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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6 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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0answers
9 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 ...
3
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0answers
21 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\\ ...
2
votes
1answer
11 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 = ...
0
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1answer
9 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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2answers
25 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
24 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 ...
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0answers
9 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 ...
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0answers
19 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
vote
1answer
420 views

Bar symbol over a matrix

So I am reading a paper (not online) and I come across a definition: $$\mathbb E=R\bar R$$ Where R is a complex matrix. I am thinking that it means complex conjugate, but I honestly have never seen ...
0
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1answer
15 views

Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$. [on hold]

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$.
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1answer
25 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
19 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
26 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
21 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 ...
0
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1answer
72 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 ...
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4answers
160 views
+50

Number of matrices with trace N

Is there a better method than bruteforcing, to find out the number of possible matrices of order 2x2 that have trace $N$. The contraints are that all elements in a matrix must be positive integers and ...
0
votes
2answers
22 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
votes
1answer
267 views

Finding reflection of a matrix

To 2 decimal places, what is the value of the lower-right entry in the reflection matrix $Q_a $if a = 1.05? Not even sure where to begin, is there a formula? This is what I could find in my textbook ...
0
votes
0answers
16 views

What changes where made on this Gaussian-Elimination?

in the Internet I have found the following use of the Gaussian Elimination method: $z \in \mathbb{R}, \ n\in\mathbb{N}, n \ge 2$ and $\begin{pmatrix} z & 1 & \dots & 1 & 1 \\ 1 & ...
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0answers
13 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)} \\ ...
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0answers
21 views

Solving Matrix Value and Optimal Strategy (Matrix Games)

How would I solve this matrix game ? I'd like to find the value of the matrix and the optimal strategies for each player. $$ \left[ \begin{array}{cccc} 0 & 3 & -2 & 2 \\ -3 & 0 ...
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2answers
42 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
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1answer
22 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
32 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
votes
3answers
34 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 ...
2
votes
0answers
43 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 ...
1
vote
0answers
16 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 $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
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vote
2answers
25 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
votes
3answers
29 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 ...
0
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1answer
23 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: $-sx+sy=0$ ...
5
votes
2answers
360 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 ...
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1answer
19 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
67 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 ...
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votes
1answer
23 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 ...
4
votes
2answers
113 views

How prove this matrix inequality $\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$

Question: Let $A_{n\times n},B_{n\times n}$ is positive Hermite matrix show that $$\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$$ I know this $$\det(A+B)\ge \det(A)+\det(B)$$ But My problem I ...
2
votes
1answer
33 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 ...
0
votes
1answer
12 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 ...
1
vote
1answer
42 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
1answer
48 views

Verify that $\det (A) = \det (A^T)$ for two matrices [on hold]

(a) $$A = \begin{bmatrix} -2& 3 \\ 1& 4 \\ \end{bmatrix}$$ (b) $$A = \begin{bmatrix} 2& -1& 3 \\ ...
0
votes
3answers
37 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: ...
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2answers
30 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
44 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
29 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
24 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
20 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
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0answers
36 views

Calculating the null space of a matrix [on hold]

I am sorry for maybe this is a duplicate question but I really need someone to help me with this I am trying to calculate the null space of this matrix, but I really don't know how and I also have ...
0
votes
1answer
21 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
63 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
20 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 ...