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 of a Matrix
Answers to Questions (FAQ)
What is an eigen value of a matrix? (Definition)
Eigenvalues are numbers that characterize a matrix. These numbers are important because, associated with their eigenvectors, they make it possible to express the matrix in a simplified form, which facilitates the calculations.
for any square matrix $ M $ of size $ m \times m $ (2x2, 3x3, 4x4, etc.), eigenvalues are generally called lambda $ \lambda $ and associated with an eigenvector $ v $ if $$ M.v = \lambda v \iff (M-\lambda I_m).v = 0 $$ with $ I_m $ the identity matrix (of size $ m $).
Practically, the eigenvalues of $ M $ are the roots of its characteristic polynomial $ P $.
An eigenvalue of a matrix is always associated with an eigenvector. Use the eigenvectors calculator proposed by dCode.
How to calculate eigen values of a matrix?
To find the eigenvalues of a matrix, calculate the roots of its characteristic polynomial.
Example: The 2x2 matrix $ 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 \mbox{ 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 $
Why eigenvalues are sometimes complex numbers?
If the roots of the characteristic polynomial do not have values on $ \mathbb{R} $ then they are calculated on $ \mathbb{C} $ which introduces complex eigenvalues.
This case can occur even if the values of the matrix are all real numbers.
Why eigen- in eigenvalues?
Eigenvalues are called eigen because it is as German word which means proper, characteristic.
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