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
3answers
90 views

are 12 different rotation matrix the same?

If I want to rotate a vector $V$ from coordinate system $A$ to $B$, I could use the rotation matrix by $V_B=R\cdot V_A$, where $R$ is the rotation matrix. There are many rotation sequences for $R$, ...
2
votes
3answers
58 views

Am I correct about this standard matrix and other definitions?

I have this linear transformation $T:\mathbb R^2\to\mathbb R^3$ such that $T\left[\begin{matrix}1\\0\end{matrix}\right]=\left[\begin{matrix}2\\3\\1\end{matrix}\right]$ and ...
1
vote
1answer
224 views

projection matrices with transposes

Is the identity matrix a projection matrix, and if so, is it the only projection matrix which is invertible? Also when considering $2\times 2$ matrices which satisfy $A^2=A^T$ what satisfies this? How ...
3
votes
0answers
128 views

Binary optimization

Let me first make my background clear. I am a PhD student with not much knowledge in optimization but I need to do some optimization as a part of my research work. My problem is as follows: There are ...
1
vote
4answers
2k views

Understanding regular matrices

A regular matrix $A$ is described as a square matrix that for all positive integer $n$, is such that $A^n$ has positive entries. How then would I prove something is regular? I mean I can prove ...
0
votes
0answers
1k views

Find the values of a,b,c such that a matrix has infinite, unique, and no solutions.

Find the values of a,b and c such that a matrix has infinite, unique, and no solutions. $$x+y=0$$ $$y+z=0$$ $$x+z=0$$ $$ax+by+cz=0$$ We can't use determinants so I turned the equations into an ...
5
votes
2answers
283 views

Sparse basis for linear subspace

Suppose I have a linear subspace of some vector space, e.g. described as the column space of some big matrix. How would I algorithmically find a basis of that same subspace where the basis matrix is ...
2
votes
2answers
59 views

Determinants of related matrices.

Given: det $\begin{bmatrix} r & s & t \\ u & v & w \\ x & y & z \\ \end{bmatrix}=4$, compute det ...
1
vote
1answer
414 views

Determinant of any squared submatrix of incidence matrix of any graph is 0, $(-2)^i$ or $2^i$?

Why the below proposition is true? For any graph, determinant of any submatrix of its incidence matrix is 0, $(-2)^i$ or $2^i$. ($i \in \mathbb{Z}$)
3
votes
1answer
130 views

smallest singular value

I know this question is a difficult one, but any advice/tip/reference/heuristic is welcome. Is there any good lower bound (other than $0$) on the smallest singular value of a matrix? It is easy to get ...
5
votes
2answers
611 views

Product of complex Hermitian positive semidefinite matrices with trace zero

Let $A$ and $B$ are both $n\times n$ complex Hermitian positive semidefinite matrices, then whether $\mathrm{trace}(AB)=0$ implies $AB=0$? When $A$ and $B$ are real Hermitian positive semidefinite ...
1
vote
2answers
226 views

Eigenvalues and Jordan form

I have a $5\times 5$ matrix and I need to find the Jordan form and its inverse. I know how to find the inverse. But for the Jordan form I am screwed. The matrix is $$\begin{bmatrix}3 & 0 & 0 ...
26
votes
3answers
589 views

Square matrices satisfying certain relations must have dimension divisible by $3$

I saw this tucked away in a MathOverflow comment and am asking this question to preserve (and advertise?) it. It's a nice problem! Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with ...
0
votes
1answer
88 views

Basic question about matrix algebra- notation

The representation $X=(I_p,0_{n-p\times p})$is confusing me. I get that $I_p$ is an identity matrix with $p$ rows and columns and the rest of the representation is confusing me. Can someone clarify ...
4
votes
3answers
91 views

Every p-group is isomorphic to some subgroup of U(n,p)

Let $U(n,p)$ be the group of upper diagonal matrices with elements from $\mathbb{F}_p$ and determinant $1$. Then prove/disprove that every $p$-group is isomorphic to some subgroup of $U(n,p)$. My ...
1
vote
3answers
7k views

What does it mean when all the values of a row in a matrix are 0?

I'm fairly new to linear algebra and I'm trying to make sense of what I'm being taught in class. I'm a little confused as to what happens when all the values of a matrix's row are equal to 0: does ...
7
votes
2answers
376 views

How did my professor do this?

