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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2
votes
1answer
120 views

$R[X]/(f)$ separable $\iff \Delta(f) \in R^*$?

Let $R$ be a commutative ring with $1$. Let $f \in R[X]$ be monic. I have to prove the following: $$ \text{The discriminant of } f \text{ is invertible} \quad \quad \iff \quad \quad R[X] / (f) \text{ ...
0
votes
1answer
33 views

Find the matrix of the orthogonal projection onto the line spanned by the vector $v$

Let V be the plane with the equation $x_1 + 2x_2 + 3x_3 = 0$ in $\mathbb{R}^3$. Find the matrix of the orthogonal projection onto the line spanned by vector $$ \begin{vmatrix} 1\\ -2\\ 1\\ \end{...
0
votes
3answers
45 views

How can two vectors, with three elements each, form a base of a two dimensional space?

I might have misunderstood something(most likely the case), but there's an example. Assume this matrix: $\begin{bmatrix} 1& 2&3 \\ 1& 1& 2\\ 1& 2 &3 \end{bmatrix}$ ...
-2
votes
0answers
46 views

Given det(A) = 2. Find the Determinant of this Matrix

I've run into a roadblock that my textbook doesn't seem to be able to help me with. I am not understanding how to solve these type of questions. I am assuming to receive the answer given, you do some ...
0
votes
2answers
55 views

Recursive matrix multiplication strassen algorithm

I am having a hard time doing 4x4 matrix multiplication using strassen's algorithm. First I computed the product of two 4x4 matrices using default matrix multiplication (https://matrixcalc.org) I ...
4
votes
2answers
65 views

Solve linear system with $A_{i,j} = \langle e_i, e_j\rangle^2$, edges of a triangle

I have three vectors in $e_i\in\mathbb{R}^3$ that form a triangle. Let us consider now the linear equation system $Ax=b$ with $$ A_{i,j} = \langle e_i, e_j\rangle^2,\\ b_i = \langle e_i, e_i\rangle. $$...
0
votes
0answers
108 views

Calculating probabilities in a Markov chain process

I have 3 variables A, B and C with each variable having a probability of 0.6 and 0.4 i.e. A can have states (ON) with probability of 0.6 as well as can remain in certain states (OFF) with probability ...
4
votes
2answers
60 views

Prove an equality of complex matrixes

If $A \in M_2(\mathbb{C})$ a matrix so that $$\det\left(A^2 + A + I_2\right)=\det\left(A^2 - A + I_2\right)=3 \tag1$$ then $$A^2\left(A^2 + I_2\right)=2I_2. \tag2$$ I tried to use Cayley-Hamilton ...
1
vote
0answers
44 views

Divergence when spectral radius is greater than one in an iterative map.

Let $(M_n)$ be a convergent sequence of matrices from $\mathbb{R}^p$ to $\mathbb{R}^p$. Each element of the sequence has the same spectral radius $\sigma$, and $\sigma\ge1$. Show that there exist an $...
2
votes
1answer
63 views

The variance of the expected distortion of a linear transformation

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the "average distortion" caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). ...
1
vote
1answer
26 views

What is the expected distortion of a linear transformation?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$. I am interested in the "average distortion" caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). Consider for instance the ...
4
votes
2answers
114 views

Symmetrical and skew-symmetrical part of rotation matrix

Every matrix can be decomposed to symmetrical and skew-symmetrical part with the formula: $ A=\dfrac{1}{2}(A+A^T)+\dfrac{1}{2}(A-A^T)$. However if it is known only symmetrical part (we assume here ...
1
vote
0answers
43 views

Given matrices A and B, how does one find C such that $ C^{-1}AC = B$?

Given matrices A and B, how does one find C such that $ C^{-1}AC = B$ ?
0
votes
1answer
19 views

Spectral radius of block-skew-hermitian matrix equals norm of block

$$\rho\left(\left[\begin{matrix}0 & A \\ -A^{\dagger} & 0\end{matrix}\right]\right)=\|A\|$$ where $\rho(\cdot)$ is the spectral radius, $\|\cdot\|$ is the induced 2-norm. Question: I am ...
0
votes
0answers
28 views

Linear transformation of positive definite diagonal matrix

Let $\mathbf \Psi$ denote the set of all positive definite, diagonal, nXn dimensional, real-valued matrices . Let $\mathbf \Phi$ denote the set of all positive semi-definite, diagonal, nXn dimensional,...
0
votes
0answers
17 views

Update in Matrix factorisation

For movie recommender system based on matrix factorisation, the joint latent space for movies and users are constructed from the sparse user-movie rating matrix. $\min_{q*,p*} \sum_{(u,i)∈k} (r_{ui}-...
0
votes
2answers
29 views

About matrix diagonalization in C from the characteristic polynomial.

