1
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2answers
44 views

How to show the identity relating to Matrix

Suppose that $$ A=\begin{bmatrix}a_{11}&a_{21}\\a_{21}&a_{22}\end{bmatrix}, \ \ B=\begin{bmatrix}d&-1\\1&0\end{bmatrix}. $$ and $$A=B^N$$ Show that $$a_{11}=\sum_{i=0}^{[N/2]}(-1)^i ...
-1
votes
0answers
24 views

Is there a name for this operation: $x_k = \sum_{ij} y_{ij} a_{ik} b_{jk}$?

Is there a name for this operation: $x_k = \sum_{ij} y_{ij} a_{ik} b_{jk}$ ?
0
votes
0answers
27 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
0
votes
1answer
22 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
1
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2answers
52 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
vote
1answer
32 views

Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
1
vote
1answer
35 views

How to evaluate a limit that involves matrices

I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me ...
1
vote
1answer
57 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
0
votes
0answers
41 views

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 ...
1
vote
1answer
56 views

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 ...
1
vote
4answers
226 views

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 ...
4
votes
1answer
62 views

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 ...
1
vote
2answers
40 views

combinatorics - permutations question, possibly with pigeon hole

Let $A \in Mat_n(\mathbb R)$ such that $\forall i,j: a_{ij}\geq 0$ We are given: $$\forall j: \sum_{i=1}^n a_{ij}=\sum_{i=1}^n a_{ji}=1$$ show there's a permutation $\pi \in S_n$ such that $$\forall ...
0
votes
0answers
34 views

Understanding the summation of matrix

There are a couple of things I think I have done wrong when calculating this formula, could someone please have a look at it. I think the way I have calculated the read part is right, but it might ...
0
votes
1answer
36 views

Sumation convention, proving matrix identity

On the lecture notes I found online there is the following proof that I don't understand. Let A, B be two matrices. $$(AB)^T=B^TA^T$$ I know this identity well, but I don't get this proof. Here it ...
0
votes
0answers
27 views

A question regarding notation of equation

I'm reading a research paper, and, have come across this summation equation $$ S_{2} = \sum_{N -1}^{j = 0}w_{j}^{2}\cdot $$ My question is: if j = 0..... N-1 do I ...
0
votes
1answer
26 views

Fundamental Matrix with Sums

Let $$\Phi(t)=\begin{bmatrix} x_{11}(t) & x_{12}(t)\\ x_{21}(t) & x_{22}(t) \end{bmatrix} $$ be a fundamental matrix for $$x'=A(t)$$ where $$A=\begin{bmatrix} a_{11}(t) & a_{12}(t)\\ ...
0
votes
2answers
136 views

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 ...
1
vote
1answer
35 views

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
0
votes
2answers
112 views

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 ...
1
vote
2answers
158 views

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.
1
vote
1answer
67 views

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 ...
0
votes
1answer
28 views

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 ...
2
votes
1answer
51 views

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 ...
0
votes
0answers
21 views

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.
4
votes
3answers
118 views

Proof with binomial coefficient and kronecker delta

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 $i=n$. ...
2
votes
1answer
40 views

Summing infinite numbers of matrices

Let $\mathbf{I}$ be a identity matrix, and $\mathbf{A}$ be a symmetric matrix in which every entry $a_{ij}$ follows $0 \le a_{ij} < 1$. I want to get $\mathbf{S} = \mathbf{I} + \mathbf{A} + ...
1
vote
1answer
57 views

a set of equalities about the covariance of two vectors

I am reading an article about circle fitting, I have two vectors $\alpha$ and $\beta$, they have $n$ components, the authors wrote that: We note that for any vectors $(\alpha_i)$ and $(\beta_i)$, ...
2
votes
1answer
104 views

Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
5
votes
1answer
205 views

eigenvalues of the sum of a matrix with known eigenvalues and a diagonal matrix

Suppose $B = A+D$, where all the eigenvalues of $A$ are already known and $D$ is a diagonal matrix, how to compute the eigenvalues of B without diagonalizing $B$ directly?
4
votes
2answers
1k views

Smallest eigenvalues of Sum of Two Positive Matrices

Let $C = A + B$, where $A$, $B$, and $C$ are positive definite matrices. In addition, $C$ is fixed. Let $\lambda (A)$, $\lambda (B)$, and $\lambda (C)$ be smallest eigenvalues of $A$, $B$, and $C$, ...