Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

Filter by
Sorted by
Tagged with
1
vote
2answers
45 views

Why is the solution set of the reduced row echelon form of A equal to the solution set of A?

One way of solving a system of linear equations is to express it in an augmented matrix. Then, we can perform elementary row operations in order to bring the matrix into RREF (reduced row echelon form)...
3
votes
1answer
28 views

If matrix $A=[a_{ij}]_{4 \times 4}$ such that…

If matrix $A=[a_{ij}]_{4 \times 4}$ such that $a_{ij}= \begin{cases} 2&\text{if}\, i=j\\ 0&\text{if}\, i \not= j\\ \end{cases}$ , then $\{\frac{det(adj(adjA))}{7}\}$ is ($\{x\}$ ...
1
vote
0answers
26 views

Multiplication by an extrinsic matrix

I need to project a 2D point taken by a Kinect camera to real world coordinates, without using the official Kinect SDK. I found this answer: https://stackoverflow.com/questions/47348266/color-space-to-...
3
votes
1answer
51 views

Showing a matrix identity

Let $A$ and $B$ be symmetric, positive definite matrices. Consider the following equations: $$ B = XAX, \qquad A = YBY,$$ where $X$ and $Y$ are also symmetric and positive definite matrices. These ...
2
votes
0answers
21 views

Defective Matrices Transforms

Defective Matrix:$\begin{bmatrix}1&1\\0&1\end{bmatrix}$ I am unable to wrap my head around what kind of transforms yield defective matrices or matrices where geometric multiplicity is less ...
0
votes
1answer
22 views

Strong separation of symmetric rank $1$ matrices

Let $V$ be the inner product space of $n \times n$ symmetric matrices where the inner product is given by $\langle A, B \rangle = trace(AB)$ and let $A, B \in V$ be such that they have eigenvalue $1$ ...
3
votes
2answers
65 views

Same Eigenvalues = Similar Matrices?

While watching Strang's lecture on similar matrices he stated that if any two matrices,$A$ and $B$, have the same eigenvalues then they can be put in the form $A=P B P^{-1}$ . This is very easy to see ...
2
votes
1answer
27 views

A question on Matrix Norms

Suppose $\mathbf{A}\in\mathbb{R}^{n\times n}$, and $\mathbf{B}\in\mathbb{R}^{n\times n}$ be a matrix with strictly positive entries. Define ${\Vert\mathbf{A}\Vert}_{\mathbf{B}}=\max\limits_{1\leq i,j \...
1
vote
0answers
23 views

symmetric matrix can be written as a matrix exponential of symmetric matrix

If I have a symmetric matrix $\boldsymbol{B} = e^{\boldsymbol{\Theta}t}e^{\boldsymbol{\Theta}^{T}t}$ where $\boldsymbol{\Theta}$ is a matrix with transpose $\boldsymbol{\Theta}^{T}$, is it possible ...
7
votes
2answers
104 views
+50

Every matrix of the centralizer of the centralizer of a matrix is a polynomial in that matrix

Let $V=M(n,\mathbb C)$. For a subset $S \subseteq V$, let $C(S):=\{A \in V | AB=BA, \forall B \in S \}$ . How to prove that for every $A\in V$, we have $C(C (\{A\})) \subseteq \{ p(A) | p(t) \in \...
0
votes
2answers
34 views

If $\operatorname{Tr}(H)\ge 0$, is $\operatorname{Tr}(PH)\ge 0?\ $ $H$ is Hermitian, $P$ is positive definite.

