Let A be an n×n matrix and let X∈ℂn be a nonzero vector for which
AX=λX
for some scalar λ. Then λ is called an eigenvalue of the matrix A and X is called an eigenvector of A associated with λ, or a λ-eigenvector of A.
The set of all eigenvalues of an n×n matrix A is denoted by σA and is referred to as the spectrum of A.
Let A be an n×n matrix and suppose det λI − A = 0 for some λ∈ℂ. Then λ is an eigenvalue of A and thus there exists a nonzero vector X∈ℂn such that AX=λX.
The expression detλI−A is a polynomial (in the variable x) called the characteristic polynomial of A, and det λI − A = 0 is called the characteristic equation. For this reason, we may also refer to the eigenvalues of A as characteristic values.
If the characteristic equation of a matrix A be λ2−λ−1, then
A−1 exist but cannot be determined from the data.
A−1=A−1
A−1does not exist.
A−1=A+1
Let A be an n×n matrix with characteristic polynomial given by det λI − A . Then, the multiplicity of an eigenvalue λ of A is the number of times λ occurs as a root of that characteristic polynomial.
For example, suppose the characteristic polynomial of A is given by λ−22. Solving for the roots of this polynomial, we set λ−22=0 and solve for λ. We find that λ=2 is a root that occurs twice. Hence, in this case, λ=2 is an eigenvalue of A of multiplicity equal to 2.
Let A be an n×n matrix.
To verify your work, make sure that AX=λX for each λ and associated eigenvector X.
Find the eigenvalues and eigenvectors of the matrix 4007.
The field below accepts a list of numbers or formulas separated by commas. For example, 2, 4, x+1, x−1.The order of the list does not matter.
Eigenvalues =
Eigenvector(w.r.t smaller eigen value) =
Eigenvector(w.r.t larger eigen value) =
Find the multiplicity of the largest eigenvalue for the matrix 612036002.