For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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1answer
19 views

Linear Algebra Subspace test

I'm currently studying Subspace tests in my linear Algebra module at uni, but am struggling to understand it, can anyone explain how to conduct a SubSpace test?
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1answer
17 views

finding if a linear transformation exists, and proving it.

We just started the topic of linear transformations and I have this hw question that I just don't understand. Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, ...
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1answer
17 views

Elementary row operations in matrices

This is really such a lovely math community, I am working on some differential equations hw and my teacher didn't teach this topic yet so I am a little confused. My first question is pertaining to ...
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2answers
32 views

Find a polynomial $f(Z)$ of degree less than 2 such that $e^{tA}=f(A)$

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. As the question says I need a polynomial $f(Z)$ of degree less than 2 such that $e^{tA}=f(A)$. Should my polynomial just be the first 2 terms ...
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0answers
13 views

Find a basis for the four fundamental subspaces.

Find a basis for the four fundamental subspaces of: $$A=\begin{bmatrix}1 & -1 & 0 & 2 \\ 0 & 0 &1 &1 \\ 0 &0 &0 &0\\0 &0 &0 &0\end{bmatrix}$$ I'm ...
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0answers
10 views

LU decomposition of a $n\times n$ matrix

I would like to find the LU decomposition of the following matrix: $$\begin{equation} a_{i,j}=\left\{ \begin{split} 1\quad \mbox{if}\qquad i=j \quad\mbox{or}\quad j=n\\ ...
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0answers
8 views

Prove that the row sums of $T$ satisfy the following formula.

Consider the lower triangular matrix defined by the recurrence: $$T(n,1) = 1$$ $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = \sum _{i=1}^{n-1} T(n-i,k-1)+y \sum _{i=1}^{n-1} T(n-i,k) \; ...
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1answer
27 views

Prove that this statement about A and B is true.

$A,B \in \mathbb{R}^{2}$, If $AB - BA = A^2$ Prove that $ (B - A)^{2014} = B^{2013}(B-2014A)$
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0answers
9 views

Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
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0answers
10 views

Matrix representation of rotation proof?

C is for Cos, S is for sine To find the matrix representation, we just apply R n to each of the standard basis vectors, as in Equation 3.3, and then place the resulting vectors into the rows of a ...
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0answers
11 views

Recomputing the Gram Matrices

Recompute the following Gram matrix for the weighted inner product $\langle x,y\rangle=x_1y_1+\frac{1}{2}x_2y_2+\frac{1}{3}x_3y_3+\frac{1}{4}x_4y_4$: ...
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1answer
12 views

How to prove $V*V^T=I$ in SVD? [duplicate]

How to prove $V*V^T=I$ in SVD: $M=U*S*V^T$? It's easy to understand $V^T*V=I$. It seems $V*V^T=I$, but how to prove it?
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1answer
12 views

Prove that the product of two positive semidefinite and symmetric matrices has non-negative eigenvalues

How can I prove the following fact: If $A$ and $B$ are two positive semi-definite and symmetric matrices then all eigenvalues of $AB$ are non-negative.
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0answers
24 views

How to find integer solutions of a linear system?

I have a $M$ equation and $N$ variables like this : $ \begin{bmatrix} 3 & 0 & 1 & 0 & -1 & -3 & 2\\ 1 & 2 & 0 & 4 & 0 & 0 & -1\\ 1 & 1 & 0 ...
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0answers
35 views

How to show that e.g. $E(\mathbf{w}) = \ldots \Rightarrow \frac{1}{2}(\Phi\mathbf{w} - \mathbf{t})^T(\Phi\mathbf{w} - \mathbf{t}) $

We have to show for a few formulas that they can also be written in matrix notation. For example: For $\mathbf{x}=(x_0,x_1,\ldots,x_n)$,xi∈R $\sum_{n=1}^ix_n^2 = ...
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vote
1answer
23 views

Why is there two versions of the rotation matrix?

Why is there two versions..for example, I got some matrices for the x axis rotation This is for the X AXIS the other one is for the x axis also, it is.. I think it could be from going ...
1
vote
1answer
14 views

Optimisation over matrix entries

I was looking to write the KKT conditions to solve this optimisation problem. $$\min_{\substack{\sum_j x_{ij}\le k_i \\ i=1,2,\ldots N}} a^\top (I-X)^{-1} b $$ Since there are $N^2$ decision ...
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0answers
17 views

Help with a matrix problem

I'm stuck with the following matrix problem: Consider $A = $$\{ X \in \mathcal{M}_2(\mathbb{C})\ \mid X = \left( \begin{array}{ccc} a & 0 \\ 0 & b \end{array} \right); a, b \in \mathbb{C}; ...
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0answers
8 views

Bounds on the coherence of very flat matrices (that are more tight than the Welch bound)

I am studying the coherence of matrices in the context of sparse recovery. Let us say I have a matrix $\mathbf \Phi$ of size $M \times N$ with, say, unit Euclidean norm columns ${\mathbf \varphi}_n$. ...
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2answers
42 views

Find the $n^{th}$ power of a $2$x$2$ matrix.