Old: Given $\operatorname{Tr}(DH)\ge 0$ for positive real diagonal $D$ and Hermitian $H$, is $\operatorname{Tr}(H)\ge 0$? Let $D$ be positive real diagonal and $H$ be Hermitian such that $H$ may have ...
0
votes
0answers
21 views

minimization containing a large matrix

Consider a minimization problem with regard to $\theta$: $ Q(\theta) = [y-f(\theta)]'[y-f(\theta)]$ where $y,(m \times 1)$ is a data vector, $\theta$ is a $p \times 1$ vector to be estimated and $...
2
votes
0answers
32 views

Properties of Pauli Matrices

Using the three Pauli spin matrices $$\boldsymbol{\sigma}_{1}=\left(\begin{array}{ll}{0} & {1} \\ {1} & {0}\end{array}\right), \quad \boldsymbol{\sigma}_{2}=\left(\begin{array}{cc}{0} & {-...
0
votes
3answers
39 views

Is it okay to take basis from rref matrix as well as original matrix in both column space and row space?

Let $A$ be the given matrix of order $m \times n$. I want to find the basis for both row and column spaces of $A$. I transformed the matrix A in to its row reduced echelon form i.e., $rref(A)$. ...
2
votes
1answer
40 views

Is there a formula to calculate the euclidean distance of two matrices?

Wiki gives this formula ${\displaystyle {\begin{aligned}d(\mathbf {p} ,\mathbf {q} )=d(\mathbf {q} ,\mathbf {p} )&={\sqrt {(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}+\cdots +(q_{n}-p_{n})^{2}}}\\[8pt]&...
0
votes
0answers
20 views

Homogeneous transformation [on hold]

I'm confused with the general matrices for homogeneous transformation for example the matrix used for reflection: if we had say a vector V = 5i -4j 3k perpendicular, would i have to find the unit ...
0
votes
1answer
22 views

A Trace Bound Identity of Matrix Products (known for reals) in the Complex Space

Please refer to this beautiful paper on trace inequalities for matrix products. Theorem $3$ of the article (rephrased) states: For any real $n\times n$ matrix $A$ and any real symmetric $B$ of the ...
1
vote
0answers
45 views

Inner product preserving mapping

Let $A \in \mathbb{R}^{n\times n}$ and $B \in \mathbb{R}^{m\times m}$ be positive definite symmetric (pds) matrices. For a matrix $M_1, M_2 \in \mathbb{R}^{n \times m}$, I would like to know that ...
0
votes
0answers
40 views

Boolean matrix factorization when a set of possible factors is known

Suppose I have a Boolean matrix $M$ which is a $m \times m$ square matrix. And there is a set of $m \times m$ Boolean matrices $M_1, M_2, M_3,\ldots, M_7$ such that $M$ is guaranteed to be a product ...
0
votes
1answer
29 views

Show that $\text{trace}(A^TA) \ge 0$ and $\text{trace}(A^TA)$ if and only if $A = O$ [duplicate]

Problem Show that $\text{trace}(A^TA) \ge 0$ and $\text{trace}(A^TA)=0$ if and only if $A = O$ when $A \in \mathbb{R}^{n \times n}, n \in \mathbb{N}$. Symbol $O$ denotes zero matrix. Attempt to show ...
8
votes
1answer
86 views

Products of Determinantal Ideals

Let $X=(x_{ij})_{1\leq i\leq m, 1\leq j\leq n}$ where the $x_{ij}$ are variables. Now, for all $d$, we consider the ideal $I_d$ of the polynomial ring $S=\mathbb{C}[x_{ij}:\, 1\leq i\leq m, 1\leq j\...
1
vote
1answer
26 views

When is there an inner product making a given matrix unitary?

The unitarity of a given operator/matrix $A$ depends on the underlying inner product, so I guess it in principle possible for a given non-unitary $A$ to act as a unitary with respect to a different ...
1
vote
1answer
40 views

Compute the inverse of a matrix with partially known values

I have an odd problem. I have a matrix $A$ of size 16 per 16. As an example, we will consider the 4 by 4 matrix below. $$A = \begin{pmatrix}50 & 3 & 10 & 2\\\ 3 & 60 & 7 & 1\\\...
0
votes
1answer
22 views

Taking positive part commutes with conjugating with $Y\geq 0$ on hermitian matrices?

