To understand eigenvalues and eigenvectors, we first need to know what these words mean in linear algebra.

Eigenvalues (we use the symbol $\lambda$) are special numbers linked to a square matrix. They help us learn more about what the matrix can do. When a matrix interacts with a vector, the eigenvalue shows how much that vector is stretched or squished along a certain direction.

An eigenvector ($\mathbf{v}$) is a non-zero vector that changes only by a constant factor (the eigenvalue) when the matrix acts on it.

The connection between a matrix $A$, its eigenvalues, and eigenvectors is shown in this equation:

$A \mathbf{v} = \lambda \mathbf{v}.$

This means that applying the matrix $A$ to the vector $\mathbf{v}$ scales it by the factor $\lambda$.

To find the eigenvalues of a square matrix, we follow these steps:

1. Set up the matrix: We start with a square matrix $A$. The identity matrix $I$ is the same size as $A$ and has 1s in a diagonal line and 0s everywhere else.

2. Create the characteristic equation: The characteristic polynomial is found using:

   $\det(A - \lambda I) = 0.$

   This means we take the determinant of $A - \lambda I$ and set it equal to zero to find a polynomial equation in terms of $\lambda$.

3. Solve for eigenvalues: We solve this polynomial equation to find the eigenvalues. The number of solutions matches the size of matrix $A$. For a $2 \times 2$ matrix, we get a quadratic equation; for a $3 \times 3$, we have a cubic equation, and so on.

4. Look for repeated eigenvalues: If the polynomial has repeated answers, that tells us that there are multiple eigenvalues with the same value. We call this algebraic multiplicity.

After finding the eigenvalues, the next important step is to discover the eigenvectors. Here’s how we do it:

1. Use eigenvalues in calculations: For each eigenvalue $\lambda$, we put it back into the equation $A - \lambda I$. This creates a new matrix.

2. Solve the system of equations: We then solve this equation:

   $(A - \lambda I)\mathbf{v} = 0$

   This is a system of equations, and we want to find solutions where the vector $\mathbf{v}$ is not zero.

3. Use row reduction: We can apply row reduction techniques (like Gaussian elimination) to the matrix $(A - \lambda I)$. This will help us get a system of equations in a different form.

4. Find the eigenvectors: The solutions we find will create a basis for what's called the eigenspace related to the eigenvalue $\lambda$. Each unique solution vector we discover contributes to this space, and there can be multiple independent eigenvectors for the same eigenvalue.

5. General eigenvector form: Any eigenvector can be multiplied by a non-zero number, leading to many valid answers. We often express these vectors in a simple form, especially when we need normalized vectors for applications.

Now, let’s talk about what eigenspaces mean in real life. An eigenvector points in a specific direction, and the eigenvalue tells us how much that direction stretches or shrinks. This is super useful in many fields, like stability studies, differential equations, and even statistics.

If a matrix $A$ has $n$ independent eigenvectors, it can be simplified or diagonalized like this:

$A = PDP^{-1},$

where $D$ is a diagonal matrix with the eigenvalues, and $P$ contains the eigenvectors. Diagonalization makes it much easier to work with the matrix later, like calculating its powers or analyzing systems.

In summary, finding eigenvalues and eigenvectors is really important in linear algebra. It involves understanding determinants and linear transformations and helps us in various areas like physics, engineering, and data science. Eigenvalues and eigenvectors are key to understanding complex systems and how they interact.