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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3
votes
1answer
18 views

Least squares / residual sum of squares in closed form

In finding the Residual Sum of Squares (RSS) We have: \begin{equation} \hat{Y} = X^T\hat{\beta} \end{equation} where the parameter $\hat{\beta}$ will be used in estimating the output value of input ...
1
vote
1answer
15 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 ...
0
votes
0answers
6 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
5
votes
1answer
43 views

Matrix Help: Combinations

Given a 10 by 10 matrix filled with 0s and 1s, how many possible outcomes are there? It sounds easy enough as a combination of $2^{100}$. The kicker to the question is there MUST be exactly five 1's ...
0
votes
1answer
28 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 ...
2
votes
2answers
31 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 ...
0
votes
0answers
25 views

Geometrical interpretation of the following transformation

I am learning linear algebra from Linear Algebra by Hadely and I came across this question that I do not have any idea how to solve Interpret geometrically the transformation produced on $E^2$ by ...
10
votes
1answer
62 views

Largest determinant of a real $3\times 3$-matrix

What is the largest determinant of a real $3\times 3$-matrix with entries from the interval $[-1,1]$ ? A result of John Williamson says that the largest value is equal to $4$, if the entries are just ...
0
votes
0answers
11 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
votes
1answer
22 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 ...
3
votes
1answer
34 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
votes
1answer
23 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
votes
1answer
20 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 ...
0
votes
1answer
17 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
votes
2answers
35 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
45 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
22 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
25 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 $A(V)$, the ring of linear transformations on $V$, satisfies a polynomial of degree $2$ over $\mathbb F$.
2
votes
1answer
29 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
votes
1answer
25 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
31 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
votes
0answers
19 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 & ...
1
vote
0answers
20 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
votes
0answers
25 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 ...
0
votes
1answer
27 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 ...
4
votes
1answer
33 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\\ ...
1
vote
1answer
43 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
votes
1answer
36 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 ...
1
vote
0answers
20 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
0answers
48 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
31 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
votes
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: $-sx+sy=0$ ...
2
votes
1answer
26 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 ...
0
votes
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 ...
-1
votes
1answer
28 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
13 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 ...
0
votes
1answer
51 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 \\ ...
3
votes
1answer
41 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
32 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
vote
2answers
31 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
2answers
58 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 ...
1
vote
1answer
44 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 ...
1
vote
0answers
22 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 ...
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, ...
0
votes
0answers
38 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 = ...
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 ...
4
votes
2answers
65 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
15 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 ...