Eigenvalues and Eigenvectors

Topics: Algebra - Linear Transformation

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(definition)

Let $T:V\to V$ with $V$ a vector space on $\mathbb{K}$.

A non-null $x\in V$ is called an eigenvector (vector propio) of $T$ if there exists a scalar $\lambda$ such that:

$$T(x)=\lambda x$$

We call $\lambda$ the eigenvalue (valor propio) that corresponds to the eigenvector $x$.

Eigenvectors and eigenvalues can be calculated by means of other theorems.

The Set of Eigenvectors is Linearly Independent §

(theorem)

Let $T:V\to V$ and let ${\lambda }_1,\dots ,{\lambda }_k$ be distinct eigenvalues of $T$.

If $x_1,\dots ,x_k$ are eigenvectors of $T$ such that $x_j$ is the eigenvector that corresponds to ${\lambda }_j$ (given $1\le j\le k$), then:

$$\{x_1,\dots ,x_j\}$$

…is a linearly independent set.

(corollary)

If $T:V\to V$ has $n$ distinct eigenvalues (with $\dim (V)=n)$, then $T$ is diagonalisable.

Quick way to calculate $\det (A-\lambda )$

Let’s remember that, if $A$ is a superior triangular matrix:

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ 0& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& a_{nn}\end{array}\right)$$

Then:

$$\det (A-\lambda I)=\det \left(\begin{array}{cccc}{a}_{11}-\lambda & a_{12}& \dots & a_{1n}\\ 0& a_{22}-\lambda & \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& a_{nn}-\lambda \end{array}\right)$$

$$\det (A-\lambda I)=(a_{11}-\lambda )(a_{22}-\lambda )(a_{nn}-\lambda )$$

Set of Eigenvectors as a Basis §

The set that contains only the eigenvectors of a Linear Transformation $T:V\to V$ is a basis for $V$. When using this basis to build the associated matrix of $T$, the resulting matrix is diagonal.

Eigenvectors and Diagonalisation §

Concerning the diagonalisation of a linear transformation, we can build the $Q$ matrix that diagonalises $A$ (the associated matrix of the linear transformation) by using the eigenvectors as its columns.

A condition necessary for diagonalisation is the existence of eigenvalues.

For an example, refer to the uses of the eigenvalues of this example.