Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 22/refcontrol

Relations between eigenspaces

We have seen in fact that the eigenspace to ${}0$ is the kernel of the endomorphism. More general, the following characterization holds.

LemmaCreate referencenumber

Let ${}K$ be a field,MDLD/field ${}V$ a ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Let ${}\lambda \in K$. Then

${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {ker} {\left(\lambda \cdot \operatorname {Id} _{V}-\varphi \right)}\,.$

Proof

Let ${}v\in V$. Then ${}v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}$ if and only if ${}\varphi (v)=\lambda v$, and this is the case if and only if ${}\lambda v-\varphi (v)=0$ holds, which means ${}{\left(\lambda \cdot \operatorname {Id} _{V}-\varphi \right)}(v)=0$.

$\Box$

In particular, ${}\lambda \in K$ is an eigenvalue of ${}\varphi$ if and only if ${}\lambda \operatorname {Id} _{V}-\varphi$ is not injective. For a given ${}\lambda$, this property can be checked with the help of a linear system (or the determinant), and the eigenspace can be determined. However, it is not a linear problem to decide whether ${}\varphi$ has eigenvalues at all and how those can be determined. We will continue to study a linear mapping ${}\varphi \colon V\rightarrow V$ by considering the differences to homotheties ${}\lambda \operatorname {Id} _{V}-\varphi$ for various ${}\lambda$.

For an ${}n\times n$-matrix ${}M$, we have to determine the kernel of the matrix ${}\lambda E_{n}-M$. If, for example, we want t know whether the matrix ${}{\begin{pmatrix}4&7\\-2&5\end{pmatrix}}$ has the eigenvalue ${}3$, then

${}3E_{2}-{\begin{pmatrix}4&7\\-2&5\end{pmatrix}}=3{\begin{pmatrix}1&0\\0&1\end{pmatrix}}-{\begin{pmatrix}4&7\\-2&5\end{pmatrix}}={\begin{pmatrix}-1&-7\\2&-2\end{pmatrix}}\,$

shows that this is not the case.

LemmaCreate referencenumber

Let ${}K$ be a field,MDLD/field ${}V$ a ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Let ${}\lambda _{1}\neq \lambda _{2}$ be elements in ${}K$. Then

${}\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\cap \operatorname {Eig} _{\lambda _{2}}{\left(\varphi \right)}=0\,.$

Proof

Let ${}v\in \operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\cap \operatorname {Eig} _{\lambda _{2}}{\left(\varphi \right)}$. Then

${}\lambda _{1}v=\varphi (v)=\lambda _{2}v\,.$

Therefore,

${}(\lambda _{1}-\lambda _{2})v=0\,,$

and this implies, because of ${}\lambda _{1}\neq \lambda _{2}$, ${}v=0$.

$\Box$

LemmaCreate referencenumber

Let ${}K$ be a field,MDLD/field ${}V$ a ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Let ${}v_{1},\ldots ,v_{n}$ be eigenvectorsMDLD/eigenvectors for (pairwise) different eigenvaluesMDLD/eigenvalues ${}\lambda _{1},\ldots ,\lambda _{n}\in K$. Then ${}v_{1},\ldots ,v_{n}$ are

linearly independent.MDLD/linearly independent

Proof

We prove the statement by induction on ${}n$. For ${}n=0$, the statement is true. Suppose now that the statement is true for less than ${}n$ vectors. We consider a representation of ${}0$, say

${}a_{1}v_{1}+\cdots +a_{n}v_{n}=0\,.$

We apply ${}\varphi$ to this and get, on one hand,

${}a_{1}\varphi (v_{1})+\cdots +a_{n}\varphi (v_{n})=\lambda _{1}a_{1}v_{1}+\cdots +\lambda _{n}a_{n}v_{n}=0\,.$

