Eigen vector of a matrix A is a vector represented by a matrix X such that when X is multiplied with matrix A, then the direction of the resultant matrix remains same as vector X.
Mathematically, above statement can be represented as:
AX = λX
where A is any arbitrary matrix, λ are eigen values and X is an eigen vector corresponding to each eigen value.
Here, we can see that AX is parallel to X. So, X is an eigen vector.
Method to find eigen vectors and eigen values of any square matrix A
We know that,
AX = λX
=> AX – λX = 0
=> (A – λI) X = 0 …..(1)
Above condition will be true only if (A – λI) is singular. That means,
A – λI = 0 …..(2)
(2) is known as characteristic equation of the matrix.
The roots of the characteristic equation are the eigen values of the matrix A.
Now, to find the eigen vectors, we simply put each eigen value into (1) and solve it by Gaussian elimination, that is, convert the augmented matrix (A – λI) = 0 to row echelon form and solve the linear system of equations thus obtained.
Some important properties of eigen values

Eigen values of real symmetric and hermitian matrices are real

Eigen values of real skew symmetric and skew hermitian matrices are either pure imaginary or zero

Eigen values of unitary and orthogonal matrices are of unit modulus λ = 1

If λ_{1, }λ_{2}…….λ_{n} are the eigen values of A, then kλ_{1}, kλ_{2}…….kλ_{n} are eigen values of kA

If λ_{1, }λ_{2}…….λ_{n} are the eigen values of A, then 1/λ_{1}, 1/λ_{2}…….1/λ_{n} are eigen values of A^{1}

If λ_{1, }λ_{2}…….λ_{n} are the eigen values of A, then λ_{1}^{k}, λ_{2}^{k}…….λ_{n}^{k} are eigen values of A^{k}

Eigen values of A = Eigen Values of A^{T }(Transpose)

Sum of Eigen Values = Trace of A (Sum of diagonal elements of A)

Product of Eigen Values = A

Maximum number of distinct eigen values of A = Size of A

If A and B are two matrices of same order then, Eigen values of AB = Eigen values of BA
This article has been contributed by Saurabh Sharma.
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