Articles of eigenvalues eigenvectors

Solving a system of linear equations to find two eigenvectors.

So I have \begin{align*} x – 2y + z & = 0 \\ -2x + 4y – 2z & = 0 \\ x – 2y + z & = 0 \end{align*} I know I need to find two eigenvectors for the eigenspace with eigenvalue 2 as I know the matrix is diagonalizable and I’ve already […]

If $\lambda$ is an eigenvalue of a nonnegative symmetric matrix with zero diagonal, is $-\lambda$ also an eigenvalue?

If $A$ is an $n \times n$ with $n\geq 2$, non-negative (i.e. no negative elements), symmetric matrix with zero diagonal, is it necessarily the case that the set of all eigenvalues of $A$ can be written $L=\{\pm\lambda_0,\pm\lambda_1,\cdots\}$?

Do endomorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an endomorphism. If $V$ is finite-dimensional, we know that the characteristic polynomial $\chi_f$ has a zero $\lambda$, which is an eigenvalue of $f$. However, does a similar argument hold if $\dim V=\infty$?

3 nonzero distinct eigenvalues, part 2

This is an attempt to generalize the answer to a previous question Consider the $n \times n$ matrix $$A=\left[ \begin{array}{cccc} 0 & \frac{1}{n-1} & … & \frac{1}{n-1} \\ 1 & 0 & & 0 \\ \vdots & & \ddots & \\ 1 & 0 & & 0% \end{array}% \right] $$ (A has $n-1$ entries equal […]

Finding eigenvalues and basis for linear transformation $T: P_{100} \to P_{100}$

Consider the linear transformation $T: P_{100} \to P_{100}$ given by $ T(p(t)) = p(t) + p(2-t) $ Find all eigenvalues and a basis for each eigenspace of T. So a standard basis for the $P_{100}$ is {$1,t,t^2,t^3,…,t^{100}$} $T(1) = 1 + 1 = 2$ $T(t) = t + (2-t) = 2$ $T(t^2) = t^2 + […]

Finding the eigenvectors (& describing the eigenspace) of a Householder transformation matrix

If one is asked to find the eigenvector(s) for a Householder transformation matrix, but one is not given the values of or dimensions of the unit vector $u$. So if $H = I_n – 2uu^T$ where $I_n$ is the n x n identity matrix and has length/norm $||u||^2 = 1$. It can easily be shown […]

Eigenvalues of $A^{T}A$

Let $\lambda_{i}(M)$ denote the $i$th eigenvalue of the square matrix $M$, and $T$ denote the matrix transpose. Is it true that $|\lambda_{i}(A^{T}A)|=|\lambda_{i}(A)|^{2}$ for every square matrix $A$? Thank you very much!

Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution. $$u”=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi”+\lambda\phi=0$$ to get the eigenfunction is $$\phi=A\cos x\sqrt{\lambda}+B\cos x\sqrt{\lambda}$$ but how should I decide A and B? Is it by system$$\phi'(0)=\alpha, \phi'(1)=\beta$$ or it should be $$\phi'(0)=0, \phi'(1)=0$$ and why? After getting eigenvalue and eigenfunctions, what should I […]

Finding the eigenvalues and a basis for the eigenspaces of a $3\times3$ matrix.

For the matrix $A \in M_{3\times3}(\mathbb{R})$ below, I need to find the eigenvalues and a basis for the corresponding eigenspaces: $$\begin{bmatrix}\ 1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4 \\ \end{bmatrix}$$ I have tried to use the formula $\det(I\lambda – A) = 0$ but I ended […]

If $A$ is a real symmetric matrix, then $A$ has real eigenvalues.

I am looking at a the following proof: If $A$ is a real symmetric matrix, then $A$ has real eigenvalues. Suppose that $\lambda$ is an eigenvalue of $A$ and $x$ is a corresponding eigenvector, where we allow for the possibility that $\lambda$ is complex and $x \in \mathbb{C}^n$. Thus, $$Ax=\lambda x$$ where $x \ne 0$. […]