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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0
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0answers
29 views

Change of Basis between linear Transformations

I am trying to get a better understanding in change of basis with matrices and linear transformations, therefore I am using several linear Transformations $^{i-1}A_i=\begin{bmatrix} \cos\theta_n ...
0
votes
1answer
31 views

Rationnal canonical form of the matrix $A$

Let the matrix \begin{equation} A=\begin{bmatrix} 2 & 1 & 2 \\ -2 & -1 & -4 \\\ 1 & 1 & 3 \end{bmatrix}. \end{equation} So far I found the characteristic polynomial ...
0
votes
0answers
25 views

How to formalise a procedure involving Cartesian products of sets of vectors and transformation in matrices?

I am asking for an help to formalise with the correct notation the following procedure. Let $n\in \mathbb{N}$. Let $\{0,1\}^{n-1}$ be the set of vectors of dimension $(n-1)\times 1$ with each ...
0
votes
0answers
17 views

Covering number of the set of $n_1\times n_2$ matrices of rank at most $r$

What is the covering number of the set of $n_1\times n_2$ matrices of rank at most $r$? We know that the dimension of the set is $r(n_1+n_2-r)$. Thus, the covering number $N(\rho)\le C ...
5
votes
2answers
72 views

If $BA = I$, prove that $AB = I$ (using determinants)

I've seen this problem around here, but I wanted to check if this particular solution is right. So, if $BA = I$, then $det(B)det(A) = 1$, meaning neither $det(B)$ or $det(A)$ are equal to $0$. ...
-1
votes
0answers
18 views

Let $M=A^{T}A$ be a positive semidefinite matrix. Is it true that for $i \neq j$, $|m_{ij}| \leq \frac{m_{ii}+m_{jj}}{2}$ [duplicate]

Let $M=A^{T}A$ be a symmetric positive semidefinite matrix. Is it true that for $i \neq j$, $|m_{ij}| \leq \frac{m_{ii}+m_{jj}}{2}$? Where $m_{ij}$ is an element of matrix $M$, and $i$ represents the ...
0
votes
1answer
20 views

Square root of a complex symmetric matrix?

Is it possible to express a complex symmetric matrix $A$ as square of a matrix $B$ (i.e. $A = B^2$)? If $A$ were Hermitian, we could use Spectral Theorem to get $A = UDU^{-1}$ where $D$ has diagonal ...
0
votes
1answer
16 views

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$.

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$. $A$, $B$, and $C$ are square matrices of the same size. $\hat{c}_j$ is the $j$th column of $C$, $\hat{a}_k$ are the columns of $A$, ...
0
votes
0answers
31 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
1
vote
2answers
29 views

Find a matrix $M$ such that $M^TAM = I$

I have that my matrix $A = \begin{bmatrix}4&0&3\\0&1&0\\3&0&4\end{bmatrix}$, I've done diagonalization but now finding a matrix $M$ and its transpose acting as a conjugate for ...
0
votes
0answers
10 views

In a transformation matrix, why is $Y$-axis ($-\sin$) in left most column as opposed to right like X and Z [duplicate]

$4 \times 4$ Transform Matrix with axis columns $XYZ$ left to right $X$-axis rotation: $$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & \cos & -\sin & 0 \\ 0 & \sin & \cos ...
2
votes
1answer
15 views

determinant inequality for Hermitian matrix

$A \in \mathbb{C}^{M \times M}$ is a positive semidefinite matrix with all diagonal entries being $1$. and the vector $\mathbf{y} \in \mathbb{C}^{M}$ has entries $|y_{i}| < 1$. Prove that $$2 ...
1
vote
0answers
18 views

$k$-th diagonal element of an inverse matrix $(\textbf{H}^{\dagger}\textbf{H})^{-1}$

Let $\textbf{W}=\textbf{H}^{\dagger}\textbf{H}$ with $\{\cdot\}^{\dagger}$ being conjugate transpose operator. In some materials I have read so far, there is a common statement that the $k$-th ...
3
votes
2answers
39 views

Property of group of $k \times k$ orthogonal matrices

Does the group of $k \times k$ orthogonal matrices lie on an $(k^2 - 1)$-sphere? If so, of what radius? If not, does it lie on some sort of other object?
0
votes
0answers
23 views

Cardinality of stabilizer in $PSL(2,\mathbb{Z}_7)$

Let $v$ be a non-zero vector in $\mathbb{Z}_7^2$ (up to scaling), and let P be the stabilizer of $v$ (up to scaling) in the Projective special linear group $PSL(2,\mathbb{Z}_7)$. Find the Cardinality ...
0
votes
0answers
43 views

Why isn't the identity/unit matrix upright?

