How to Find Eigenvalues and Eigenvectors
The matrix equation Ax=b{\displaystyle A\mathbf {x} =\mathbf {b} } involves a matrix acting on a vector to produce another vector. In general, the way A{\displaystyle A} acts on x{\displaystyle \mathbf {x} } is complicated, but there are certain cases where the action maps to the same vector, multiplied by a scalar factor.
Eigenvalues and eigenvectors have immense applications in the physical sciences, especially quantum mechanics, among other fields.
Steps

Understand determinants. The determinant of a matrix detA=0{\displaystyle \det A=0} when A{\displaystyle A} is noninvertible. When this occurs, the null space of A{\displaystyle A} becomes nontrivial  in other words, there are nonzero vectors that satisfy the homogeneous equation Ax=0.{\displaystyle A\mathbf {x} =0.}

Write out the eigenvalue equation. As mentioned in the introduction, the action of A{\displaystyle A} on x{\displaystyle \mathbf {x} } is simple, and the result only differs by a multiplicative constant λ,{\displaystyle \lambda ,} called the eigenvalue. Vectors that are associated with that eigenvalue are called eigenvectors.
 Ax=λx{\displaystyle A\mathbf {x} =\lambda \mathbf {x} }
 We can set the equation to zero, and obtain the homogeneous equation. Below, I{\displaystyle I} is the identity matrix.
 (A−λI)x=0{\displaystyle (A\lambda I)\mathbf {x} =0}

Set up the characteristic equation. In order for (A−λI)x=0{\displaystyle (A\lambda I)\mathbf {x} =0} to have nontrivial solutions, the null space of A−λI{\displaystyle A\lambda I} must be nontrivial as well.
 The only way this can happen is if det(A−λI)=0.{\displaystyle \det(A\lambda I)=0.} This is the characteristic equation.

Obtain the characteristic polynomial. det(A−λI){\displaystyle \det(A\lambda I)} yields a polynomial of degree n{\displaystyle n} for n×n{\displaystyle n\times n} matrices.
 Consider the matrix A=(1432).{\displaystyle A={\begin{pmatrix}1&4\\3&2\end{pmatrix}}.}
 1−λ432−λ=0(1−λ)(2−λ)−12=0{\displaystyle {\begin{aligned}{\begin{vmatrix}1\lambda &4\\3&2\lambda \end{vmatrix}}&=0\\(1\lambda )(2\lambda )12&=0\end{aligned}}}
 Notice that the polynomial seems backwards  the quantities in parentheses should be variable minus number, rather than the other way around. This is easy to deal with by moving the 12 to the right and multiplying by (−1)2{\displaystyle (1)^{2}} to both sides to reverse the order.
 (λ−1)(λ−2)=12λ2−3λ−10=0{\displaystyle {\begin{aligned}(\lambda 1)(\lambda 2)&=12\\\lambda ^{2}3\lambda 10&=0\end{aligned}}}

Solve the characteristic polynomial for the eigenvalues. This is, in general, a difficult step for finding eigenvalues, as there exists no general solution for quintic functions or higher polynomials. However, we are dealing with a matrix of dimension 2, so the quadratic is easily solved.
 (λ−5)(λ+2)=0λ=5,−2{\displaystyle {\begin{aligned}&(\lambda 5)(\lambda +2)=0\\&\lambda =5,2\end{aligned}}}

Substitute the eigenvalues into the eigenvalue equation, one by one. Let's substitute λ1=5{\displaystyle \lambda _{1}=5} first.
 (A−5I)x=(−443−3){\displaystyle (A5I)\mathbf {x} ={\begin{pmatrix}4&4\\3&3\end{pmatrix}}}
 The resulting matrix is obviously linearly dependent. We are on the right track here.

Rowreduce the resulting matrix. With larger matrices, it may not be so obvious that the matrix is linearly dependent, and so we must rowreduce. Here, however, we can immediately perform the row operation R2→4R2+3R1{\displaystyle R_{2}\to 4R_{2}+3R_{1}} to obtain a row of 0's.
 (−4400){\displaystyle {\begin{pmatrix}4&4\\0&0\end{pmatrix}}}
 The matrix above says that −4x1+4x2=0.{\displaystyle 4x_{1}+4x_{2}=0.} Simplify and reparameterize x2=t,{\displaystyle x_{2}=t,} as it is a free variable.

Obtain the basis for the eigenspace. The previous step has led us to the basis of the null space of A−5I{\displaystyle A5I}  in other words, the eigenspace of A{\displaystyle A} with eigenvalue 5.
 x1=(11){\displaystyle \mathbf {x_{1}} ={\begin{pmatrix}1\\1\end{pmatrix}}}
 Performing steps 6 to 8 with λ2=−2{\displaystyle \lambda _{2}=2} results in the following eigenvector associated with eigenvalue 2.
 x2=(−43){\displaystyle \mathbf {x_{2}} ={\begin{pmatrix}4\\3\end{pmatrix}}}
 These are the eigenvectors associated with their respective eigenvalues. For the basis of the entire eigenspace of A,{\displaystyle A,} we write
 {(11),(−43)}.{\displaystyle \left\{{\begin{pmatrix}1\\1\end{pmatrix}},{\begin{pmatrix}4\\3\end{pmatrix}}\right\}.}
Tips
 The determinant of a triangular matrix is easy to find  it is simply the product of the diagonal elements. The eigenvalues are immediately found, and finding eigenvectors for these matrices then becomes much easier.
 Beware, however, that rowreducing to rowechelon form and obtaining a triangular matrix does not give you the eigenvalues, as rowreduction changes the eigenvalues of the matrix in general.
 We can diagonalize a matrix A{\displaystyle A} through a similarity transformation A=PDP−1,{\displaystyle A=PDP^{1},} where P{\displaystyle P} is an invertible changeofbasis matrix and D{\displaystyle D} is a matrix with only diagonal elements. However, if A{\displaystyle A} is an n×n{\displaystyle n\times n} matrix, it must have n{\displaystyle n} distinct eigenvalues in order for it to be diagonalizable.
 In our case, A=(1−413)(500−2)(1−413)−1.{\displaystyle A={\begin{pmatrix}1&4\\1&3\end{pmatrix}}{\begin{pmatrix}5&0\\0&2\end{pmatrix}}{\begin{pmatrix}1&4\\1&3\end{pmatrix}}^{1}.}
 There are a few things of note here. First, the diagonal elements of D{\displaystyle D} are the eigenvalues that we found. Second, the columns of P{\displaystyle P} are the eigenspace of A.{\displaystyle A.} Third, D{\displaystyle D} is similar to A{\displaystyle A} in the sense that they have the same determinant, eigenvalues, and trace.
 When diagonalizing, the eigenbases in P{\displaystyle P} that correspond to their eigenvalues must line up  in other words, you must be consistent with the ordering. In the example above, you cannot switch the columns of P{\displaystyle P} without switching the positions of the diagonal elements in D.{\displaystyle D.}
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