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
votes
1answer
25 views

If $A$ and $B$ are $n\times n$ matrices, then $((AB)^{-1})^T=(A^{-1})^T (B^{-1})^T$

Please help me to solve this. Prove that if $A$ and $B$ are $n\times n$ matrices, then $((AB)^{-1})^T=(A^{-1})^T (B^{-1})^T$. a problem involve transpose and inverse of matrices. check the ...
11
votes
0answers
263 views
+50

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
2
votes
1answer
59 views

Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
102
votes
5answers
4k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
0
votes
0answers
14 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
-1
votes
2answers
17 views

Finding the values of the rank and nullity

$Q$ is a $3 \times 3$ matrix that is not invertible. What are all the possible values of the rank and nullity of $Q$ ?
1
vote
2answers
41 views

Linear maps of polynomials, the bases of the space and their corresponding matrix.

Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by: $$Tp(x)=(x^2p(x))',$$ for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} ...
7
votes
1answer
52 views

A is a product of two self-adjoint matrices if and only if A is similar to adjoint of A?

How can we prove that $A$ is a product of two self-adjoint matrices $X, Y$ if and only if $A$ is similar to $A^\ast$? I'm thinking about proving it but no useful techniques come to my mind. Thanks!
0
votes
1answer
56 views

Theorems restricting the eigenvalue of a matrix

I have a square matrix $C$, whose entries I will denote by $c_{ij}$, and I would like to bound the magnitude of its eigenvalues. Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows: ...
0
votes
1answer
20 views

Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$

Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$, then find the value of $Det(A-kadj(A))$ My attempt: $Det(A+kadj(A))=0$ ...
1
vote
0answers
36 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
0
votes
1answer
19 views

Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify ...
0
votes
0answers
26 views

If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all $p$ positive integer?

I have got some questions regarding matrix norms and inequalities. We only consider square, nonsingular matrices in the following. If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all ...
1
vote
1answer
49 views

Integral of exponential with linear term

$$x \in \mathbb{R^n},$$ M is a positive symmetrical nonsingular nxn Matrix and j is an arbitrary vector in $$\mathbb{R}^n.$$ The following has to be calculated: $$Z(j) = \int_\mathbb{R^n} ...
0
votes
2answers
28 views

System of linear equations - Resolution

$$ \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) = \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) \begin{bmatrix} 0.6 & 0.3 & ...
3
votes
2answers
48 views

Intuition behind: Integral operator as generalization of matrix multiplication

So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications. In ...
0
votes
1answer
34 views

Find the dimension and a basis of a subspace

Let $U$ is the set of all commuting matrices with matrix $A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}$. Prove ...
0
votes
2answers
23 views

Finding the general solution of a system of linear equations

so I've come across this question in preparation for an exam: Let $A$ be a $4\times 4$ matrix where $rank(A)=3$. The vectors $(1,2,0,-1),(0,2,1,1)$ are solutions to the system ...
14
votes
3answers
4k views

Are one-by-one matrices equivalent to scalars?

I am a programmer, so to me $[x] \neq x$—a scalar in some sort of container is not equal to the scalar. However, I just read in a math book that for $1 \times 1$ matrices, the brackets are often ...
0
votes
1answer
24 views

Multiplication of block matrices

Let $J_{m \times n}$ be an $m \times n$ matrix of $1$'s (and to abbreviate we write $J_m=J_{m \times m}$) and let $M=\begin{pmatrix} 0_n & J_{n \times m} \\ J_{m \times n} & 0_m ...
0
votes
1answer
20 views

How to calculate the variation of a matrix?

Suppose we have two diagonal matrices $$ A_{\mu \nu}=\left(\begin{array}{cccc} \rho(t) & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0 ...
2
votes
1answer
40 views

Group isomorphism between two groups .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group. Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} ...
2
votes
1answer
26 views

Check if two square matrices are similar.

