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
190 views

A question about Hadamard matrices

Is it possible to find a matrix $A$ such that: $$\exp(A)+\exp(A^{-1})=H_2$$ with $A$ a $2\times 2$ matrix and $H_2$ a Hadamard matrix? The result can be extended to every Hadamard matrix $H_N$ with ...
1
vote
1answer
92 views

Relation between Hadamard product and scalar product

Is there a known relation/formula for $$(A\circ B, C)$$ where $\circ$ is the Hadamard product and $(\cdot, \cdot)$ is the scalar (euclidean) product? In particular, I have a vector $y$ and a two ...
0
votes
0answers
34 views

Determine general form of control function andthus show this coul have been achieved earlier [duplicate]

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
6
votes
3answers
438 views

Every invertible matrix can be written as the exponential of one other matrix

I'm looking for a proof of this claim: "every invertible matrix can be written as the exponential of another matrix". I'm not familiar yet with logarithms of matrices, so I wonder if a proof exists, ...
2
votes
1answer
164 views

Prove that if matrix $A$ is an $m\times n$ and $B$ is $n\times p$, then $\operatorname{rank} AB$ is less than or equal to $\operatorname{rank} B$

Prove that if $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, then $\operatorname{rank} AB$ is less than or equal to $\operatorname{rank} B$. The hint is: prove that if the $k$th ...
2
votes
3answers
106 views

Lesson : Solving System of Equations using matrices

I have a matrix $$ A = \begin{pmatrix} a & 0 & 0 \\ 2 & b & 5 \\ -3 & 1 & b \end{pmatrix} $$ in my try, I came up with $$ bx1 = 0,\quad x2 + 5/a x3 = 0,\quad x2 + a x3 ...
5
votes
2answers
79 views

If $x\perp \mathrm{span}\{r_1,\dots,r_p\}$, can we prove $x\notin\mathrm{span}\{v_1,\dots,v_p\}$?

Notations: For a scalar $a\in\mathbb{R}$, denote $$\mathrm{sgn}(a)=\left\{ \begin{array}{l l} 1 & \mbox{if } a>0\\ 0 & \mbox{if } a=0\\ -1 ...
2
votes
1answer
60 views

How can I solve a polynominal of degree 2 with more than one variable?

(Sorry if the title is not informative) How can I find the value of matrices $F$ and $d$, in the following equation: $$y'Ay+b'y+c'c = (y-d)'F(y-d)$$ Given $A:n \times n$, which is positive definite ...
5
votes
3answers
114 views

Extension of a continuous map on $ {\mathbf{GL}_{n}}(\mathbb{R}) $ to $ {\mathbf{M}_{n}}(\mathbb{R}) $.

I was reading in my analysis textbook that the map $ f: {\mathbf{GL}_{n}}(\mathbb{R}) \to {\mathbf{GL}_{n}}(\mathbb{R}) $ defined by $ f(A) := A^{-1} $ is a continuous map. I also saw that $ ...
4
votes
2answers
141 views

What are the generators for $SL_n(\mathbb{R})$ (Michael Artin's Algebra book)

The book asks you to prove that $SL_n(\mathbb{R})$ is generated by elementary (row operation) matrices in which one nonzero off-diagonal entry is added to the identity matrix. For example, $$ ...
1
vote
2answers
91 views

Using Neumann series to compute $T^{-1}$

Need help on how to show that $S$ satisfies the necessary condition for Neumann series. Here is what is given. $T\in B(X,X)$ where $X$ is a Banach space. Let $T: \mathbb R^3 \rightarrow \mathbb R^3$ ...
1
vote
0answers
56 views

2 positive decomposable maps

A positive map $\phi:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ is said to be $k$-positive if the natural extension ...
4
votes
1answer
67 views

How to compute determinant of $A$ such that $A=(I+\ [c_ic_j])\in M_n(\mathbb R)$ ,$c_i\in\mathbb R$

assume $\ [c_ic_j]_{n\times n}\in M_n(\mathbb R)$ such that $c_1,c_2,\ldots,c_n\in\mathbb R$ and $I$ be identity matrix how compute $$\det (I+\ [c_ic_j])=?$$ Thanks in advance
1
vote
1answer
89 views

Calculating determinant of a matrix whose non-zero elements are two sub-matrices on its diagonal

Given a $m \times m$ square matrix $M$: $$ M = \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} $$ $A$ is an $a \times a$ and $B$ is a $b \times b$ square matrix; and of course $a+b=m$. All the ...
1
vote
1answer
86 views

Derivative of a scalar function with resepct to a Matrix

I need help with the following differentiation $$ \text{trace}((aI+bXX^T)^{-1}(aI+XX^T)) $$ with respect to $X$, where $a,b$ are some positive constants, and $I$ is the identity matrix. Thank you
4
votes
1answer
162 views

What is the difference between first and second right eigenvectors of a row stochastic matrix and their meaning?

