Intereting Posts

Finite Group generated by the union of its Sylow $p_i$-subgroups
Do these arithmetic rules work? They extend the number system by a zero not based on the empty set that is a divisor with unique quotients.
The “general algorithm” for primary decomposition
Tensors of rank two in physics and mathematics
Is every linear ordered set normal in its order topology?
Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?
How to derive this interesting identity for $\log(\sin(x))$
Fibonacci proof by induction
On Dirichlet series and critical strips
Simultaneous Diophantine approximation: multiple solutions required
Solving $x\; \leq \; \sqrt{20\; -\; x}$
Representation of symmetric functions
Is $\sum \sin{\frac{\pi}{n}}$ convergent?
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
Maximal and Prime ideals of cross product of rings

Let $A$ be an $n\times n$ matrix, with $A_{ij}=i+j$. Find the eigenvalues of $A$. A student that I tutored asked me this question, and beyond working out that there are 2 nonzero eigenvalues $a+\sqrt{b}$ and $a-\sqrt{b}$ and $0$ with multiplicity $n-2$, I’m at a bit of a loss.

- Prove that $ AA^T=0\implies A = 0$
- How to deal with misapplying mathematical rules?
- Every endomorphims is a linear combination of how many idempotents in infinite dimensions?
- Convert a piecewise linear non-convex function into a linear optimisation problem.
- Linear independence of images by $A$ of vectors whose span trivially intersects $\ker(A)$
- Is the product of symmetric positive semidefinite matrices positive definite?
- Working out a concrete example of tensor product
- Deriving the inverse of $\mathbf{I}$+Idempotent matrix
- Commuting matrices and simultaneous diagonalizability
- Distance/Similarity between two matrices

Let $u=(1,1,\ldots,1)^T$ and $v=(1,2,\ldots,n)^T$. Then $A=uv^T+vu^T$. As $u$ is not parallel to $v$ and they are nonzero vectors, $A$ has rank 2 and every eigenvector for a nonzero eigenvalue $\lambda$ must lie in the span of $u$ and $v$. Let $(\lambda,\,pu+qv)$ be such an eigenpair. Then

$$

\lambda(pu+qv)=(uv^T+vu^T)(pu+qv)=u[v^T(pu+qv]+v[u^T(pu+qv)].

$$

Since $u$ and $v$ are linearly independent, by comparing coefficients on both sides, we get

$$

\lambda\pmatrix{p\\ q}=\underbrace{\pmatrix{v^Tu&v^Tv\\ u^Tu&u^Tv}}_B\pmatrix{p\\ q}.

$$

Therefore the nonzero eigenvalues of $A$ are exactly the two eigenvalues of $B$. Since the characteristic polynomial of $B$ is

$$

\lambda^2-2(u^Tv)\lambda+[(u^Tv)^2-(u^Tu)(v^Tv)]

\equiv (\lambda-u^Tv)^2-(u^Tu)(v^Tv),

$$

it follows that

$$

\color{red}{\lambda=u^Tv\pm\sqrt{(u^Tu)(v^Tv)}.}

$$

In your case, by putting $u=(1,1,\ldots,1)^T$ and $v=(1,2,\ldots,n)^T$, we get

$$

\lambda = \frac{n(n+1)}2\pm\sqrt{\frac{n^2(n+1)(2n+1)}6}.

$$

- When is something “obvious”?
- If $a=\langle12,5\rangle$ and $b=\langle6,8\rangle$, give orthogonal vectors $u_1$ and $u_2$ that $u_1$ lies on a and $u_1+u_2=b$
- Convolution product on the linear dual of the polynomial algebra
- What are some easy to understand applications of Banach Contraction Principle?
- Fubini's Theorem for Infinite series
- Calculating $\frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{-\alpha x^2 + \beta x} \right) $
- Generalisation of alternating functions
- what is the area of the region ?
- What is the fallacy in this proof?
- Remainder with double exponent?
- What can be said if $A^2+B^2+2AB=0$ for some real $2\times2$ matrices $A$ and $B$?
- Extended Euclidean Algorithm in $GF(2^8)$?
- Dimension of $\text{Hom}(U,V)$
- How can I do a constructive proof of this:
- Derived subgroup where not every element is a commutator