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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0answers
35 views

Rigid Deformation

I'm trying to parse through this paper on using the method of moving least squares for rigid transformations - http://www.cs.rice.edu/~jwarren/research/mls.pdf Under section 2.3, the author mentions ...
0
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1answer
39 views

Construct matrix of ones and zeros based on sequences

We are given ($a_1,a_2,....,a_m$) and ($b_1, b_2,....,b_n$) sequences with non-negative integers. Decide whether it's possible and if it is construct a matrix $\Re^{m x n}$ of ones("1") and ...
4
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1answer
93 views

if $tr(A)=0$,then we have $A=BC-CB?$

if for any matrix $A_{n\times n}$,and such $tr(A)=0$,show that there exist matrix $B$ and $C$ such $$A=BC-CB$$ I know prove this: if $A=BC-CB$,then we have $tr(A)=0$ because $$tr(BC)=tr(CB)$$ ...
0
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1answer
41 views

matrix derivative of 3 multiplied matrices

I have the following function $L=\mu_w^T\Sigma_w^{-1}\mu_w$ where both $\mu_w$ and $\Sigma_w$ are functions of a vector $w$. How do I differentiate this wrt. $w$. $\Sigma$ is a positive definite ...
0
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1answer
40 views

Simple expression for the number of elements in a diagonal half of a matrix

Consider the following matrix: $\begin{bmatrix} 0 & 0 & 0 & 0 \\ I & 0 & 0 & 0 \\ A & I & 0 & 0 \\ A & A & I & 0 \\ A & A & A & I ...
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1answer
35 views

Differentiate matrix quadratic

I wish to differentiate $x^TAx$ wrt. $x_i$ where $x_i$ is the i-th element in the vector $x$. I realise when differentiating wrt. $x$ alone the answer is $2Ax$. How would this change when its $x_i$
2
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1answer
58 views

Derivative of a Matrix to a Power

Fix a positive interger $k$ and let $F: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times n}$ be the map on $n \times n$ matrices defined by $F(A)= A^k$. Show that $F$ is differentiable at ...
2
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1answer
95 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
0
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0answers
35 views

PSD matrix and non-negative polynomial

So I'm trying to prove that if there exists a $5 \times 5$ matrix $Q$ such that $$Q \succeq0,\,\, a_{l-1} = \sum\limits_{i+j=l} Q_{ij} , l=1,\ldots,5$$ then there exists a fourth degree polynomial ...
2
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1answer
96 views

Boolean Least Squares semidefinite relaxation

So I'm working on the Boolean least squares problem that comes up a lot in circuit design. In its raw form, it looks like this, $$\phi = \min \operatorname{trace}(A^TAX) - 2b^TAx + b^Tb$$ s.t. $$X = ...
2
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1answer
48 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
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3answers
37 views

How are specific linear maps defined?

I'm revising for exams and a question that often crops up is: given a linear map $\mathcal{T}:\;\mathbb{R}^n\to\mathbb{R}^m$, describe how to represent $\mathcal{T}$ as a matrix relative to bases ...
1
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0answers
48 views

markov chain computation

I consider a 2 state Markov chain: $X = \{1,2\}$, transitions are $M(i,j)$ and the matrix has a unique stationnary distribution $\pi$: $$ \pi(1) = \frac{M(2,1)}{2-M(1,1) - M(2,2)} \\ ...
0
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1answer
87 views

proof of positive semi-definiteness of the precision matrix (inverse of the covariance matrix)

I would like to know how to prove that the inverse of a covariance matrix $\Sigma^{-1}=\Omega$ is positive semi-definite too. My second question can we prove that for any matrix $A\in\cal{M}_{nm}$ we ...
5
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1answer
206 views

What are eigenvalues of higher order finite differences matrices?

I know (at first empirically, then read somewhere) that for for second order finite differences matrices like $$\begin{pmatrix} -2&1&0&0&0\\ 1&-2&1&0&0\\ ...
3
votes
1answer
78 views

Matrices rank problem

$X\in \text{Mat}_n (\mathbb{R} )$ and $|X|\neq 0$. $X$ has column vectors $X_1,X_2,\ldots ,X_n$. $Y$ is a matrix that consists of column vectors $X_2,X_3,\ldots ,X_n,0$. Let $A=YX^{-1}$ and ...
0
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1answer
70 views

Determinant of Matrix is different than product of diagonal

(sorry in advance, but I can't find a page on how to format math equation/structures) I'm having a bit of an issue with this matrix and finding its determinant. I know what the correct determinant is ...
2
votes
2answers
71 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
2
votes
1answer
104 views

How to FAST calculate 2 norm / spectral norm of a matrix.