I'm trying to figure out how my professor got to the step circled in red in the image below: How did he get the values of the first row to become completely positive, and how did he derive the ...
3
votes
1answer
159 views

Question on generalized eigenspaces of commuting matrices

The following question came up as a though while I was reading. I cannot see how to proceed on it. Let us have $M_1,\ldots,M_n$ be commuting matrices. I know that that the generalized eigenspaces are ...
3
votes
1answer
214 views

Matrix, Ranks and Rows

Let $f:V \rightarrow W$ be a linear transformation. Given bases $\{v_i\}_{1\leq i \leq n}$ and $\{w_j\}_{1\leq j \leq m}$ of V and W, respectively, $f$ has an associated $m \times n$ matrix $A$. I am ...
4
votes
2answers
45 views

What is the sufficient and necessary condition for $U$ and $V$ to be the same in the SVD?

As we know, SVD decomposites any matrix $M$ into the form: $$M=U\Sigma V^*,$$ where $U$ and $V$ are normally different. In here Wikipedia says that a matrix A is normal if and only if $U=V$. But in ...
3
votes
1answer
115 views

Matrix trace based formulation of least-squares

How can the following function be represented in a matrix form using matrix trace? $||y-X\beta||^2 + \lambda \beta^T S \beta$ Note that $y, \beta$ are real vectors and $\lambda$ is a real scalar ...
2
votes
0answers
334 views

Derivative/Jacobian of the matrix logarithm

I need help finding the Jacobian of the matrix logarithm function, i.e. $\log{M} = R$ defined by $e^R = M = V\begin{bmatrix}e^{\lambda_1} & & \\ & \ddots & \\ & & e^{\lambda_n} ...
4
votes
1answer
72 views

Matrix with rank $1$

Let $A=(a_{ij})_n$ a symmetric matrix with positive coefficients. We suppose that there is $\alpha>0$ such that, for all permutation $\sigma$ of $\{1,\ldots,n\}$, we have ...
4
votes
4answers
17k views

Find the standard matrix for a linear transformation.

If T: $\Bbb R$3→ $\Bbb R$3 is a linear transformation such that $$ T \Bigg (\begin{bmatrix}-2 \\ 3 \\ -4 \\ \end{bmatrix} \Bigg) = \begin{bmatrix} 5\\ 3 \\ 14 \\ \end{bmatrix}, T \Bigg ...
3
votes
5answers
1k views

What does positive definite matrix mean?

What do we mean by a matrix is positive or negative definite? Does it have any analogy with a positive real number?
0
votes
1answer
493 views

Proving that matrix is positive definite

I have a set of real variables $\{a_1,a_2,a_3,...,a_n\}$. Let $u=\sqrt{a_1^2+a_2^2+...+a_n^2}$. I am trying to prove that the following matrix is positive definite: $A\in \mathbb{R}^{n\times n}$, ...
2
votes
2answers
62 views

Are transformation matrices invariant over row operations?

The title says it all. I'm currently taking an introductory course in linear algebra and this issue has not been adressed spesifically. What I'm wondering is this: Given a transformation matrix $A$ ...
2
votes
1answer
139 views

Derivative of the off-diagonal $L_1$ matrix norm

We define the off-diagonal $L_1$ norm of a matrix as follows: for any $A\in \mathcal{M}_{n,n}$, $$\|A\|_1^{\text{off}} = \sum_{i\ne j}|a_{ij}|.$$ So what is $$\frac{\partial ...
2
votes
0answers
59 views

positive definite matrix and double non-negative matrix with 0-1 entries

If we have a positive definite (strictly) matrix $A$ and $M$ semi-positive definite with entries $0$ or $1$ and diagonal all ones, What are the conditions to have $\operatorname{max eigenvalue}(A ...
3
votes
2answers
542 views

How to compute a matrix for rotating and centering rectangle in viewport?

I have a rectangle given by 4 points. I'm trying to compute a transformation matrix such that the rectangle will appear straight and centered within my viewport. I'm not even sure where to begin. ...
2
votes
2answers
685 views

square root of a real matrix

I want to compute the square root of a real symmetric positive definite matrix $S\in \mathcal{M}_{m,m}$ such that $S^{1/2}S^{1/2}=S$ and it's well known that this decomposition is unique. My question ...
1
vote
1answer
109 views

Counting degrees of freedom

Why does a tensor with 3 indices in $n$ dimensions, such that if you swap any two of the indices gives the same value, has degree of freedom equal to $$n+2 \choose 3$$? I would have thought that it's ...
3
votes
1answer
60 views

How to construct non-square isometry matrix

How can we construct a non-square isometry matrix $U\in \mathcal{M_{n,m}}$; that is, all columns of $U$ are orthonormal and $U U^T=I_{n,n}$?
1
vote
1answer
657 views

Find 3D rotation vector and angle to transform a rectangle into a given quadrilateral

I have a given rectangle that I need to transform into a given quadrilateral shape that resulted from a rotation and translation in 3D space, and subsequent projection. ...
2
votes
2answers
89 views

Rank of a bilinear form

I have to prove that a bilinear form $B$ has full rank, and I would like to know some ideas on how to prove that. Can anyone give an idea?
2
votes
1answer
298 views

Matrix norms involving singular values

The induced 2-norm of a $m \times n$ matrix $A$ is known as $$ ||A||_2 = \underset{||x||=1}{\text{max}} ||Ax||_2 = \sigma_1,$$ where $\sigma_1$ is the largest singular value of $A$. Then, is it ...
0
votes
0answers
395 views

Prove that if its reduced row echelon form is [R c] then R is the reduced row echelon form of A.

