Articles of trace

Show $\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A,A\in M(2,\mathbb{C})$

Show $$\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A$$ for $A\in M(2,\mathbb{C})$. In addition, $\operatorname{trace}(A)=0$. Can anyone give me a hint how this can connect with cosine and sine? Thanks!

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for the derivative of the determinant wrt a scalar (e.g. see wikipedia): $$ \frac{\partial}{\partial \alpha} \det(\mathbf{X}) = \det(\mathbf{X}) \operatorname{tr} \left( \mathbf{X}^{-1} \frac{\partial \mathbf{X}}{\partial \alpha} \right) $$ […]

Derivative involving the trace of a Kronecker product

I’m stuck trying to solve a derivative that looks like this: $$\frac{\partial}{\partial X} \mbox{Tr} \{ A(X^{-1} \otimes I_{n} )B \},$$ where A is a $N\times 2n$ matrix, B is a $2n \times N$ matrix, and X is a $2 \times 2$ matrix. Thank you!

Consider the trace map $M_n (\mathbb{R}) \to \mathbb{R}$. What is its kernel?

The map is the trace map. I.e, it takes any $n$ by $n$ matrix and associates to that matrix, a number of the form $\mathrm{Tr}(A) = \sum_{i=1}^n a_{ii}$, where $A \in M_n (\mathbb{R})$. I need to find the kernel of this map, give a basis and its dimension (which is easy once I have the […]

Differential and derivative of the trace of a matrix

If $X$ is a square matrix, obtain the differential and the derivative of the functions: $f(X) = \operatorname{tr}(X)$, $f(X) = \operatorname{tr}(X^2)$, $f(X) = \operatorname{tr}(X^p)$ ($p$ is a natural number). To find the differential I thought I could just find the differential of the compostion function first and then take the trace of that differential. Am […]

Generalized Poincaré Inequality on H1 proof.

let’s see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable functions in $\Omega$ given by the equivalence relation $u\sim v \iff u(x)=v(x)\, \text{a.e.}$ being a.e. almost everywhere, in other words, two functions belong to the same equivalence […]

Convexity of a trace of matrices with respect to diagonal elements

Can we prove that $\mbox{trace}({\bf A} ({\bf P}+{\bf Q})^{-1} {\bf A}^T)$ is a jointly convex function of positive variables $[q_1,q_i,…,q_N]$, where ${\bf Q}=\mbox{diag}(q_1,…,q_N)$, $q_i>0 , \forall i$, and ${\bf P}$ is a positive definite matrix.

Proof that the characteristic polynomial of a $2 \times 2$ matrix is $x^2 – \text{tr}(A) x + \det (A)$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $C_{A}(x) := \det(xI-A)$ be the characteristic polynomial of A. Show that $$C_{A}(x)=x^2-\text{tr}(A)x+\det(A).$$ I know that $\text{tr}(A)=a_{11}+a_{22}$ and $\det(A)=a_{11}a_{22}-a_{21}a_{12}$. Plugging this into the above equation I get $$C_{A}(x)=x^2-(a_{11}+a_{22})x+a_{11}a_{22}-a_{21}a_{12}.$$ I’m not sure how to get past this. As you can tell, I’m not too good at […]

Can you check my proof on the characterization of the trace function?

The following is my proof: Theorem: If $W$ is the space of $n \times n$ matrices over the field $F$ and if $f$ is a linear functional on $W$ such that $f(AB)=f(BA)$ for each $A$ and $B$ in $W$, then $f$ is a scalar multiple of the trace function. Proof: *$\mathcal{B}$ is the standard ordered […]

Finite field question involving the trace and a permutation.

Let $q$ be a power of a prime $p$, and $m,l$ positive integers with gcd$(l,q^m-1)=1$. Denote $Tr$ to be the trace of $GF(q^m)$ over $GF(q)$. Suppose that there exists a nonzero $\gamma \in GF(q^m)$ such that $$ Tr(x)=0 \Leftrightarrow Tr(\gamma x^l)=0, \;\text{ for all } x\in GF(q^m). \;\;\;\;\;\;\text{(1)}$$ I’m pretty sure that Equation $(1)$ implies […]