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. Using Lagrange's interpolation compute $A^n$ for $n\in\mathbb{N} $ So far I've worked out the minimum polynomial of $A$ to be $(x-2)(x+1)$ but ...
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1answer
20 views

finding the inverse of a matrx

In order to decrypt a cipher text using hill cipher, we must first find the inverse matrix of a given matrix. From this link http://en.wikipedia.org/wiki/Hill_cipher, ...
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2answers
28 views

How to calculate row sums of a power of a matrix

Let $P $ be an $n\times n$ matrix whose row sums $=1$.Then how to calculate the row sums of $P^m$ where $m $ is a positive integer?
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1answer
16 views

Derivative involving inner product

How would I take the derivative of a function $$f(x) = < x,x >=x^{T}x?$$ The answer seems to be 2x but I don't know how to explicitly show this other than saying "there are 2 x's being operated ...
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1answer
40 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
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1answer
15 views

How Do I Find The Permanent of a Double Stochastic Matrix n * n size

I am reading a book on Stochastic Models, and I don't understand this practice questions: A doubly stochastic n × n matrix S has all entries equal to 1/n. The permament of a n × n matrx A is ...
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4answers
99 views

Why is the volume of a parallelepiped equal to the square root of $\sqrt{det(AA^T)}$

Why is the $\sqrt{det(AA^T)}$ equal to the volume of a parallelepiped? Is is somehow related to the fact that $det(A) = det(A^T)$? EDIT: To clarify, the parallelepiped is spanned by the columns of ...
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0answers
17 views

Is there a single matrix norm such that for all stochastic $P$, $\| P \| = 1$?

By a stochastic matrix I mean a square real non-negative matrix with rows summing to one. Denote the set of all such matrices $\mathcal{S}$. By matrix norm I mean a norm in the vector space of ...
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2answers
16 views

Proving a theorem about trace of matrix which involving generalized inverse matrix

can you prove that theorem for me: Let A be mxn matrix of rank r then, $\ tr[I-A(A'A)^-A'] = m-r $  .   $\ A' $(transpose of A) ,$\ A^- $(generalized inverse of A)
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1answer
22 views

Norm of the sum of inverse matrices

Let $A,B$ be two invertible matrices. Is there a way to compute $\|A^{-1} -B^{-1}\|$ in terms of $\|A-B\|$?
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3answers
27 views

How do I prove matrix irreversibility without determinants?

I have to prove that if matrix has two identical rows or columns then it is not a reversible matrix. I know that in such scenario matrix determinant is equal zero, but I cannot use determinants in my ...
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1answer
18 views

Does there exist a Unit Matrix for a m x n matrix?

By definition, a Unit/Identity matrix (I) is a matrix such that, I A = A I = A If the matrix A is of dimension m x n, then unit matrix in IA must be of dimention m x m, while in A I should be of ...
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0answers
17 views

Solve the following system using Gauss-Jordan elimination

$4x - 8y = 12$ $3x - 6y = 9$ $-2x + 4y = -6$ So the augmented matrix will be: $$ \begin{bmatrix} 4 && -8 && 12\\ 3&& -6 && 9\\ -2 && 4 && -6 ...
3
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0answers
44 views

Proving that table totals can always be preserved with ceiling and floor

$\begin{array}{|c|c|c|c|} \hline 11.998& 9.083 & 2.919 & &24 \\ \hline 12.983&10.872&3.145&&27\\ \hline 1.019&2.045&0.936&&4\\ \hline & & ...
1
vote
1answer
29 views

Is $vv^{T} - v^{T}vI$ non-singular? [on hold]

Is $vv^{T} - v^{T}vI$ non-singular ? Why? $v$ is vector
0
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1answer
26 views

Derivation of Trace

I am curious about a formula in http://zh.wikipedia.org/wiki/%E8%B7%A1 $$\frac{\partial\text{tr}(A^{-1})}{\partial A}=-(A^{-2})^T$$ I have tried to prove this. We have $A^{-1}=\frac{A^*}{|A|}$, and ...
0
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1answer
23 views