Ok the excercise is: You have one characteristic polynomial, it's: $\lambda^4 + \lambda^2$ Find two matrixes with this polynomial, one of them diagolalizable in C and the other one not. so the ...
3
votes
2answers
95 views

Derivative of $X_u A X B X_u^T$ w.r.t. $X_u$

How to solve this $\frac{d X_u A X B X_u^T}{d X_u}$, where $X, A, B \in \mathbb{R}^{n \times n}$ and $X_u$ is the $u$-th row in $X$?
4
votes
2answers
73 views

A generalization of holomorphic functions

Let's fix a matrix $A\in M_{2}(\mathbb{R})$. Assume that the following vector space of smooth functions is closed under complex multiplication: $$\mathcal{S}_{A}=\{f:\mathbb{C}\to \mathbb{C}\...
0
votes
0answers
21 views

Making a particular matrix expression positive-definite

I have the real matrices $S_1\in\mathbb{R}^{k\times n}$, $S_2\in\mathbb{R}^{k\times n}$, $V\in\mathbb{R}^{k\times k}$. I know that $V$ is symmetric positive definite. What properties do $S_1$ and $S_2$...
6
votes
2answers
91 views

Bridges across a tiled floor

A few years back, a friend of mine did a seminar on "Bridges across a tiled floor". A "bridge" was defined as a row or column of an $n \times n$ binary matrix consisting entirely of $1$'s, for ...
7
votes
3answers
109 views

Inverse of the Pascal Matrix

Let $P_n$ be the $(n+1) \times (n+1)$ matrix that contains the numbers of Pascal's triangle in the upper triangle. For example in the case of $n=3$ $$ P_3 = \begin{pmatrix} 1 & 1 & 1 & 1 \...
1
vote
1answer
69 views

$3\times 3$ matrix with eigenvalues are given

If $\displaystyle P=\begin{bmatrix} 0 & -2 & 3\\ -1& 1& -1\\ a & 2 & b \end{bmatrix}$ for some $a,b\in \mathbb{R},$ suppose $1$ and $2$ are eigen values of $P$ and $\...
1
vote
0answers
35 views

Prove that the determinant of a triangular matrix only has one non zero permutation term

Using permutations explain how for a triangular matrix only one term can be non zero. Please do not include any proofs using the cofactor method. Edit (OP's attempt as written in the comment ...
1
vote
1answer
24 views

Interpreting results of matrix transform

This is my first post on Mathematics Exchange, so I hope you'll be easy on me! I'm trying to project points in one 2-d coordinate space into another 2-d coordinate space using a simple matrix ...
0
votes
2answers
27 views

Differences of matrix exponentials

Let $T:V\rightarrow W$ be a linear map of inner product spaces with $T^\ast$ the dual map. I am to calculate $f(\lambda)=\operatorname{tr}e^{-\lambda T^\ast T}-\operatorname{tr}e^{-\lambda TT^\ast }$. ...
1
vote
1answer
29 views

Factorization of 2x2 rational matrices

I'm trying to show that any $2\times 2$ rational matrix $M$ with positive determinant can be factored as $M=SB$ where $S$ has integer matrices, $\det S = 1$, and $B$ is upper triangular. I haven't ...
1
vote
2answers
28 views

Complex matrix 2x2 eigenvalues determination.

So i have a matrix A: \begin{pmatrix} i & i\bar i \\ 1-i &\bar i \end{pmatrix} How can i calculate the eigenvalues of a complex matrix like this? I already know how to do if the matrix is ...
2
votes
1answer
81 views

$\hat{e_\mu } \cdot \hat{e^\nu } \neq \delta _{\mu} ^{\nu}$? Tensor algebra question.

Let $\hat{e_{\mu }}$ and $\hat{e^{\mu }}$ be the co- and contravariant basis vectors, respectively, for an arbitrary coordinate system Is it true that sometimes, $\hat{e_\mu } \cdot \hat{e^\nu } \neq \...
0
votes
1answer
45 views

minimal polynomial of an easy $3\times 3$ matrix.

So I have a $3\times 3$ matrix, lets say... $$ A= \begin{pmatrix} 1 &1 &0 \\ 0 &1 &1 \\ 0 &0 &1 \end{pmatrix}.$$ If I calculate the characteristic polynomial, I get $(1-...
2
votes
1answer
40 views

Is this how eigenvalues of some matrix $A$ are related to the inverse of $A$?

Let $A$ be an invertible $n\times n$ matrix. If $$Av = \lambda v \qquad (1)$$ for some $v$ and $\lambda$ then $\lambda$ is an eigenvalue of $A$ and $v$ a corresponding eigenvector. Equation $(1)$ may ...
1
vote
0answers
79 views

How does the error estimate change in the Kalman filter?