Check if matrices $A= \begin{bmatrix} 1 & 1 & 5 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 7 ...
0
votes
1answer
34 views

What is the derivative of matrix vector product $(A^Tx)$ with respect to A?

What is the derivative of a vector with respect to a matrix? Specifically, $\frac{d(A^Tx)}{dA} = ? $, where $ A \in R^{n \times m}$ and $x \in R^n$.
0
votes
1answer
27 views

How to show that $M_{2\times 2}(\mathbb{R})=W_1\oplus W_2$ based on the following assumption?

Let the subspaces $W_1=\{\begin{pmatrix}a&b\\-b&a \end{pmatrix}|a, b\in \mathbb{R}\}$ and $W_2=\{\begin{pmatrix}c&d\\d&-c \end{pmatrix}|c, d\in \mathbb{R}\}$ of $M_{2\times ...
3
votes
2answers
59 views

Derivative of trace of matrix

I'm new to matrix calculus and I have a problem with my assignment. Following is a function of trace of matrices: $$ f = \mathrm{tr}[\mathbf{X} \mathbf{X^T}] - \mathrm{tr}[\mathbf{X} \mathbf{H^T} ...
1
vote
2answers
24 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...
0
votes
3answers
33 views

How can you enlarge a shape about a point other than (0,0), using matrices?

If I want to enlarge a shape, $A$, by scale factor $k$ about $\left(0,0\right) $ I multiply each point (in the form $\begin{bmatrix}x\\y\end{bmatrix}$) by $kI$. However, I can't work out a general ...
0
votes
0answers
31 views

In matrix algebra, what's the name for the inverse operation of pre- or post- multiplication?

For example, in this typical equation: $$\mathbf{Mv}-\lambda \mathbf{v}=\mathbf{0}$$ (where $\mathbf{M}$ is a symmetric matrix, $\mathbf{v}$ is a vector, $\lambda$ is a scalar, and $\mathbf{0}$ is a ...
2
votes
1answer
27 views

For every integer $n>1$ , does there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $AD=DA $ holds only if $A$ is diagonal?

Is it true that for every integer $n>1$ , there exist a diagonal matrix $D \in M(n,\mathbb R)$ such that $A \in M(n,\mathbb R)$ and $AD=DA \implies A$ is also a diagonal matrix ?
3
votes
3answers
35 views

Find the Jordan normal form of a nilpotent matrix $N$ given the dimensions of the kernels of $N, N^2, N^3$

Let $N\in \text{Mat}(10 \times 10,\mathbb{C})$ be nilpotent. Furthermore let $\text{dim} \ker N =3 $, $\text{dim} \ker N^2=6$ and $\text{dim} \ker N^3=7$. What is the Jordan Normal Form? The only ...
0
votes
1answer
710 views

gradient of vector 2-norm

I have a function $f(\Theta) = \frac{1}{2N}\| y-\mathcal{X}(\Theta)\|_2^2$. Matrix $\Theta\in\mathbb{R}^{m_1\times m_2}$, $y=[y_1,\cdots,y_N]^T\in\mathbb{R}^N$ is the observation vector, and we use ...
0
votes
1answer
43 views

If $\operatorname{rank}A=k$ then $A=A_1+…+A_k$ such that $\operatorname{rank}A_i=1$ [closed]

Let $A\in M_n$ and $\operatorname{rank}A=k$. Is the following true? There are $A_i\in M_n$ ($i=1,...,k$), such that $\operatorname{rank}A_i=1$ and $A=A_1+....+A_k$.
1
vote
2answers
56 views

How to do QR Factorisation of a matrix

given A = $\begin{bmatrix}1 & 0 & 3\\2 & -6 & 3\\ -2 & 3 & -3\end{bmatrix}$ How would i find the QR factorisation? Well i have a guide on how to do this and have attempted ...
0
votes
1answer
27 views

Demonstration of the uniquenes of a QL matrix factorisation

One of my teacher say to us that the QL decomposition is unique. I am not convinced. How can I demonstrate the decomposition is unique?
0
votes
2answers
38 views

Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
0
votes
2answers
29 views

how to find the index of following subgroup?

if I denotes the principal congurence group of level 2 i.e. $I=\{ M \in SL(2,Z) ; \:M \:\:\text{congruent to I} \mod(2)\}$. or I= ...
1
vote
0answers
17 views

$SU(n)$ generators

What is the generalization of the Pauli matrices and Dirac matrices in higher dimensions? I am actually looking for $\sqrt{\mathbb{I}}$ but I can't use the principal root which is just $\mathbb{I}$. ...
1
vote
0answers
35 views

Is $U(1)$ a normal subgroup of $U(2)$ and does the question even make sense?

I have been wondering whether $U(1)$, defined as the group of complex phases (edit for clarity: I mean complex numbers of unit absolute value, such as $e^{i\alpha}$ with $\alpha \in \mathbb{R}$) with ...
1
vote
0answers
39 views

Getting The Inverse Of A Positive Definite Matrix By Mutiplying It On A Diagonal One

Is the following true ? The inverse of a positive definite matrix is also positive definite and since symmetric then we could write the following: $A=PP^T, \space B = A^{-1} = P\Lambda P^T$ since ...
1
vote
1answer
49 views

Find “almost inverse” of positive definite bilinear form

Let $A$ be a positive definite $d \times d$ matrix, and define $A(x,x)=x^TAx$. Let $x$ be a point such that $\vert x^T\xi\vert^2\leq \xi^T A\xi$ for all $\xi\in\mathbb{R}^d$. Is this somehow ...
0
votes
2answers
30 views

How can we know if the minimal polynomial of a matrix has no multiple products?

If the characteristic polynomial $f_A(x)$ has multiples of the same product, for example $f_A(x)= (x+2)^2(x-1)$ so $(x+2)$ has a multiple of $2$, then is there a condition on $A$ such that we know ...
0
votes
0answers
10 views

how to write amatlab code for document representation using second order tensor [closed]

My project is on document representation using tensor.How i will represent a document using second order tensor in matlab?
1
vote
2answers
143 views

The formula $\DeclareMathOperator{tr}{tr}\mathrm{adj}(A)=\tfrac{1}{2}[(\tr A)^2-\tr(A^2)]I_3-[\tr A]A+A^2$ for the adjoint of a $3\times 3$ matrix

Let $A$ be a square matrix of order $3$. Prove that $$ \operatorname{adj}(A) = \tfrac{1}{2} \bigl[ (\operatorname{tr} A)^2 - \operatorname{tr}(A^2) \bigr] I_3 - [\operatorname{tr} A] A + A^2 ...
0
votes
1answer
28 views

System of differential equation (Matrix form)

I'm trying to solve this system $$ M\ddot{X}(t) = KX(t) $$ where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = ...
0
votes
1answer
26 views

Is $X'X$ positive definite a necessary condition for $X'X$ to have full rank?

Let $X$ be a $T \times K$ matrix. $X'X$ positive definite means that for all $c \not = 0$ $c'X'X c >0$, so then $(Xc)' Xc > 0$ which implies $(X\cdot c)\cdot (X\cdot c)$ (I'm not sure what ...
0
votes
1answer
27 views

How can a non-zero matrix $A$ be found such that Adj$(A) = 0$? [closed]

Is it possible to find a $3 \times 3$ matrix $A$ such that it's adjoint is $0$?
1
vote
0answers
40 views

How to find out generators of the following free group?

The following is the quotient group of SL($2,\mathbb{Z}$). Consider $(H/\{-1,1\} \cap H)$ where $H=\begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}$ How do ...
0
votes
1answer
28 views

Find upper triangular matrix C such that Cx=y

In the image above, how does one know that $c=e$ and $c$ is not equal to $f$? and $e$ is not equal to $f$? How does one know that $b=d$?
0
votes
0answers
24 views

Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...