I realize this is more of a typesetting problem then a mathematical one. I've already tried the TeX stack exchange and the question got canned. In ISO 80000-2:2009, variables and running numbers are ...
1
vote
1answer
37 views

Vector space endomorphisms in $\mathbb{R}[x]$ commuting with $E:f\mapsto f+f'$

I am wondering if every vector space endomorphism in $\mathbb{R}[x]$ that commutes with $E:f\rightarrow f+f'$ is invertible. (denoting $f'$ the derivative of $f$) To begin with, $E$ is invertible ...
-3
votes
0answers
19 views

Properties of Determinants - True or False [on hold]

Can you help me answer these true or false questions for an n x n matrix A? I think that 3 and 10 are actually false Picture of the problem The determinant of a lower-triangular matrix A is the sum ...
0
votes
0answers
29 views

Divisibility of dimension by matrix equations.

Let ${\bf M}$ and ${\bf N}$ be two real $k\times k$ matrices such that ${\bf M}^2+{\bf N}^2={\bf M}{\bf N}$. Show that if $\det\left({\bf N}{\bf M}-{\bf M}{\bf N}\right)\neq 0$, then $3\mid k$.
0
votes
1answer
17 views

Diagonal block matrices of a positive definite block matrix

Let $R=\begin{bmatrix} R^{11} & R^{12} & R^{13} \\ R^{21} & R^{22} & R^{23}\\ R^{31} & R^{32} & R^{33} \end{bmatrix}$ be a symmetric positive definite matrix where $R^{ii}$, ...
2
votes
0answers
29 views

If $v^T A^{-1} u = -1$, then the matrix $A + uv^T$ isn't invertible

Let $A \in M_n(\mathbb{R})$ be an invertible matrix and $u,v \in \mathbb{R^n}$. By the Sherman Morrison formula, we know that if $v^TA^{-1}u \neq -1$ then $(A + uv^T)^{-1}$ exists. I want to prove ...
1
vote
1answer
14 views

Finding the inverse of a map given in vector form.

The question asks me to find the inverse map $ \mathbf\Phi^{-1} $, of: $$ \mathbf{\Phi}(\mathbf{x})= \mathbf{n\lor(x \lor n)} + \alpha\mathbf{(n \cdotp x)n} $$ for $\alpha$ such that the inverse ...
2
votes
6answers
38 views

Determine whether $w$ is in the $Span\{v_1, v_2, v_3\}$

my question is how to determine whether a vector $w$ is in the $span\{v_1, v_2, v_3\}$. In this case: $w = \begin{bmatrix} 9 \\ 6 \\ 1 \\ 9 \\ ...
2
votes
2answers
65 views

When this matrix is diagonalizable?

When this matrix is diagonalizable? ($a_i \in \mathbb{R}$) $$ \begin{pmatrix} &&&a_1\\ &&a_2&\\ &\ddots&&\\ a_n&&&\\ \end{pmatrix} $$ I think I should ...
0
votes
0answers
35 views

Let A=$\tiny\begin{pmatrix}1&1&1\\1&2&2\\ 1 & 2 &3 \end{pmatrix}$ and B=$\tiny\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 &0 \\ 1 & 1 &1 \end{pmatrix}$

Then (A) there exists a matrix C such that A = BC = CB (B) there is no matrix C such that A = BC (C) there exists a matrix C such that A = BC, but A $\neq$ CB (D) there is no matrix C such that A ...
0
votes
1answer
10 views

Inverse of a correlation matrix when all the correlations are equal

Let's have variables that multivariate normally distributed and have the same correlation among each other (different variance). Can we analytically derive the inverse of the correlation/covariance ...
0
votes
1answer
30 views

Prove: If $[T]$ is nilpotent of degree $k$, a $v$ exists such that $\{v,T(v),T^2(v),…,T^{k-1}(v)\}$ is linearly independent

I really need a hint on this one. I tried to look at nilpotent matrices but could not find a v that satisfies.
1
vote
0answers
21 views

Solving a matrix equation problem

I've been trying to solve this matrix equation but I just can't find the way to do it. The task is: Find matrix $\mathbf{X}$ from the equation in the photo where $\mathbf{A, B, C}$ are given matrices ...
3
votes
1answer
65 views

Result about Matrices of form $B(AB)^{-1}A$

I am trying to prove the following result. So far my only idea was to try using the formula for inversion of block matrices, but that did not get me very far. Any help will be much appreciated. ...
0
votes
1answer
18 views

Consider the following change of basis matrix [on hold]

Consider the $n$ dimensional vector space $\mathbb{V}$ over the reals with two basis $B$ and $B'$. Show that the transition matrix maps coordinate vectors of the basis $B'$ to coordinate vectors in ...
0
votes
1answer
22 views

Gradient Chain Rule: Applying Gradient in the case of a Series of Matrix operations (Neural Net Gradient Calculation)

I have the following situation: I need to calculate the gradient of the Error of a CNN a few layers deep by hand. Starting with the Error function, The $\operatorname{Error}[readoutX]= -\sum_i ...
0
votes
2answers
42 views