I am currently learning about the Kalman filter. A typical exam question is In which steps does the uncertainty $P$ get bigger / smaller? This is how I think the Kalman filter works / which ...
0
votes
1answer
33 views

how would you describe the range of T, space of $2\times 2$ real matrices.

hope you can help me, I've got stuck on this, I'm really new to linear algebra. Let $V$ be the set of complex numbers regarded as a vector space over the field of real numbers. We define a function $...
2
votes
1answer
42 views

To distinguish among the various subsets of $M_n(\Bbb R)$

I am having problem in doing a certain type of problems relating to matrices: To distinguish among the various subsets of $M_n(\Bbb R)$ such as symmetric, diagonal, diagonalizable, upper triangular, ...
1
vote
0answers
99 views

Find the limit of the mediant sequence $\frac{p_{n+1}}{q_{n+1}}=\frac{ap_n+bk_n}{aq_n+bm_n},~\frac{k_{n+1}}{m_{n+1}}=\frac{cp_n+dk_n}{cq_n+dm_n}$

How to find the general formula for the limit of the sequence: $$\frac{p_{n+1}}{q_{n+1}}=\frac{ap_n+bk_n}{aq_n+bm_n}$$ $$\frac{k_{n+1}}{m_{n+1}}=\frac{cp_n+dk_n}{cq_n+dm_n}$$ $$a,b,c,d>0$$ $$...
0
votes
1answer
16 views

Complexity of LUP decomposition of tri-diagonal matrix to solve an equation?

Doing LU decomposition of tri-diagonal matrix and then solving the eqn by using forward substitution followed by backward substitution is done is O(n) time. http://www.cfm.brown.edu/people/gk/chap6/...
1
vote
1answer
32 views

How to prove the equivalence of these optimization problems?

I am reading some lecture notes and in one procedure step it is stated that: $$\min_{\mathbf{x}}\; \langle \mathbf{H}, \mathbf{Rx-Z}\rangle + \frac{\lambda}{2} \|\mathbf{Rx-Z}\|_F^2$$ is equivalent to ...
-1
votes
1answer
30 views

Calculating the eigenvalues [closed]

I'm trying to understand the dynamics of the eigenvectors and the eigenvalues. My question is about formula for finding the eigenvalues. At 4:15(the athor starts the calculating at 1:30) of the given ...
2
votes
1answer
56 views

Express this sum as a matrix equation

Suppose I have matrices $G,H\in \mathbb{R}^{N\times K}$ and vector $w\in\mathbb{R}^K$. I want to express $$ \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^K w_k g_{ik}^2 h_{jk}^2 $$ as a matrix equation as ...
1
vote
2answers
44 views

What is $e^{A}$ where A is an anti-diagonal matrix

I am trying to get a closed form for the matrix produced by the following operation: $$e^A$$ where $A$ is an anti diagonal matrix, say, of size $2\times 2$: $$A=\begin{pmatrix} 0 &b \\ c &0 \...
0
votes
1answer
38 views

Matrix projection along another matrix

I vaguely have this idea mentioned to me by somebody, but neither can I find anything exactly like it anywhere no can I prove/disprove it. Given two matrices $U \in \mathbb{R}^{m \times n},Z \in \...
3
votes
1answer
69 views

nullity of infinite matrix A equals nullity of $A^T$?

Suppose you have an infinite matrix A with real entries. I know the dimension of the null space of A. Question 1) if the dimension of the null space is a finite number k, is the dimension of the null ...
8
votes
4answers
119 views

Matrix equation $A^2+A=I$ when $\det(A) = 1$

I have to solve the following problem: find the matrix $A \in M_{n \times n}(\mathbb{R})$ such that: $$A^2+A=I$$ and $\det(A)=1$. How many of these matrices can be found when $n$ is given? Thanks in ...
7
votes
3answers
155 views

How much can we tell about $\det(X)$ if we know $\det(I + X)$?

What can we tell about $\det(X)$ if we know $\det(I + X)$? Will it give some kind of bound for $\det(X)$? In general, if we know the determinant of matrix $A + X$, where $A$ is a constant matrix, how ...
0
votes
2answers
42 views

transition matrix for urn model

There are slides regrading to urn model I have two questions 1.if a Species A dies and a Species A is born, the original text says the probability is 0.4*0.4, but since a Species A has died , ...
1
vote
1answer
22 views

Calculating adjoint operator for standard inner product of matrices

Let $V=M_2(\mathbb R )$ with $ \left\langle A,B \right\rangle =\operatorname{tr}(B^tA)$. Define $$ T\begin{pmatrix}a& b\\c& d\end{pmatrix}=\begin{pmatrix}3d& 2c\\-b& 4a\end{pmatrix}. ...
0
votes
0answers
10 views

Wishart plus scalar multiple of identity

Is the sum of a Wishart distributed matrix and a scalar multiple of identity matrix, another Wishart distributed matrix? I guess it is not. If not, what is the distribution called and can its density ...
0
votes
0answers
25 views

Representing matrix of orthogonal projection

In an exercise I was given an 3d inner product space and a basis for a subspace and had to orthogonalize it and complete it into an orthogonal basis for the whole space. Then, I was told to find the ...
1
vote
0answers
34 views

Similar matrix for numerical computations

I compute numerically a symmetric matrix $W$ from the flow of a ode. I have to check numerically if this matrix is definite-positive. Two cases: either I use the Cholesky algorithm :ok or I ...
0
votes
1answer
51 views

Prove that determinant of a 2x2 symmetric positive definite matrix is positive by “completing the square” method.

From my understanding, determinant = product of Eigen values. Since it is a positive definite matrix, the eigen values are positive and hence, the determinant is ...