$ $-$ $λ$ $$ $]$

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3. Find the Determinant

Compute the determinant of $A - \lambda I$ $A$ $-$ $λI$ . For a 2x2 matrix, the determinant is calculated as:

\text{det} = (a - \lambda)(d - \lambda) - (b \cdot c)

$det$ $=$ $($ $a$ $-$ $λ$ $)$ $($ $d$ $-$ $λ$ $)$ $-$ $($ $b$ $\cdot$ $c$ $)$

For our example:

\text{det}(A - \lambda I) = (4-\lambda)(3-\lambda) - (2 \cdot 1)

$det$ $($ $A$ $-$ $λI$ $)$ $=$ $($ $4$ $-$ $λ$ $)$ $($ $3$ $-$ $λ$ $)$ $-$ $($ $2$ $\cdot$ $1$ $)$

Simplify:

\text{det}(A - \lambda I) = (4-\lambda)(3-\lambda) - 2 = 12 - 7\lambda + \lambda^2 - 2 = \lambda^2 - 7\lambda + 10

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

4. Solve the Characteristic Equation

Set the determinant to 0 and solve for $\lambda$ $λ$ :

\lambda^2 - 7\lambda + 10 = 0

$λ$ $2$ $-$ $7$ $λ$ $+$ $10$ $=$ $0$

Use factoring, completing the square, or the quadratic formula to find $\lambda$ $λ$ . In this case, factoring works:

\lambda^2 - 7\lambda + 10 = (\lambda - 5)(\lambda - 2) = 0

$λ$ $2$ $-$ $7$ $λ$ $+$ $10$ $=$ $($ $λ$ $-$ $5$ $)$ $($ $λ$ $-$ $2$ $)$ $=$ $0$

So, the eigenvalues are:

\lambda = 5 \text{ and } \lambda = 2

$λ$ $=$ $5$ $and$ $λ$ $=$ $2$

Key Points to Remember

Eigenvalues are solutions to $\text{det}(A - \lambda I) = 0$ $det$ $($ $A$ $-$ $λI$ $)$ $=$ $0$ .
The number of eigenvalues is equal to the size of the matrix (e.g., a 3x3 matrix has three eigenvalues).
Eigenvalues can be real or complex numbers.

Applications of Eigenvalues

Physics: Understanding vibrations and stability of systems.
Computer Science: Principal Component Analysis (PCA) for dimensionality reduction.
Engineering: Analyzing stress and strain on structures.

Eigenvalues Of A Matrix FAQs

Are all Eigenvectors Orthogonal 100% of the time?

For any network, the eigenvectors are not symmetrical all of the time. Nonetheless, in an asymmetric framework, where eigenvalues are generally genuine, the relating eigenvalues are symmetrical all the time. A framework A increased with its render, yields a symmetric grid wherein the eigenvectors are symmetrical all of the time. The foremost part examination is applied to the symmetric network, henceforth the eigenvectors will be symmetrical 100% of the time.

Would a Real Matrix be able to have Complex Eigenvectors?

In the event that a n × n network has generally genuine qualities, it isn't required that the eigenvalues of the lattice are largely genuine numbers. The eigenvalues are gotten from arrangements of a quadratic polynomial. It isn't generally vital for a quadratic polynomial to yield genuine qualities. For instance: If a lattice has an eigenvalue like t2+1, then, at that point, it will yield a nonexistent outcome.

What is a left eigenvector?

Assuming an Eigenvector is addressed as far as a line vector, it is then called a left eigenvector and the condition that it follows is AXL =λXL, here 'A' is the provided network having request 'n' and 'λ' is one of the eigenvectors of A. 'XL' signifies a column vector. Along these lines, by the above definition, a line vector can be known as a left eigenvector.

What Are Eigenvalues?

Steps to Determine Eigenvalues

Key Points to Remember

Applications of Eigenvalues

Eigenvalues Of A Matrix FAQs

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