Let $X,Y\in\mathbb C^{n\times n}$ be hermitian and $Y$ positive semi-definite. Does $$ (YXY)^+=YX^+Y $$ hold, where $(\cdot)^+$ denotes the positive part of the respective hermitian matrix (i.e. if $A=...
6
votes
2answers
91 views

There does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F) $ for any field $\mathbb F.$

For a positive integer $n$, I have to show that there does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F) $ for any field $\mathbb F.$ If it ...
0
votes
1answer
37 views

Trouble derivating an equation with matrices and vectors.

I have the following equation $p=(1+r)(pA+wl)$ where w, r are scalars; p, l are vectors; A is a matrix. I want to prove that an increase in w would lead to a decrease in A or l (i.e., p is constant). ...
1
vote
2answers
41 views

Convert Rotation matrix to Euler angles $~zyz~ (y$ convention$)$ analytically.

The rotation matrix of Euler angle $ZYZ$ is: $$ R_{z1}=\left[\cos(\psi),\sin(\psi),0;-\sin(\psi),\cos(\psi),0;0,0,1 \right]; $$ $$ R_y=[\cos(\theta),0,-\sin(\theta);0,1,0;\sin(\theta),0,\cos(\theta)]; ...
2
votes
2answers
143 views

Use properties of determinant and show

Let $n$ be a positive integer and \begin{align} M = \begin{pmatrix} n! & (n+1)! & (n+2)! \\ (n+1)! &(n+2)! & (n+3)! \\ (n+2)! & (n+3)! & (n+4)! \\ \...
0
votes
1answer
20 views

How can one write $x^TAx=v^Ty$, where $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{n\times n}$ and $v,y$ are vectors of equal dimension.

I wish to express, $x^TAx$ ($x \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n\times n}$) which is a scalar quantity as product of two vectors, i.e, $v^Ty$, where $v$ should only contain elements of $x$ ...
0
votes
1answer
44 views

Approximation of a matrix finite geometric series [on hold]

So I would like to approximate or find an upper bound for the norm of a matrix finite geometric series of the type: $$ \left\| \sum_{i=0}^{v-1} (N+M)^i \right\|$$ where $N$ and $M$ are two matrices. ...
0
votes
3answers
42 views

Study the behavior of a 2x2 dynamical system with parameters

I need to study the behavior of this discrete dynamical system: $$\left\{ \begin{array}{c} x_{n+1}= y_n \\ y_{n+1}= \frac{1}{a}x_n-\frac{1}{a}y_n \\ \end{array} \right.$$ in function of the ...
1
vote
1answer
37 views

System of equations involving complex eigenvalues

Consider the following equation: $$ x_{n+2}-2ax_{n+1}+x_n=0$$ a) Define a new auxiliary variable $y_n = x_{n+1}$ and rewrite the previous equation as a discrete, two-equation dynamical system. b) What ...
0
votes
1answer
73 views

Lower bound $\operatorname{Tr}(MB)$ where $M>0$ and $B$ is arbitrary

Let $M, B$ be complex and square matrices. $M$ is positive definite (pd) and $B$ is arbitrary. If this is relevant, there is an afore-knowledge that $\operatorname{Tr}(MB)\ \ge \ 0$. Please help to ...
0
votes
1answer
22 views

The regular representation of finite group is always reducible.

I have to prove that the regular representation of a finite group is always reducible. This is equivalent to saying that the G invariant subspaces of it is the trivial subspace and the full subspace.I ...
2
votes
1answer
28 views

Right-inverse approximation in Frobenius Norm

Let $A \in \mathbb{R}^{m\times n}$, with $m \geq n$, be a matrix of rank $r$, and suppose we have a SVD decomposition $A = U\Sigma V^t$. We define the pseudo-inverse of $A$ as $A^{\dagger} := V\...
2
votes
0answers
21 views

Why are arithmetical operations on during row reduction counted in this way?

