# Tagged Questions

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### Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
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### If $tr(A+B)>tr(A)$, does it hold that $tr((A+B)^k)>tr(A^k)$ for all $k\geq 1$

I wonder whether the following holds and if so how it could be proved: Let $A, B$ be (non-commuting) positive semi-definite matrices, If $tr(A+B)>tr(A)$, does it hold that ...
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### Algebra-sum of entries in each column of a sqaure matrix = constant

This is a question from an algebra homework and I am just looking for some tips. The question is: We have: $M$: an $n\times n$ matrix with real entries $c$: a real constant the ...
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### Show that if $tr(A+B) > tr(A)$ then $tr((A+B)^k)\geq tr(A^k)$ for any $k\geq 1$

This may be a stupid question, but I am completely stuck, I don't even know where to start. I have to show that if $tr(A+B) > tr(A)$ then $tr((A+B)^k)\geq tr(A^k)$ for any $k\geq 1$, where $A$ and ...
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### Raising $e$ to the power of a matrix

Does there exist a definition for matrix exponentiation? If we have, say, an integer, one can define $A^B$ as follows: $$\prod_{n = 1}^B A$$ We can define exponentials of fractions as a power of a ...
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### Could someone explain this partial sum expression to me?

I found this in one of my programming exercises that asks for the sum of each column so that the result vector V of size m is defined like so: What exactly is this telling me? Thanks for any help
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### Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
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### Notation for summing all elements under the diagonal of a square matrix

I have a simple question: What is the notation for summing all elements under the diagonal of a square matrix? I appreciate your help.
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### Least square proof, Notation sum matrices

I have spent weeks trying to understand a "proof" in my textbook. However I am not able to get what is going on. The "proof" goes like this:(I have marked the numbers in red) This is how I have ...
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### Need to find N value where each sum A+B is different

I need to find N value (in this case 12, but next time they could more o less) and I need that every sum of two value is a unique number. In the picture below you can see an easy matrix where there ...
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### How to show that the order in which multiple sums are performed does not matter

So let $A=(a_{ij}) \in M_{nm} (R)$ I need to show that: $\sum\limits_{i=1}^n$($\sum\limits_{j=1}^m a_{ij}$)= $\sum\limits_{j=1}^m$($\sum\limits_{i=1}^n a_{ij}$) (the order in which multiple sums ...
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### Matrices inner produce

I had a question about matrices. Why is $1^T*X = \sum X_i$ ? [ Here X is also a matrix] Basically, why is $1^T*X$ the sum of all the elements in the matrix X.
I want to prove that $$\sum_{k=i}^n \binom{n}{k}\binom{k}{i}(-1)^{n-k}=\delta_{n,i}$$ Where $\delta_{n,i}$ is the Kronecker Delta, i.e. $\delta_{n,i}=0$ if $n \neq i$ and $\delta_{n,i}=1$ if $i=n$. ...