Intereting Posts

Show that $p \Rightarrow (\neg(q \land \neg p))$ is a tautology
Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian?
How does one create a partition of unity for a complex manifold?
Isomorphism of $\mathbb{Z}/n \mathbb{Z}$ and $\mathbb{Z}_n$
Existence of vector space complement and axiom of choice
Centralizer/Normalizer of abelian subgroup of a finite simple group
Why is every selfadjoint operator closed?
Real Analysis Qualifying Exam Practice Questions
Riemann vs. Stieltjes Integral
Cauchy-Schwarz Inequality in a $C^∗$-algebra
Asymptotic behaviour of sums of consecutive powers
Complex numbers system of equations problem with 5 variables
Is the coordinate ring of SL2 a UFD?
How many combinations can I make?
If $9^{x+1} + (t^2 – 4t – 2)3^x + 1 > 0$, then what values can $t$ take?

Let $A \in M_{n \times 1} (\mathbb K)$. I’m asked to proof that $AA^t$ is diagonalizable.

My attempt: If $A = 0, \, AA^t = 0$ is diagonal. Let $A = \begin{bmatrix}

a_1\\\vdots

\\

a_n

\end{bmatrix} \neq 0$, then $AA^t = \begin{bmatrix}

a_1\\\vdots

\\

a_n

\end{bmatrix} \begin{bmatrix}

a_1&…

&

a_n

\end{bmatrix} = \begin{bmatrix}

a_1 a_1 & \cdots & a_1 a_n \\

\vdots & & \vdots \\

a_n a_1 & \cdots & a_n a_n

\end{bmatrix}$. Each column $v_i = \begin{bmatrix}

a_ia_1\\ \vdots

\\

a_ia_n

\end{bmatrix} = a_i \begin{bmatrix}

a_1\\ \vdots

\\

a_n

\end{bmatrix}$. So, $A$ has rank 1 what implies in $A$ diagonalizable or Nilpotent.

How can I see that $A$ is not Nilpotent?

- Diagonalization of a projection
- Prove that $ND = DN$ where $D$ is a diagonalizable and $N$ is a nilpotent matrix.
- Rank of square matrix $A$ with $a_{ij}=\lambda_j^{p_i}$, where $p_i$ is an increasing sequence
- Eigenvalues of a tridiagonal stochastic matrix
- Is the matrix diagonalizable for all values of t?
- diagonalisability of matrix few properties

Help?

- Why $\displaystyle f(z)=\frac{az+b}{cz+d}$, $a,b,c,d \in \mathbb C$, is a linear transformation?
- Does the set of matrix commutators form a subspace?
- Relationships between $\det(A+B)$ and $A+B$
- $A,B,C \in M_{n} (\mathbb C)$ and $g(X)\in \mathbb C$ such that $AC=CB$- prove that $A^jC=CB^j$ and $g(A)C=Cg(B)$
- What can we say about two graphs if they have similar adjacency matrices?
- For matrices $A$ and $B$, $B-A\succeq 0$ (i.e. psd) implies $\text{Tr}(B)\geq \text{Tr}(A)$ and $\det(B)\geq\det(A)$
- Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?
- Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues
- Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$
- Find the value of k, (if any), for which the system below has unique, infinite or no solution.

Also: let $u$ be the column vector

$$u=\begin{pmatrix}a_1\\\vdots \\ a_n\end{pmatrix}$$

and $B=AA^t$. Note that $Bu=\|u\|^2u$ and $Bx=0\cdot x$ when $\langle x,u\rangle=0$. So, the eigenvectors generate the whole space and therefore $B$ is diagonalizable.

You cannot prove that, because $AA^t$ can be nilpotent but nonzero.

In fact, when $A\ne0$, the rank-one matrix $AA^t$ is nilpotent if and only if its nilpotency index is $2$. As $(AA^t)^2=A(A^tA)A^t=(A^tA)\cdot(AA^t)$ (where the dot denotes a scalar multiplication), we see that $AA^t$ is nilpotent (hence non-diagonalisable) precisely when $A^tA=0$. And this can indeed happen, such as when $\mathbb K=\mathbb C$ and $A=\pmatrix{1\\i}$.

- $f(z_1 z_2) = f(z_1) f(z_2)$ for $z_1,z_2\in \mathbb{C}$ then $f(z) = z^k$ for some $k$
- How to plot a phase potrait of a system of ODEs (in Mathematica)
- A riddle for 2015
- Negative factors of a number
- Convergence types in probability theory : Counterexamples
- A question about the equivalence relation on the localization of a ring.
- Closed form for integral of integer powers of Sinc function
- How to show that $A^3+B^3+C^3 – 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?
- Inscribed kissing circles in an equilateral triangle
- If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent
- Continued Fraction
- $\Delta^d m^n =d! \sum_{k} \left { {k+n} \brace m + d}(-1)^{m+k}$ Is this a new formula?
- Travelling with a car
- Do there exist totally ordered sets with the 'distinct order type' property that are not well-ordered?
- What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?