On the other hand, we multiply the equation with ${}\lambda _{n}$ and get

${}\lambda _{n}a_{1}v_{1}+\cdots +\lambda _{n}a_{n}v_{n}=0\,.$

We look at the difference of the two equations, and get

${}(\lambda _{n}-\lambda _{1})a_{1}v_{1}+\cdots +(\lambda _{n}-\lambda _{n-1})a_{n-1}v_{n-1}=0\,.$

By the induction hypothesis, we get for the coefficients ${}(\lambda _{n}-\lambda _{i})a_{i}=0$, ${}i=1,\ldots ,n-1$. Because of ${}\lambda _{n}-\lambda _{i}\neq 0$, we get ${}a_{i}=0$ for ${}i=1,\ldots ,n-1$, and because of ${}v_{n}\neq 0$, we also get ${}a_{n}=0$.

$\Box$

CorollaryCreate referencenumber

Let ${}K$ be a field,MDLD/field ${}V$ a finite-dimensionalMDLD/finite-dimensional (vs) ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping Then there exist at most ${}\dim _{K}{\left(V\right)}$ many eigenvaluesMDLD/eigenvalues

for ${}\varphi$.

Proof

See exercise.

$\Box$

In particular, an endomorphism on a finite-dimensional vector space has only finitely many eigenvalues.

Geometric multiplicity

The restriction of a linear mapping to an eigenspace is the homothety with the corresponding eigenvalue, thus a very simple linear mapping. For a diagonal matrix

${\begin{pmatrix}d_{1}&0&\cdots &\cdots &0\\0&d_{2}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&0\\0&\cdots &\cdots &0&d_{n}\end{pmatrix}},$

the standard basis has the property that every basis vector is an eigenvector for the linear mapping given by the matrix. In this case, it is easy to describe the eigenspaces, see example; the eigenspace for ${}d$ consists of all linear combinations of the standard vectors ${}e_{i}$, for which ${}d_{i}$ equals ${}d$. In particular, the dimension of the eigenspace equals the number how often ${}d$ occurs as an diagonal element. In general, the dimensions of the eigenspaces are important invariants of an endomorphism.

Definition

Let ${}K$ be a field,MDLD/field ${}V$ a ${}K$-vector spaceMDLD/vector space and

$\varphi \colon V\longrightarrow V$

a linear mapping.MDLD/linear mapping For ${}\lambda \in K$ we call

$\dim _{K}{\left(\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\right)}$

the geometric multiplicity

of ${}\lambda$.

In particular, a number ${}\lambda \in K$ is an eigenvalue of ${}\varphi$ if and only if its geometric multiplicity is at least ${}1$. It is easy to give examples where the geometric multiplicity of an eigenvalue is any number between ${}1$ and the dimension of the space.

LemmaCreate referencenumber

Let ${}K$ denote a field,MDLD/field and let ${}V$ denote a ${}K$-vector spaceMDLD/vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping.MDLD/linear mapping Then the sumMDLD/sum (linear subspaces) of the eigenspacesMDLD/eigenspaces ${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}$ is direct,MDLD/direct (sum) and we have

${}\sum _{\lambda \in K}\dim _{K}{\left(\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\right)}\leq \dim _{K}{\left(V\right)}\,.$

Proof

This follows directly from fact.

$\Box$

Diagonalizability

Definition

Let ${}K$ denote a field,MDLD/field let ${}V$ denote a vector space,MDLD/vector space and let

$\varphi \colon V\longrightarrow V$

denote a linear mapping.MDLD/linear mapping Then ${}\varphi$ is called diagonalizable, if ${}V$ has a basisMDLD/basis (vs) consisting of eigenvectorsMDLD/eigenvectors

for ${}\varphi$.

TheoremCreate referencenumber

Let ${}K$ denote a field,MDLD/field and let ${}V$ denote a finite-dimensionalMDLD/finite-dimensional (fgvs) vector space.MDLD/vector space Let

$\varphi \colon V\longrightarrow V$

denote a

linear mapping.MDLD/linear mapping Then the following statements are equivalent.