In an $n\times n$ non negative row stochastic matrix (rows sum up to 1). The entries of the stochastic matrix I have represent directed links between countries. Why is the first right eigenvector a ...
1
vote
1answer
103 views

Condition of a unitary matrix

If a square matrix $\mathbf{A}$ satisfies $\mathbf{A}^{H}\mathbf{A}=\mathbf{I}$, then is it a unitary matrix? Or, is there a counter example with $\mathbf{A}\mathbf{A}^{H}\neq\mathbf{I}$?
3
votes
1answer
401 views

Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity (for AB and BA not square)?

I know that if A and B are square nxn matrices, then AB and BA have the same characteristic polynomial and thus the same eigenvalues (and same algebraïc multiplicity). I'm wondering though if this ...
-3
votes
1answer
263 views

How should I prove this bilinear form is symmetric and positive definite?

I am trying to prove this problem: Let M be the space of all 2 × 2 complex matrices, satisfying 〖(X)bar〗^t = -X (skew-hermitian). Consider M as a vector space over R. Define a bilinear form B ...
4
votes
2answers
635 views

How to make a matrix positive semidefinite

We have a symmetric matrix $A$, with some entries specified and others not. We are trying to find the values of the unspecified entries so that the matrix $A$ becomes positive semidefinite. How can I ...
3
votes
3answers
289 views

Raising a matrix to a large power when the values are fractions (precision problem)

I have a matrix $M$ where various elements may be in the form of $x/y$. If I use the decimal form of that number, I lose precision if I raise $M$ to a large power. My question: is it possible to do ...
1
vote
3answers
242 views

Is Induction Independent of the Other Axioms of PA?

I am trying to come up with a model of first order Peano Arithmetic (PA) where induction fails. Let $PA^{-IND}$ have the same axioms as PA except the first order induction axiom schema is replaced ...
1
vote
1answer
66 views

A bound on the norm of the sum of two index-disjoint matrices

Given two matrices, it is well known that $\parallel A+B \parallel _2 \leq \parallel A \parallel _2+\parallel B \parallel_2$. Now, suppose that the nonzero indices are disjoint (i.e., $A$ is nonzero ...
5
votes
2answers
225 views

Diagonally dominant matrix with matrix similarity

Applying similarity transform to a matrix $A$ gives: $$M=P^{-1}AP$$ $M$ and $A$ have same eigenvalues. What is the way to to find $P$ such that $M$ is diagonally dominant case of $A$? $M$ is ...
2
votes
1answer
136 views

Transforming matrix-equation to overdetermined minimum problem

i have broken down my problem to plainmath and could really use some help. Basis: I have an image. In this image I have several UV-XYZ pairs. So i know the 3d position of serveral Pixels. Given the ...
1
vote
2answers
1k views

Calculating formula to store location of Lower Triangular Matrix

I am struggling with a problem from this textbook. The question is as follows: Determine a formula h = f(i,j) to store location MATRIX[i][j] in h. Ensure to only store nonzero elements. Then it asks ...
1
vote
2answers
252 views

Computing exponential of a $2\times 2$ matrix using only its trace and determinant.

I want to compute the exponential of an arbitrary $2\times 2$ matrix over $\mathbb{R}$ only using its trace and determinant. I've shown that for a traceless matrix $A$ there is the following formula: ...
3
votes
3answers
37 views

Solve for $X$ in a simple $2\times2$ equation system.

I posted a similar question recently but I still have problem with this problem and would appreciate any help! $$\left[ \begin{array}{cc} 9 & -3\\ 5 & -5\end{array} \right] - X \left[ ...
1
vote
0answers
223 views

Complexity of SVD

I am working with $SVD$. Here they noted that, it's complexity is $O(n^3)$. I know that we need to find three matrices $ U,V,S $. i.e., If A is a matrix of order $m*n$ then $svd $ of A is ...
4
votes
2answers
37 views

Solve for $X$ in a simple equation system.

I cannot really understand how to read this question, so please help me out here. $$\left[ \begin{array}{ccc} 0 & -6 & 4\\ 1 & 2 & 7\end{array} \right] = 4X + 5 \left[ ...
0
votes
0answers
161 views

Solving an overdetermined system of inequalities using null-space arguments

The solutions to a linear system of equations: $$A\cdot x = b$$ (where $x$ is a $(n\times 1)$ column vector, $b$ is a $(m\times 1)$ column vector and $A$ is $(m\times n)$ matrix) can all be ...
4
votes
1answer
104 views

Why does $ (A^T x) · y = x · (A y) $ hold?