I meant reduced 2 norm, the largest singular value. My current approach is applying the SVD decomposition of A via "?gesdd" in MKL, and then taking the largest singular value. I think there should ...
0
votes
3answers
83 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
1
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1answer
207 views

Solve Coupled System of Equations via Matrix

I have a coupled system of three equations that I am trying to solve via matrices and I am having trouble figuring out how to write out my matrices. My three equations are as follows: $-sx+sy=0$ ...
2
votes
1answer
75 views

Is the smallest singular value able to measure the similarity between two matrices?

I came across an interesting statement. Given two matrices $A$ and $B$, with orthogonal unit column vectors of the same length. $A$ and $B$ are not necessarily square matrices. One would use ...
1
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1answer
79 views

Can someone please provide an intuition behind cramer's rule?

See question. I usually get concepts like this very quickly (no studying required), but this one looks like Chinese. Can someone please help me understand a brief intuition behind Cramer's rule for ...
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1answer
2k views

How to find a transition matrix?

let $a= \{a_1, a_2, a_3, a_4\}$ and $b=\{b_1,......,b_4\}$ and $r = \{r_1,...,r_4\}$ Also, $b_1 = 4a_1$ $b_2 = 8a_1 + 7a_2$ $b_3 = 4a_1 + 4a_2 + 4a_3$ $b_4 = 9a_1 + 5a_2 + 8a_3 + 5a_4$ and ...
1
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1answer
25 views

Duality and Optimality Conditions

I have seen the solution and it involves adding a $x_5$ and $x_6$ to the inequalities. I really do not understand why this happens? I have not seen any questions like this yet. Any pointers would ...
3
votes
2answers
314 views

3x3 matrices completely determined by their characteristic and minimal polynomials

How do you show that two 3x3 matrices with the same characteristic and minimal polynomials both conjugate to the same Jordan normal form, assuming no knowledge of the eigenspaces? I know that it is ...
2
votes
1answer
76 views

Proving that if $n\times n$ Hadamard matrix exists, then 4 divides $n$

Im looking for an explanation of the following: a standard way to prove that, if there exists Hadamard matrix of dimension $n > 2$, then $4|n$, is to suppose that without loss of generality every ...
1
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2answers
89 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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2answers
172 views

Cholesky Decomposition for positive semidefinite separation

Cholesky decomposition is a common way to test positive semi definiteness of a symmetric matrix $A$. If the algorithm "goes wrong" trying to take a square root of a negative number, I know the matrix ...
2
votes
1answer
73 views

Find an $n\times n$ integer matrix with determinant $1$ and $n$ distinct positive eigenvalues

I feel pretty stupid for doing this, but here goes anyway. Earlier today I asked: Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues. As it turns out, for my problem I ...
0
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0answers
38 views

3-Species Population Model [closed]

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
3
votes
1answer
89 views

Inverse of product of matrices

Let $n>m$ and let $A$ and $B$ be $m\times n$ and $n\times n$ matrices. $B$ is invertible. If $A$ was square and invertable, then obviously $$ \left(ABA^T\right)^{-1} = A^{-T}B^{-1}A^{-1} $$ But, ...
0
votes
1answer
22 views

Error Correction in Matrices

I have a matrix for which I am supposed to find the solutions to Ax=0, however Linear Algebra was some time ago and I cannot remember how to do this. Any help would be appreciated. $A = ...
0
votes
2answers
22 views

How to expand $\mathbf{r}^T\mathbf{A}^{-1}\mathbf{r}=(\mathbf{y}-\mathbf{Ax})^T\mathbf{A}^{-1}(\mathbf{y}-\mathbf{Ax})$

I need to expand: $$\mathbf{r}^T\mathbf{A}^{-1}\mathbf{r}=(\mathbf{y}-\mathbf{Ax})^T\mathbf{A}^{-1}(\mathbf{y}-\mathbf{Ax})$$ I believe that $\mathbf{AB}\neq\mathbf{BA}$, $\mathbf{AA}^{-1}=1$, and ...
4
votes
2answers
109 views

Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
0
votes
1answer
17 views

Does $(\mathbf{y}-\mathbf{Ax})^T(\mathbf{y}-\mathbf{Ax}) = \mathbf{x}^T\mathbf{AAx}-2\mathbf{x}^T\mathbf{Ay}+\mathbf{y}^T\mathbf{y}$