1. ${}\varphi$ is diagonalizable.MDLD/diagonalizable (vs)
2. There exists a basis ${}{\mathfrak {v}}$ of ${}V$ such that the describing matrixMDLD/describing matrix ${}M_{\mathfrak {v}}^{\mathfrak {v}}(\varphi )$ is a diagonal matrix.MDLD/diagonal matrix
3. For every describing matrix ${}M=M_{\mathfrak {w}}^{\mathfrak {w}}(\varphi )$ with respect to a basis ${}{\mathfrak {w}}$, there exists an invertible matrixMDLD/invertible matrix ${}B$ such that

   $BMB^{-1}$

   is a diagonal matrix.

Proof

The equivalence between (1) and (2) follows from the definition, from example, and the correspondence between linear mappings and matrices. The equivalence between (2) and (3) follows from fact.

$\Box$

If ${}\varphi$ is diagonalizable and if the eigenvalues together with their geometric multiplicities are known, then we can simply write down a corresponding diagonal matrix: just take the diagonal matrix such that, in the diagonal, the eigenvalues occur as often as the geometric multiplicities. In particular, the corresponding diagonal matrix of a diagonalizable mapping is, up to the ordering of the diagonal entries, uniquely determined.

Example Create referencenumber

We continue with example. There exists the two eigenvectorsMDLD/eigenvectors ${}{\begin{pmatrix}{\sqrt {5}}\\1\end{pmatrix}}$ and ${}{\begin{pmatrix}-{\sqrt {5}}\\1\end{pmatrix}}$ for the different eigenvaluesMDLD/eigenvalues ${}{\sqrt {5}}$ and ${}-{\sqrt {5}}$, so that the mapping is diagonalizable,MDLD/diagonalizable due to fact. With respect to the basisMDLD/basis (vs) ${}{\mathfrak {u}}$, consisting of these eigenvectors, the linear mapping is described by the diagonal matrix

${\begin{pmatrix}{\sqrt {5}}&0\\0&-{\sqrt {5}}\end{pmatrix}}.$

The transformation matrix,MDLD/transformation matrix from the basis ${}{\mathfrak {u}}$ to the standard basis ${}{\mathfrak {v}}$, consisting of ${}e_{1}$ and ${}e_{2}$, is simply

${}M_{\mathfrak {v}}^{\mathfrak {u}}={\begin{pmatrix}{\sqrt {5}}&-{\sqrt {5}}\\1&1\end{pmatrix}}\,.$

The inverse matrixMDLD/inverse matrix is

${}{\frac {1}{2{\sqrt {5}}}}{\begin{pmatrix}1&{\sqrt {5}}\\-1&{\sqrt {5}}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2{\sqrt {5}}}}&{\frac {1}{2}}\\{\frac {-1}{2{\sqrt {5}}}}&{\frac {1}{2}}\end{pmatrix}}\,.$

Because of fact, we have the relation

${}{\begin{aligned}{\begin{pmatrix}{\sqrt {5}}&0\\0&-{\sqrt {5}}\end{pmatrix}}&={\begin{pmatrix}{\frac {1}{2}}&{\frac {\sqrt {5}}{2}}\\{\frac {1}{2}}&{\frac {-{\sqrt {5}}}{2}}\end{pmatrix}}{\begin{pmatrix}{\sqrt {5}}&-{\sqrt {5}}\\1&1\end{pmatrix}}\\&={\begin{pmatrix}{\frac {1}{2{\sqrt {5}}}}&{\frac {1}{2}}\\{\frac {-1}{2{\sqrt {5}}}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&5\\1&0\end{pmatrix}}{\begin{pmatrix}{\sqrt {5}}&-{\sqrt {5}}\\1&1\end{pmatrix}}.\end{aligned}}$

CorollaryCreate referencenumber

Let ${}K$ denote a field,MDLD/field and let ${}V$ denote a finite-dimensionalMDLD/finite-dimensional (fgvs) vector space.MDLD/vector space Let

