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
75 views

Finding Matrix adjoint

Need someone to check my reasoning as I don't feel confident in this topic: consider $P_2$(C) with inner product $$<p(x), q(x)> = \int {q(x)p(x) dx} $$ T is defied by T(p(x)) = p'(x) + p(x) ...
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1answer
29 views

diferences of spectral decomposition of different types of matrices

For an $n \times n$ square complex matrix let say $A$ with eigenvalues $\lambda_1,\lambda_2,.....,\lambda_n$. $A$ is normal iff $A$ is unitary diagonalizable;that is there exist unitary matrix U such ...
5
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4answers
404 views

Mysterious Proof about Induced Norms (was: Uniqueness of SVD)

In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen), argues as follows: Let $\sigma$ be the first singular value of A, and $v_{1}$ the ...
2
votes
2answers
59 views

Uniqueness of Singular Values

Given a matrix A, one inductively constructs (and thus proving its very existence) the singular value decomposition as follows: take $ \sigma_{1}=||A||_{2} $, and consider a couple of vectors such ...
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1answer
74 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
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1answer
35 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 ...
11
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1answer
213 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
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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 ...
1
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0answers
31 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 ...
5
votes
1answer
187 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\\ ...
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1answer
39 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 ...
4
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1answer
86 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 $$tr(BC)=tr(CB)$$ ...
2
votes
1answer
89 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 = ...
3
votes
1answer
77 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 ...
2
votes
2answers
6k views

Singularity in matrix when inverting in Matlab

As data I get a matrix A but in my algorithm I need to work on its inverse. What I do is: C = inv(A) + B; Then in another line I update A. In the next cycles I ...
0
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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
57 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
votes
1answer
88 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
votes
0answers
34 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 ...
0
votes
1answer
38 views

Least Square with Singular Matrix

Suppose I have vector $x'=[1 $ $ $ $ x_2 $ $ $ $ x_3]$ and $x_3 = a + bx_2$ (where $a$ and $b$ are constant), and data, say $y$. In general, the least square will be $\beta = E[xx']^{-1}E[xy]$. Now, ...
2
votes
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 ...
2
votes
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 ...
0
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1answer
85 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 ...
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1answer
84 views

Peculiar Matrix

I came up with this idea recently and I want to go deeper in this, but it has been difficult to me. Hope someone can help me on this. Suppose I have a matrix of order $(n^2-1)\times (n^2-1)$ with ...
0
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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 & ...
1
vote
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)} \\ ...
1
vote
2answers
167 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 ...
0
votes
1answer
69 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 ...
0
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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 ...
1
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2answers
80 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 ...
0
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1answer
187 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: $-sx+sy=0$ ...
1
vote
2answers
653 views

Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P94, Strang P59]

From P59 of Intro to Lin Alg, 4th Ed by Strang & P94-95 of Linear Algebra and its Apps by Lay For relief, I denote all row vectors with superscripts and column with subscripts. Define ...
0
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1answer
70 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 ...
0
votes
1answer
105 views

Given its pseudo-inverse, is there a fast way to measure the degree of full-rankness of a nonsquare matrix?

update: I realized the core of question is about ill-conditioning of the matrix (aka Multicollinearity). In a computer, with floating point arithmetic, it is impossible to talk about full-rankness. We ...
-1
votes
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 ...
0
votes
1answer
23 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 ...
2
votes
1answer
70 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
votes
3answers
44 views

Find the conditions required for the values of a, b, and c that make the following matrix symmetric.

Set up the system: $$A = \begin{bmatrix} 5& a+b+c& a-b \\ 3& -7& 2\\ 1& a+c & 6 \end{bmatrix}$$ I did it like this: ...
1
vote
2answers
70 views

reduction of a skew-symmetric matrix

Birkoff and MacLane state that any real symmetric matrix $A$ has the form $ A = P^{-1}BP $ where $ B^2 $ is diagonal and they ask for a proof as an exercise. It seems to me that if $A$ is ...
2
votes
2answers
269 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 = ...
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 ...
3
votes
1answer
80 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, ...
1
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0answers
34 views

3-Species Population Model

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 ...
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 = ...
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
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 ...
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
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 ...
1
vote
3answers
78 views

Problem with LU decomposition

I have this matrix: $$ A =\begin{bmatrix}1 & -2 & 3\\ 2 & -4 & 5 \\ 1 & 1 & 2\end{bmatrix} $$ After I decomposit it, I get: $$ L = \begin{bmatrix}1 & 0 & 0\\1 & 1 ...