An Inequality in Numerical Optimization

I am reading Jorge Nocedal and Sepher J. Wright's Numerical Optimization and stuck at an exercise 4.6 in chapter 4. The Canchy-Schwarz inequality states that for any vector $u$ and $v$, we have ...
0
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1answer
8 views

Number of Distinct Elements in Set of Products of 2 Matrices

Let $X=\begin{pmatrix}\cos\frac{2\pi}{5} & -\sin\frac{2\pi}{5}\\\sin\frac{2\pi}{5} & \cos\frac{2\pi}{5}\end{pmatrix}$ and $Y=\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}$. Find the ...
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0answers
26 views

Eigenvalues with constraints?

Note: This is a short version of About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms) For a $n$-dimensional symmetric matrix A, orthogonal matrix C exists ...
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0answers
16 views

System solving with Substitution and Matrices

My class was able to produce solutions using Substitution on the following System: $$ \left\{ \begin{array}{c} x+y+z=0 \\ 2x+3y+2z=-1\\ x-y+z=2 \end{array} \right. $$ The solution was: x = 1, y = ...
0
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1answer
23 views

Linear Algebra - Give an example for $3x3$ matrix for these eigenvalues

I'm having trouble with this problem : Give an example for matrix $A$ with these eigenvalues $\lambda_1-1,\lambda_2=1,\lambda_3=0$ while : $$v_1=(0,1,1)$$ $$v_2=(1,-1,1)$$ $$v_3=(0,1,-1)$$ ...
0
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2answers
34 views

Let $A$ be a single $p\times p$ Jordan block. Find general solution to $\dfrac{dx}{dt} = Ax$

Let $A$ be a single $p\times p$ Jordan block. Find the general solution to $\,\dfrac{dx}{dt} = Ax$. What should I approach first? Please help!
4
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4answers
179 views

Question about determinants

I am working on some practice problems and I just cant see to even begin to understand how to do this question. It starts off by giving some facts such as det= 1 for the following:$$ \begin{matrix} a ...
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votes
2answers
14 views

Is the null space inside the collumn space of a matrix?

From what I've seen online, it seems that the null space isn't in the column space, but I don't understand why that is the case. If the null space is the set of all combinations that equal 0, and the ...
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2answers
22 views

Proof that theorems about trace of matrix :

Can somebody help me about proofs of this theorems A is an nxn matrix and $\ A^2$ = mA then, tr(A) = m rank(A) . A is an nxn matrix and k is a positive integer then, tr($\ A^k$) = $\sum_{i=1}^n ...
0
votes
1answer
20 views

Need to prove $(JC=0=CJ,\,JJ=nJ)\implies (C-aJ)^{-1}-(C-bJ)^{-1}=\frac{b-a}{ab n^2} J$

I can't prove that matrix $C$: $$\big(JC = 0 = CJ\text{ and } JJ = nJ\big) \implies \left((C-aJ)^{-1} - (C-bJ)^{-1} = \frac{b-a}{abn^2} J\right)$$ I know that $$(JC = 0 = CJ\text{ and }JJ = nJ) ...
2
votes
1answer
27 views

Get normalised eigenvectors

I am given the matrix: $\begin{pmatrix} a & b \\ b & -a \end{pmatrix}$ and I already calculated the eigenvalues $\lambda = \pm \sqrt{a^2+b^2}$. Now, I want to get the normalised ...
0
votes
1answer
26 views

Element matrix multiplication representation

Matrix element by element multiplication defined : $C=A*B$ $c_{ij}=a_{ij}b_{ij}$ Is this multiplication can be represented with stardant matrix multiplication or Kronecker product ?
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0answers
21 views

2D convolution: how to eyeball it?

I have a question of doing simple convolution in 2d by just "eye-balling" it without doing the actual computation. In 1D discrete time, when we have a simple input ...
2
votes
3answers
61 views

Roots of a cubic equation with coefficients based on unknown values $a$, $b$ and $c$.

I want to find the eigenvalues of the following matrix: $$ \left( \begin{array}{ccc} 0 & a & b \\ a & 0 & c \\ b & c & 0 \end{array} \right) $$ So, I found the characteristic ...
0
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0answers
23 views

Proof involving projections and column spaces

Let $A \in \mathbb{M}_{m×n}(\mathbb{R})$ with linearly independent columns. If $\overrightarrow{b} \in \mathbb{R}^m$, then prove $proj_{Col(A)}(\overrightarrow{b}) = ...