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How can I prove that if I have $n$ eigenvectors from different eigenvalues, they are all linearly independent?

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up vote 29 down vote accepted

I'll do it with two vectors. I'll leave it to you do it in general.

Suppose $\mathbf{v}_1$ and $\mathbf{v}_2$ correspond to distinct eigenvalues $\lambda_1$ and $\lambda_2$, respectively.

Take a linear combination that is equal to $0$, $\alpha_1\mathbf{v}_1+\alpha_2\mathbf{v}_2 = \mathbf{0}$. We need to show that $\alpha_1=\alpha_2=0$.

Applying $T$ to both sides, we get $$\mathbf{0} = T(\mathbf{0}) = T(\alpha_1\mathbf{v}_1+\alpha_2\mathbf{v}_2) = \alpha_1\lambda_1\mathbf{v}_1 + \alpha_2\lambda_2\mathbf{v}_2.$$ Now, instead, multiply the original equation by $\lambda_1$: $$\mathbf{0} = \lambda_1\alpha_1\mathbf{v}_1 + \lambda_1\alpha_2\mathbf{v}_2.$$ Now take the two equations, $$\begin{align*} \mathbf{0} &= \alpha_1\lambda_1\mathbf{v}_1 + \alpha_2\lambda_2\mathbf{v}_2\\ \mathbf{0} &= \alpha_1\lambda_1\mathbf{v}_1 + \alpha_2\lambda_1\mathbf{v}_2 \end{align*}$$ and taking the difference, we get: $$\mathbf{0} = 0\mathbf{v}_1 + \alpha_2(\lambda_2-\lambda_1)\mathbf{v}_2 = \alpha_2(\lambda_2-\lambda_1)\mathbf{v}_2.$$

Since $\lambda_2-\lambda_1\neq 0$, and since $\mathbf{v}_2\neq\mathbf{0}$ (because $\mathbf{v}_2$ is an eigenvector), then $\alpha_2=0$. Using this on the original linear combination $\mathbf{0} = \alpha_1\mathbf{v}_1 + \alpha_2\mathbf{v}_2$, we conclude that $\alpha_1=0$ as well (since $\mathbf{v}_1\neq\mathbf{0}$).

So $\mathbf{v}_1$ and $\mathbf{v}_2$ are linearly independent.

Now try using induction on $n$ for the general case.

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3  
I believe that you wrote $\lambda_2$ instead of $\lambda_1$ in the row before "Now take" – jacob Mar 14 '14 at 8:08

Alternative:

Let $j$ be the maximal $j$ such that $v_1,\dots,v_j$ are independent. Then there exists $c_i$, $1\leq i\leq j$ so that $\sum_{i=1}^j c_iv_i=v_{j+1}$. But by applying $T$ we also have that

$$\sum_{i=1}^j c_i\lambda_iv_i=\lambda_{j+1}v_{j+1}.$$ Hence $$\sum_{i=1}^j \left(\lambda_i-\lambda_{j+1}\right) c_iv_i=0$$ which is a contradiction since $\lambda_i\neq \lambda_{j+1}$ for $1\leq i\leq j$.

Hope that helps,

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P.S. The argument uses the well-ordering principle on the naturals (by looking at the least $j$ such that $v_1,\ldots,v_{j+1}$ is dependent). Well-ordering for the naturals is equivalent to induction. – Arturo Magidin Mar 28 '11 at 1:20
    
@Arturo: Thanks! Should be fixed now. (Didn't realize I was using well ordering in that way) – Eric Naslund Mar 28 '11 at 1:30
1  
No problem; I'll delete the other two comments since it's been fixed. In any case, many people prefer arguments along these lines to an explicit induction, even if they are logically equivalent (you can think of the argument you give as a proof by contradiction of the inductive step, with the base being taken for granted [or as trivial, since $v_1$ is nonzero]; cast that way, it may be clearer why the two arguments are closely connected). – Arturo Magidin Mar 28 '11 at 1:34
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@Eric Naslund This proofs seems very similar to the one given in Axler... – user38268 Aug 31 '11 at 7:29
    
@D B Lim: What is Axler? – Eric Naslund Aug 31 '11 at 14:42

Hey I think there's a slick way to do this without induction. Suppose that $T$ is a linear transformation of a vector space $V$ and that $v_1,\ldots,v_n \in V$ are eigenvectors of $T$ with corresponding eigenvalues $\lambda_1,\ldots,\lambda_n \in F$ ($F$ the field of scalars). We want to show that, if $\sum_{i=1}^n c_i v_i = 0$, where the coefficients $c_i$ are in $F$, then necessarily each $c_i$ is zero.

For simplicity, I will just explain why $c_1 = 0$. Consider the polynomial $p_1(x) \in F[x]$ given as $p_1(x) = (x-\lambda_2) \cdots (x-\lambda_n)$. Note that the $x-\lambda_1$ term is "missing" here. Now, since each $v_i$ is an eigenvector of $T$, we have \begin{align*} p_1(T) v_i = p_1(\lambda_i) v_i && \text{ where} && p_1(\lambda_i) = \begin{cases} 0 & \text{ if } i \neq 1 \\ p_1(\lambda_1) \neq 0 & \text{ if } i = 1 \end{cases}. \end{align*}

Thus, applying $p_1(T)$ to the sum $\sum_{i=1}^n c_i v_i = 0$, we get $$ p_1(\lambda_1) c_1 v_1 = 0 $$ which implies $c_1 = 0$, since $p_1(\lambda_1) \neq 0$ and $v_1 \neq 0$.

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For eigenvectors $\vec{v^1},\vec{v^2},\dots,\vec{v^n}$ with different eigenvalues $\lambda_1\neq\lambda_2\neq \dots \neq\lambda_n$ of a $ n\times n$ matrix $A$.

Given the $ n\times n$ matrix $P$ of the eigenvectors (with eigenvectors as the columns). $$P=\Big[\vec{v^1},\vec{v^2},\dots,\vec{v^n}\Big]$$

Given the $ n\times n$ matrix $\Lambda$ of the eigenvalues on the diagonal (zeros elsewhere): $$\Lambda = \begin{bmatrix} \lambda_1 & 0 & \dots & 0 \\ 0 & \lambda_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda_n \end{bmatrix} $$ Let $\vec{c}=(c_1,c_2,\dots,c_n)^T$

We need to show that only $c_1=c_2=...=c_n=0$ can satisfy the following: $$c_1\vec{v^1}+c_2\vec{v^2}+...= \vec{0^{}}$$ Applying the matrix to this equation gives: $$c_1\lambda_1\vec{v^1}+c_2\lambda_2\vec{v^2}+...+c_n\lambda_n\vec{v^n}= \vec{0^{}}$$ We can write this equation in the form of vectors and matrices:

$$P\Lambda \vec{c^{}}=\vec{0^{}}$$

But with since $A$ can be diagonalised to $\Lambda$, we know $P\Lambda=AP$ $$\implies AP\vec{c^{}}=\vec{0^{}}$$ since $AP\neq 0$, we have $\vec{c}=0$.

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