Tool to calculate eigenvalues of a matrix. The eigenvalues of a matrix are values that allow to reduce the associated endomorphisms.

Eigenvalues of a Matrix - dCode

Tag(s) : Matrix

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Eigenvalues for any square matrix $ M $ of size $ m \times m $ (2x2, 3x3, 4x4, etc.), are the scalar values noted with character lambda $ \lambda $ that are associated with an eigenvector $ \vec{v} $ such as $$ M.\vec{v} = \lambda \vec{v} $$

Practically, the eigenvalues $ \lambda $ of a matrix $ M $ are the roots of its characteristic polynomial $ P $ as $ (M-\lambda I_m).\vec{v} = 0 $$ (with $ I_m $ the identity matrix of size $ m $).

An eigenvalue of a matrix is always associated with an eigenvector. Use the eigenvectors calculator proposed by dCode.

To determine/find the eigenvalues of a matrix, calculate the roots of its characteristic polynomial.

__Example:__ The 2x2 matrix (or order 2) $ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} $ has for characteristic polynomial $ P(M) = x^2 − 4x − 5 = (x+1)(x-5) $. The roots of $ P $ are found by the calculation $ P(M)=0 \iff x= -1 $ or $ x = 5 $. The eigenvalues of the matrix $ M $ are $ -1 $ and $ 5 $.

NB: The eigenvectors associated are $ \begin{bmatrix} 1 \\ 2 \end{bmatrix} $ for $ 5 $ and $ \begin{bmatrix} -1 \\ 1 \end{bmatrix} $ for $ -1 $

A square matrix of size/dimension $ n $ has $ n $ eigen values. Be careful however, certain eigen values can be identical, to know the number of distinct eigenvalues (without multiplicity) then calculate the distinct roots of the characteristic polynomial of the matrix.

Eigenvalues are numbers that characterize a matrix. These numbers are important because, associated with their eigenvectors, they make it possible to determine the eigendirections of the matrix and to express it in a basis in a simplified form (see matrix diagonalization), which facilitates calculations.

To determine that a value `λ` is an eigenvalue of a matrix $ M $, show that there exists a nonzero vector $ \vec{x} $ such that $ M . \vec{x} = \lambda . \vec{x} $. If this equation has a solution for $ \vec{x} $, then $ \lambda $ is an eigenvalue of the matrix $ M $.

If the roots of the characteristic polynomial do not have values on the real set $ \mathbb{R} $ then they are calculated on the complex set $ \mathbb{C} $ which introduces complex eigenvalues.

This case can occur even if the values of the matrix are all real numbers.

Complex eigenvalues have no physical meaning and are usually excluded from mechanics/physics etc calculations.

Eigenvalues are called *eigen* because it is a German word which means *proper*, *characteristic*.

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*Eigenvalues of a Matrix* on dCode.fr [online website], retrieved on 2024-06-15,

- Eigenvalues Calculator
- Eigenvectors Calculator
- What is an eigen value of a matrix? (Definition)
- How to calculate eigen values of a matrix?
- How many eigenvalues does a matrix have?
- Why calculate eigen values of a matrix?
- How to prove that a value is an eigenvalue of a matrix?
- Why are eigenvalues sometimes complex numbers?
- Why eigen- in eigenvalues?

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