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Eigenvalues of a Matrix

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

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Eigenvalues of a Matrix -

Tag(s) : Matrix

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# Eigenvalues of a Matrix

## Eigenvectors Calculator

### 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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