$\varphi \colon V\longrightarrow V$

denote a linear mapping.MDLD/linear mapping Suppose that there exists ${}n$ different eigenvalues.MDLD/eigenvalues Then ${}\varphi$ is

diagonalizable.MDLD/diagonalizable

Proof

Because of fact, there exist ${}n$ linearly independentMDLD/linearly independent eigenvectors.MDLD/eigenvectors These form, due to fact, a basis.MDLD/basis (vs)

$\Box$

LemmaCreate referencenumber

Let ${}K$ denote a field,MDLD/field and let ${}V$ denote a ${}K$-vector spaceMDLD/vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping.MDLD/linear mapping Then ${}\varphi$ is diagonalizableMDLD/diagonalizable (ev) if and only if ${}V$ is the direct sumMDLD/direct sum (vs) of the

eigenspaces.MDLD/eigenspaces

Proof

If ${}\varphi$ is diagonalizable, then there exists a basisMDLD/basis (vs) ${}v_{1},\ldots ,v_{n}$ of ${}V$ consisting of eigenvectors.MDLD/eigenvectors Hence,

${}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\langle v_{i},\,{\text{the eigenvalue of }}v_{i}{\text{ is }}\lambda \rangle \,.$

Therefore,

${}V=\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{k}}{\left(\varphi \right)}\,.$

That the sum is direct follows from fact. If, the other way round,

${}V=\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{k}}{\left(\varphi \right)}\,$

holds, then we can choose in every eigenspace a basis. These bases consist of eigenvectors and yield together a basis of ${}V$.

$\Box$

Example Create referencenumber

We consider ${}2\times 2$-shearing matrices

${\begin{pmatrix}1&a\\0&1\end{pmatrix}}$

with ${}a\in K$. The condition for some ${}\lambda \in K$ to be an eigenvalueMDLD/eigenvalue means

${}{\begin{pmatrix}1&a\\0&1\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}=\lambda {\begin{pmatrix}x\\y\end{pmatrix}}\,.$

This yields the equations

$x+ay=\lambda x{\text{ and }}y=\lambda y.$

For ${}\lambda \neq 1$, we get ${}y=0$ and hence also ${}x=0$, that is, only ${}1$ can be an eigenvalue. In this case, the second equation is fulfilled, and the first equation becomes

$x+ay=x{\text{ and }}ay=0.$

For ${}a\neq 0$, we get ${}y=0$ and thus ${}{\begin{pmatrix}x\\0\end{pmatrix}}$ is the eigenspace for the eigenvalue ${}1$, and ${}{\begin{pmatrix}1\\0\end{pmatrix}}$ is an eigenvector which spans this eigenspace. For ${}a=0$, we have the identity matrix, and the eigenspaceMDLD/eigenspace for the eigenvalue ${}1$ is the total plane. For ${}a\neq 0$, there is a one-dimensional eigenspace, and the mapping is not diagonalizable.MDLD/diagonalizable

The product of two diagonal matrices is again a diagonal matrix. The following example shows that the product of two diagonalizable matrices is not necessarily diagonalizable.

Example Create referencenumber

Let ${}G_{1}$ and ${}G_{2}$ denote two lines in ${}\mathbb {R} ^{2}$ through the origin, and let ${}\varphi _{1}$ and ${}\varphi _{2}$ denote the reflections at these axes. A reflection at an axis is always diagonalizable,MDLD/diagonalizable the axis and the line orthogonal to the axis are eigenlines (with eigenvalues ${}1$ and ${}-1$). The compositionMDLD/composition

${}\psi =\varphi _{2}\circ \varphi _{1}\,$

of the reflections is a plane rotation,MDLD/plane rotation the angle of rotation being twice the angle between the two lines. However, a rotation is only diagonalizable if the angle of rotation is ${}0$ or ${}180$ degree. If the angle between the axes is different from ${}0,90$ degree, then ${}\psi$ does not have any eigenvector.

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