Why does $ (A^T x)· y = x ·(A y) $ hold? The proof has to do with properties of transposes. I did a proof using coordinates (which was correct) but there is an infinitely easier way to do it. A is ...
1
vote
0answers
144 views

Transforming matrices in a differential equation

This is from Dupont et al., "Simplified density-matrix model applied to three-well terahertz quantum cascade lasers", PRB 81, 205311 (2010): Equation (3) (...) can be rewritten as a 16x16 system of ...
1
vote
0answers
52 views

Issue with tridiagonal matrix factorization

So let's assume I have an arbitrary mxm tridiagonal matrix made up of real numbers. How many flops are needed to get its QR factorization (assuming I'm using the householder method?) How would one go ...
0
votes
4answers
297 views

Relationships between $\det(A+B)$ and $A+B$

When computing $\det(A+B)$ we notice that there is no relation between $\det A + \det B$. However does the $\det(A+B)$ have any relation to the matrices $A+B$ as they stand?
2
votes
1answer
88 views

When $e$ is an eigenvector to $G$ prove that $e$ is an eigenvector of $G+k I$ and $G^2$

I've just started learning matrices and I've been shown how to perform row operations, how to find an inverse matrix, how to find eigenvectors from a given matrix, reduced-echelon form, basis, dim(M), ...
0
votes
1answer
34 views

Need help interpreting an equation from an article (related to quaternions).

At this link, about half way down the page, there is an equation I don't understand http://physicsforgames.blogspot.com/2010/02/quaternions-why.html This is the equation. $$VV† = -x^2I^2 - y^2J^2 - ...
2
votes
4answers
1k views

Determinant of matrix exponential?

Suppose $A$ is a $n \times n$ constant matrix. How can I prove $\det(e^A) = e^{\displaystyle \sum_{\lambda_i\in\sigma(A)} \lambda_i}$, where $\sigma(A)$ is the multiset of eigenvalues of $A$? The ...
1
vote
2answers
4k views

Can an underdetermined system have a unique solution?

In my case, I am calling an underdetermined system as a system of linear equations where there are fewer equations than variables (unknowns). My textbook says the answer is false, however the internet ...
2
votes
2answers
425 views

How do collinear points on a matrix affect its rank?

Consider the matrix \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{matrix} what effect does $({x_1},{y_1})$,$({x_2},{y_2})$,$({x_3},{y_3})$ being ...
4
votes
3answers
301 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
0
votes
1answer
66 views

Dimensions of Matrices Range (equalities).

I’d like to find range equalities. Considering the following: $$ A=B+C \\ A=B.C^T \\ A=[ B^T C^T ]^T \\ $$ I would like to find the function $f$ for each equality above. $$ dim( R(A) ) = f( R(B) , ...
0
votes
1answer
7k views

Question regarding trivial and non trivial solutions to a matrix.

I would very much appreciate som explanations regarding trivial and non trivial solutions to a matrix (I am a beginner in studies of linear algebra). Suppose that we have two matrices $A$ and $B$. ...
1
vote
1answer
94 views

How to calculate the powers of the following matrix

I need the powers $A^n$ of the following matrix $A=\begin{bmatrix}0 & t & 1-t\\1-t & 0 & t\\ t & 1-t & 0\end{bmatrix}$ with real $t$, but I get messed up in the calculations ...
11
votes
3answers
222 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
2
votes
0answers
168 views

Low-rank approximation to the Graph Laplacian matrix of a regular grid.

As mentioned in the title, does anybody know any methods of efficient low-rank approximation $LL^T$ to the Graph Laplacian matrix $A$ corresponding to a square lattice? (except PCA)
3
votes
3answers
225 views

Rotating Matrix by $180$ degrees through another matrix

To rotate a $2\times2$ matrix by $180$ degrees around the center point, I have the following formula: $PAP$ = Rotated Matrix, where $$P =\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$$ $$A= ...
3
votes
1answer
175 views

The multiplication of 2D vectors produces what?

I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication. To avoid confusion with other types of multiplication, this is the basic form I ...
1
vote
1answer
322 views

How to calculate left and right eigenvector corresponding to the zero eigenvalue.

I'm working on $8\times8$ matrix resulting from the Jacobian of $8$ differential equation of a disease model evaluated at disease free equilibrium. I needed to get the left and right eigenvectors ...