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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) : Mathematics, Algebra, Symbolic Computation

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# Eigenvalues of a Matrix

## Eigenvalues Calculator

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

### How to calculate eigen values of a matrix?

Consider $$M$$ a square matrix of size $$m \times m$$, the eigenvalues of $$M$$ are the roots of the characteristic polynomial $$P$$ of the matrix $$M$$.

Eigenvalues are generally called $$\lambda$$ 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 $$m$$).

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

Example: $$M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \Rightarrow P(M) = x^2 − 4x − 5 = (x+1)(x-5)$$

Example: $$P(M)=0 \iff x= -1 \mbox{ or } x = 5$$ The eigenvalues of the matrix $$M$$ are $$-1$$ and $$5$$. And the eigenvector associated are $$\begin{bmatrix} 1 \\ 2 \end{bmatrix}$$ for $$5$$ and $$\begin{bmatrix} -1 \\ 1 \end{bmatrix}$$ for $$-1$$.

### Is it possible to have a zero vector?

Normally the definition of the eigenvector exclude the zero vector. However, if there are not as many independent eigenvectors as eigenvalues, dCode will display a null vector.