I am told, that if $\mathbf{A}$ is symmetric $\mathbf{A}^T=\mathbf{A}$ and: $$ (\mathbf{y}-\mathbf{Ax})^T(\mathbf{y}-\mathbf{Ax}) = ...
0
votes
1answer
19 views

Gradient Method of solving $\mathbf{Ax}=\mathbf{y}$

The problem; solve a linear system of equation: $$\mathbf{Ax}=\mathbf{y}\tag1$$ can be recast as; Find $\mathbf{x}$ to minimise the 'error residual', a column vector, $\mathbf{r}$, defined as a ...
0
votes
3answers
52 views

grouping non-zero entries in a matrix according to a rule

I have a matrix say, $a = \left[\matrix{ 0 & 1 & 0& 0& 0& 1& 0\\ 0& 0 &0 &0 &0 &1& 1\\ 1& 0 ...
0
votes
1answer
97 views

Centralizer of $SO(n)$

Given the set $M(n,\mathbb C)$ of all complex $n\times n$ matrices, what's the centralizer of $SO(n)$ in $M(n,\mathbb C)$? For $n=2$, the centralizer must be the matrices $A$ such that $RA=AR$ where ...
2
votes
1answer
77 views

Matrix exponential proof

I am solving a problem: $$A^3=\alpha^2A\implies \exp(A)=E+\frac{\mathrm{sinh}\alpha}{\alpha}A+\frac{\mathrm{cosh}\alpha-1}{\alpha^2}A^2;\, \alpha\in\mathbb{C},\,A\in\mathbb{M}_{n\times ...
0
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2answers
58 views

Why not $R_j + cR_i \rightarrow R_i$ for Elementary Row Operation of Replacement ? [Lay P6]

P6 of Linear Algebra and Its Applications, 4th Ed by David Lay says: Replacement: $\color{green}{kR_j + R_i \rightarrow R_i}$. For example (from BP P435 Example 2): $\left[\begin{array}{cc|c} ...
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1answer
62 views

Another matrix of a given operator $A \otimes A$

Let $V$ be a $4$-dimensional vector space with an ordered basis $e_{1}$, $e_{2}$, $e_{3}$, $e_{4}$, and $A: V → V$ be a linear operator given by its matrix relative to the ordered basis $(e_{1}, ...
0
votes
1answer
53 views

Is there a significance to a matrix having an eigenvector equal to a column vector within a matrix?

Consider matrix $A$: $\begin{bmatrix} 2 & -2\\ 3 & -3\\ \end{bmatrix}$ After little computing, we find the eigenvectors (and their corresponding eigenvalues) to be equal to $E_{\lambda=0}= ...
0
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1answer
33 views

Linear Algebra nullSpace and multiplication

If you have a $m \times n$ matrix $A$, and an $n \times p$ matrix $B$ and $AB=0$. Is the $dim (null A) = p$? No clue how to start this.
0
votes
2answers
40 views

discrete fourier transform proof (show equals n*I)

Let $w=e^{(-2\pi i/n)}$. Let $W$ be an $n \times n$ matrix defined by $$ W = \begin{pmatrix} 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & w & w^2 & w^3 & \cdots & ...
0
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1answer
267 views

Matlab Index out of bounds

I'm trying to call a function using ode45 from another script file and I am passing in an matrix A. Matlab throws an error and ...
2
votes
2answers
294 views

Is it possible to triangularize a matrix only by adding scalar multiples of rows to each other?

I am working on showing if $B$ is a $s \times s$ matrix, $D$ is a $t \times t$ matrix, $C$ is a $s \times t$ matrix, and $0$ is a $t \times s$ zero matrix, then $\det(A)=\det(B)\det(D)$, where $$A = ...
0
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1answer
37 views

Problem with matrices

Got one problem here where I don't really get what is being asked of me to do: "Show all matrices $A \in M_{22}(\mathbb R)$, for which following applies: ...
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vote
2answers
73 views

Is $W$ a subspace of $\mathbb{R}^{n \times n}$?

Let $W = \{A \in \mathbb{R}^{n \times n} | A_{11} \geq 0\}$ is $W$ a subspace of $\mathbb{R}^{n \times n}$? Prove or disprove. I know how to do it if it was a specific sized matrix, but I'm not sure ...
2
votes
1answer
64 views

Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...