Eigenvalues and eigenvectors are important ideas in linear algebra, and determinants are key to understanding them. Let’s break it down in simpler terms.

First, what is an eigenvalue?

An eigenvalue (let’s call it $\lambda$) of a square matrix $A$ is part of a special equation:

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

Here, $\mathbf{v}$ is called an eigenvector and it is not zero. This equation shows that when you apply the transformation represented by $A$ to $\mathbf{v}$, you get another version of $\mathbf{v}$ that is stretched or shrunk by the factor $\lambda$.

Next, to find the eigenvalues, you rearrange the equation to:

$A\mathbf{v} - \lambda \mathbf{v} = 0.$

This can also be written as:

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

where $I$ is the identity matrix that is the same size as $A$.

For this to have non-zero solutions (where $\mathbf{v} \neq 0$), the determinant of the matrix $A - \lambda I$ must be zero. This leads us to something called the characteristic polynomial:

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

Finding the solutions to this polynomial gives us the eigenvalues $\lambda$ of the matrix $A$.

So, the determinant connects linear transformations and their special aspects. It not only helps us figure out if eigenvalues exist, but it also gives us useful information about the matrix itself. For example, if a matrix has a non-zero determinant, it can be inverted, meaning it has no zero eigenvalues. On the other hand, a determinant of zero means that $A$ has at least one eigenvalue that equals zero.

To dig deeper, we can look at some important properties of determinants. One way to compute the determinant is through something called the Laplace expansion. This method helps when dealing with smaller parts of matrices that relate to eigenvalues and eigenvectors.

When we talk about matrix decomposition—like in Singular Value Decomposition (SVD) or Schur decomposition—determinants help us understand the structure of the original matrix. In SVD, we express $A$ as $A = U\Sigma V^*$, which relates the eigenvalues of $A^*A$ to important values called singular values found in $\Sigma$. These connections are key to identifying eigenvalues and understanding the geometric meaning behind the transformations represented by the matrices.

We can also look at how determinants behave when we perform operations with matrices. For two square matrices $A$ and $B$, the property is:

$\det(AB) = \det(A) \cdot \det(B).$

This means that if $A$ and $B$ are square matrices, the eigenvalues of the product $AB$ are linked to the eigenvalues of $A$