Let $[A\;b]$ be the augmented matrix of a system linear equations. Prove that if its reduced row echelon form is $[R\;c]$, then $R$ is the reduced row echelon form of $A$. How do I prove it? I mean ...
0
votes
1answer
46 views

Differential and derivative of $X^{-2}$

Determine the differential and derivative of $F(X) = X^{-2}$ in which the variable X is an n x n-matrix. I computed the differential by using the product rule. So I first wrote $$ f(X)= X^{-1} X^{-1} ...
-1
votes
1answer
142 views

Elementary lower-triangular $4\times 4$ matrices

What are the three elementary lower triangular $4 \times 4$ matrices and what does their operation do? How can I prove that for all of these, $\det(L)=1$ and $L(x)^{-1}=L(-x)$?
4
votes
2answers
78 views

A quantitative measure of rank of a matrix

The rank of a matrix is only defined as integers. Is there some other criteria that is more quantitative. E.g. $$A = \begin{bmatrix} 1 & 1\\ 1 & 0\\ \end{bmatrix} $$ $$B= \begin{bmatrix} 1 ...
8
votes
1answer
309 views

How to diagonalize this matrix?

Consider the $n\times m$ matrix $M=[M_1, \ldots, M_m]$ where the $i$-th column reads $$ M_i= \,^t(\underbrace{1,\ldots,1}_{a_i},0,\ldots,0) $$ where the $a_i$'s are given positive natural numbers. ...
4
votes
3answers
77 views

How to show $T$ is not one-one and $T$ is not ont0?

Suppose $V$ is the space of all $n \times n$ matrices with real elements. Define $T : V \to V$ by $$T (A) = AB − BA,\; A \in V,$$ where $B \in V$ is a fixed matrix. Show that for any $B \in V$, ...
1
vote
2answers
786 views

Differential and derivative of the trace of a matrix

If $X$ is a square matrix, obtain the differential and the derivative of the functions: $f(X) = \operatorname{tr}(X)$, $f(X) = \operatorname{tr}(X^2)$, $f(X) = \operatorname{tr}(X^p)$ ...
0
votes
2answers
72 views

Any idea which matrix theorem this is?

I came across a theorem that boyd uses to convert the simplex to the form of a polyhedra. I don't know anything about this theorem. Theorem states: If $B$ has rank $k$, then we can find two matrices ...
2
votes
3answers
49 views

Matrix decomposition again

If some matrix (M×N) can be expressed as product of (M×1) and (1×N) vectors: what is proper term for such kind of decomposition? how to tell if such kind of decomposition exists for given matrix? ...
3
votes
0answers
74 views

Elementary Lower Matrices

First of all forgive me for my lack of format. I want to prove that the following elementary lower triangular $nxn$ matrix $Li(x)= I-xe(i)^T$ where $x=[0 \ldots 0 x(i+1) \ldots x(n)]^T$ has the ...
-2
votes
1answer
306 views

Prove the following facts about the matrix exponential:

a) Prove that $(e^{At}-I)/t\rightarrow A$ as $t\rightarrow 0$, meaning $||(e^{At}-I)/t - A|| \rightarrow 0$ as $t\rightarrow 0$ for all $A\in\mathbb{C^{n \times n}}$. Hint: You may use the inequality ...
5
votes
3answers
937 views

Differentiate $f(x)=x^TAx$

Calculate the differential of the function $f:\Bbb R^n\rightarrow\Bbb R$ given by $f(x)=x^TAx$, with $A$ symmetric. Also differentiate this function to $x^T$. How exactly does this work in the case ...
0
votes
1answer
96 views

Lower bound on the norm of product of non square matrices

The following inequality is known: $\parallel AB\parallel\geq\parallel A\parallel \sigma_{n}(B)$. However, it is only valid where both $A$ and $B$ are square. Is there an analogue for rectangular ...
0
votes
3answers
164 views

How is a $4\times 4$ Matrix built (concerning position, translation, scale and rotation)?

I am given to understand that a $4\times 4$ matrix can contain position, translation, scale and rotation, but I don't know where all of these are in the matrix. What I have seen so far is that ...