How to prove that the square matrix $A_{n}$ matrix is nilpotent such that $A^{(n-1)}=0$

The matrix A looks like this: $$A=\begin{bmatrix} 0 & 1 & 0 & 0 & .&.&. &0\\ 0 & 0 & 2 & 0 & .&.&. &0\\ 0 & 0 & 0 & 3 ...
0
votes
2answers
62 views

If $A$ is normal and upper triangular then it is diagonal

Let $A$ be a normal matrix in Mat$_{n\times n}(\mathbb C)$, if $A$ is upper triangular then it is diagonal (Normal means $AA^*=A^*A$, where $A^*$ is the conjugate transpose of $A$) If I consider ...
0
votes
1answer
33 views

How to find out if some eigenvalues of a matrix are the same?

I know that in order for a matrix to have two equal eigenvalues, one term in the characteristic polynomial must be in the power of two. Is there any way to tell if two eigenvalues are the same? I have ...
6
votes
2answers
71 views

What is the determinant of []? [on hold]

I typed this in Matlab, but I can't understand why it returns the determinant one. A = [] det(A) ans = 1
0
votes
1answer
52 views

Square root of matrix that is a square of skew-symmetric matrix

Let's suppose we have a matrix $A$ (dimension $3\times 3$) which is the square of some skew-symmetric matrix $S$ i.e. $A=S^2$. How to obtain from $A$ its skew-symmetric square root $S$?
0
votes
2answers
20 views

Matrices Eqvialence Relation

How can I prove that $A\mathcal{R}B$ is an equivalence relation if there exists an invertible matrix $C$ such that $B = CA$? I know there there is a reflexive, symmetric, and transitive steps. ...
1
vote
3answers
49 views

Prove that the product of two invertible matrices also invertible

I am working on a homework problem, but I am lacking some understanding. Here is the problem: Let $A$ and $B$ be invertible $n \times n$ matrices with $\det(A) = 3$ and $\det(B) = 4$. I know that ...
0
votes
0answers
35 views

Are the only 2x2 real matrices with complex-conjugate eigenvalues the rotation matrices?

If so, how can I see this fact? I'm wondering if it's something fundamental that I am overlooking. Thanks,
-1
votes
0answers
46 views

Dimension of Algebraic Variety

What is the dimension of the algebraic variety formed by $2m \times n$ real matrices, where all $(r_1+1) \times (r_1+1)$ minors of the top $m \times n$ matrix vanish, all $(r_2+1) \times (r_2+1)$ ...
0
votes
0answers
16 views

Matrix Derivative/Operation of Flat(A).dot(B)?

In calculating the gradient of a convolutional neural net by hand, I am running into a snag. In the middle of the net, going forward, there is a layer where I take an array $A$, flatten it into a ...
1
vote
2answers
27 views

Expressing a function as a linear combination

Hi so I know how to express a vector as a linear combination of another vector but now I've been given a function and it's thrown me a bit, just wondering if anyone can assist. The question is: In ...
0
votes
1answer
38 views

Find the matrix X such as A . X is close to B

Consider : A an m by n matrix B an ...
0
votes
0answers
16 views

Help understanding a homework problem (Preconditioning matrices, numerical methods)

Below is a link to the problem (because I didn't want to have to go through the pain of TeXing it all out myself), the basic idea is we are supposed to be first showing that a specific matrix has a ...
0
votes
1answer
18 views

symetric matrix inverse

Is there an easy way to invert a 3x3 symmetric matrix? for example A = $\begin{pmatrix} -1& 2& 0\\ 2& -5& 0\\ 0& 0& ...
0
votes
1answer
24 views

How to denote dimensions

I am struggling with nomenclature. If I have matrix $M \in \mathbb{R}^2 \times \mathbb{R}^4$ it would be considered an element of an 8-dimensonal vector space. If I index $M$ by two indices $i$ and ...
1
vote
1answer
24 views

Does $A \succeq B A^{-1} B$ imply that $A \succeq B$?

Let $A,B$ be two symmetric matrices with equal dimensions. Suppose $A \succeq 0 $ (ie, PSD), $B \succeq 0$ and $$ A - B A^{-1} B \succeq 0.$$ Then is it true that $A-B \succeq 0 $?
-1
votes
0answers
14 views

My professor asked me to link the Singular Values of a matrix to the Input matrix ? what does he mean by that [on hold]

So i tried doing this solution Singular values = sqrt ( eigenvalues ( AA' ) ). But he said that it wasn't the correct answer . what does he mean by linking Singular Values to the Input matrix?
-1
votes
0answers
10 views

Exponential matrix using Laplace transform - reference request [on hold]

I am looking for a book that covers the concept of exponential matrix in detail using the Laplace transform plz
0
votes
0answers
12 views

Question with Gauss-Jordan elimination produces the matrix

Please help me explain the problem below: How can I use Gauss-Jordan to get all bottom roll become 0? Thank you so much!