eigenvalue

an eigenvector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. the corresponding eigenvalue, often denoted by $\text{[math]}$ , is the factor by which the eigenvector is scaled.

let $\text{[math]}$ be a linear operator on a vector space $\text{[math]}$ . a nonzero vector $\text{[math]}$ is called an eigenvector of $\text{[math]}$ if there exists a scalar $\text{[math]}$ such that $\text{[math]}$ . the scalar $\text{[math]}$ is called the eigenvalue corresponding to the eigenvector $\text{[math]}$ .
let $\text{[math]}$ be in $\text{[math]}$ . a nonzero vector $\text{[math]}$ is called an eigenvector of $\text{[math]}$ if $\text{[math]}$ is an eigenvector of $\text{[math]}$ ; that is, if $\text{[math]}$ for some scalar $\text{[math]}$ . the scalar $\text{[math]}$ is called the eigenvalue of $\text{[math]}$ corresopnding to the eigenvector $\text{[math]}$ .
the words characteristic vector and proper vector are also used in place of eigenvector. the corresponding terms for eigenvalue are characteristic value and proper value.
[cite:;taken from @algebra_insel_2019 chapter 5.1 eigenvalues and eigenvectors]

the eigenvector of a linear operator can be defined to be the eigenvector of a transformation matrix and vice versa. this is because matrices and linear transformations can be represented using one another. so when we refer to an eigenvalue (or eigenvector), it may be in reference to the eigenvalue (or eigenvector) of a linear transformation, or that of a matrix. it should be clear from context which is correct.

let $\text{[math]}$ be an eigenvalue of a linear operator or matrix with characteristic polynomial $\text{[math]}$ . the multiplicity (sometimes called algebraic multiplicity) of $\text{[math]}$ is the largest positive integer $\text{[math]}$ for which $\text{[math]}$ is a factor of $\text{[math]}$ .
[cite:;taken from @algebra_insel_2019 chapter 5 diagonalization]

the geometric multiplicity of an eigenvalue $\text{[math]}$ is $\text{[math]}$ which equals $\text{[math]}$ .

a set of eigenvectors $\text{[math]}$ that correspond to different eigenvalues is linearly independent.

if the characteristic polynomial of $\text{[math]}$ can be decomposed into a multiplication of different linear operators then $\text{[math]}$ is diagonalizable. in other words, if $\text{[math]}$ has $\text{[math]}$ distinct eigenvalues then $\text{[math]}$ is diagonalizable.

the free coefficient in $\text{[math]}$ equals $\text{[math]}$ .

$\text{[math]}$ is invertible iff the free coefficient isnt equal to 0 iff $\text{[math]}$ iff $\text{[math]}$ for all $\text{[math]}$ .

for every eigenvalue $\text{[math]}$ , $\text{[math]}$ .

for every eigenvalue $\text{[math]}$ , $\text{[math]}$ ( $\text{[math]}$ ).

an eigenvalue $\text{[math]}$ is a matrix's special scaling factor, and its eigenvector $\text{[math]}$ is a vector that just gets stretched, not changed in direction. when you find an eigenvalue, there are two ways to measure its importance, which can be thought of as a story of "expectation vs. reality." the algebraic multiplicity is the expectation set by the algebra--it's how many times the eigenvalue shows up as a solution. the geometric multiplicity is the reality found in the geometry--it's the actual number of independent eigenvector directions you find for that eigenvalue. a matrix is only diagonalizable when, for every single eigenvalue, the reality perfectly matches the expectation.
here is the table that breaks down that intuition:

	algebraic multiplicity ( $\text{[math]}$ )	geometric multiplicity ( $\text{[math]}$ )
the source	the characteristic polynomial.	the actual eigenvectors / eigenspace.
the question it answers	"how many times does the algebra tell me this eigenvalue is a solution?"	"how many independent directions are actually associated with this eigenvalue?"
the intuition	the expectation. the algebra suggests that this eigenvalue is "important" enough to potentially contribute $\text{[math]}$ dimensions.	the reality. this is what you actually get. it's the dimension of the line, plane, etc., that is simply stretched by the matrix.

to find eigenvalues and eigenvectors of an operator $\text{[math]}$ :

choose any basis $\text{[math]}$ for $\text{[math]}$ .
find the matrix representation $\text{[math]}$ .
find eigenvalues $\text{[math]}$ and eigenvectors (coordinate vectors) $\text{[math]}$ of $\text{[math]}$ .
the eigenvalues of $\text{[math]}$ are $\text{[math]}$ .
the eigenvectors of $\text{[math]}$ are $\text{[math]}$ (obtained from $\text{[math]}$ ).