I am reading the section on counting arithmetic operations in Strang's book. It says: [The] operations are of two kinds. We divide by the pivot to find out what multiple (say $\ell$) of the pivot ...
1
vote
1answer
47 views

Solving recursive matrix system not fully correct

I'm having a problem with a personal project in which I want to determine the dynamics of a population. Suppose the following transition matrix is given by \begin{equation} T = \begin{pmatrix} 0.9 &...
0
votes
1answer
26 views

What are the necessary conditions for a matrix to have a complete set of orthogonal eigenvectors?

From my lecture notes, I learned that for a $N\times N$ real symmetric matrix $\mathbf{A}$, it is known that it has a complete set of $N$ orthogonal eigenvectors $\hat{e}^{k}$, with $k=1 \ldots N$ ...
0
votes
0answers
25 views

Condition on matrix to give a cylinder

The question is to put a condition on the 2x2 matrix $H$ which is in the form $$ \left( \begin{matrix} z & w\\ -\bar{w} & \bar{z}\\ \end{matrix} \right) $$ where z and w are complex numbers....
2
votes
2answers
41 views

Show that $\alpha=||x(t)||_2$ where $t> 0$.

Consider the initial value problem: $x^{'}(t)=Ax(t),x(0)=x_0$ where $t\ge 0$ Suppose that $A$ is a skew symmetric matrix and $\alpha=||x_0||_2$. Show that $\alpha=||x(t)||_2$ where $t> 0$. ...
0
votes
0answers
23 views

Block matrix multiplication example

Let \begin{equation} A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}, \qquad B = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 3 & 1 \\ 1 & 1 & 1 \end{pmatrix} \end{...
1
vote
2answers
42 views

Nonhomogenous matrix differential equation

In this paper, I've stumbled unpon the solution they give for an non-homogeneous matrix differential equation. Basically, the equation is: $\frac{\partial\boldsymbol{\psi(t)}}{\partial t} = \...
-3
votes
2answers
43 views

Prove of disprove.There exists 3x3 real valued matrix such that … [closed]

Prove of disprove this statement. There exists 3x3 real valued matrix B such that $B^2=A$. $$ A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\1 & 1& 0 \\ \end{bmatrix}$$
0
votes
1answer
43 views

Gradient of $X^{T}AX$ wrt matrix X

I need to calculate the gradient of $X^{T}AX$ with respect to $X$, where $X$ and $A$ are $nxn$ matrixes. Using the matrix cookboox here (pag 9), I see that the gradient is obtained for each element $...
1
vote
2answers
44 views

How can I find the rank of this linear transformation

Suppose $Q \in M_{3 \times 3}\mathbb(R)$ is a matrix of rank $2$. Let $T : M_{3 \times 3}\mathbb(R) \to M_{3 \times 3}\mathbb(R)$ be the linear transformation defined by $T(P) = PQ$. Then rank of T ...
2
votes
2answers
83 views

Prove that this determinant equals zero

I recently came up with a problem in which I need to use the fact that the determinant below is equal to zero : \begin{vmatrix} 0 & a_{21} & 0 & a_{41} & 0 & \dots&0 &a_{...
1
vote
1answer
37 views

To prove a property of a positive semidefinite matrix with the zero first entry.

I want to prove the following: If the matrix $$ M=\begin{pmatrix} 0&\vec{q}^T \\ \vec{q}&N \end{pmatrix} $$ is PSD, then $\vec{q}=\vec{0}$. The only three properties of a positive ...
1
vote
0answers
38 views

Comparing condition numbers

I have two matrices, $A$ and $B$ where $A,B \in R_{mxn}$ ($m>>n$). I need to answer the question: Which one is better-conditioned among $A$ and $B$ ? However, I will be making these comparisons ...
0
votes
1answer
11 views

start T in matrix equation

In many start equations I see T (transpose) or -1 (inverse). Why is there using T, but not original matrix? Example https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula ($\mathbf{A}$ + $\...
0
votes
2answers
41 views

Similarity transformation into symmetric matrix

I have a matrix of the form: $$ \begin{bmatrix} 0 & q & 0 & 0 & 0 & 0 & \cdots \\ p & 0 & q & 0 & 0 & 0 &...