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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1answer
22 views

Inverse of $\{a_1 A_1,…,a_n A_n\}$

$a_1,...,a_n\in \mathbb{R}$ $A_1,...,A_n$ are the rows of the invertible matrix A I am trying to find a regular formula for this. Is it possible? Thanks for help!
0
votes
1answer
157 views

Backward stable algorithm

Assume we have fixed unitary matrices $Q_1, \dots, Q_k \in \mathbb{C}^{m,m}$ and a matrix $A \in \mathbb{C}^{m,n}$ which can be perturbed. How can we proof that the algorithm on computing the product ...
0
votes
2answers
43 views

Matrix norm inequality involving max and stacked matrices

In a paper I found the following inequality for matrices $A$ and $B$: $\max\left\{||A||, ||B||\right\} \le \left\| \begin{align}A \\ B\end{align} \right\|_2 $ I suspect that this is a well-known ...
1
vote
1answer
172 views

Historical reason to define a matrix vector product the way it is

What is the reason why we defined a matrix vector product (a transformation) this way: $$\begin{pmatrix} a_1 & a_2 \\ a_3 & a_4 \\ \end{pmatrix}\cdot ...
1
vote
1answer
2k views

Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
0
votes
2answers
31 views

Is $A$ s.t $A_{i, j} = x^T_i x_j$ semi-positive definite?

Let $x_1, x_2, \ldots, x_k \in \mathbb{R}^n$ and set define a $k$ by $k$ matrix $A$ by setting $A_{i, j} = x^T_i x_j$. Is $A$ semi-positive definite? If so, how can I show it?
4
votes
2answers
960 views

Derivative of inverse quadratic function of a matrix

I have been stuck with the following derivative for some time: $$ \frac{\partial\,\mathbf{b}^\mathrm{T}(\mathbf{X}\mathbf{C}\mathbf{X}^\mathrm{T})^{-1}\mathbf{b}}{\partial\,\mathbf{X}} $$, where ...
2
votes
0answers
154 views

Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$ \Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}. $$ Applying the Woodbury matrix identity ...
0
votes
0answers
89 views

Matrices that commute with nonderogatory matrices

Let $A$ be a nonderogatory $n \times n$ matrix (with complex entries) and let $B$ be a matrix such that $AB=BA$. Show that there exists a polynomial $q(x)$ such that $\sigma(B) = \{ p(\lambda) \mid ...
0
votes
0answers
46 views

The differentiation of the trace of complex matrix

Condition: all the matrices are complex. $\dagger$ denotes the conjugate transpose, $*$ denotes the conjugate, $\mathop{Trace}$ denote the trace of a matrix. What is the differentiations of the ...
1
vote
1answer
94 views

Roots of the characteristic polynomial of a symmetric matrix

I'm looking for a proof (using basic tools : definition of the characteristic polynomial and its basic properties) of the following fact : The roots of the characteristic polynomial of a symmetric ...
0
votes
1answer
180 views

Differentiation of a unitary matrix

Let $\mathbf{U}$ be a unitary matrix ($\mathbf{UU}^\dagger=\mathbf{1}$). What does this implies for $d( \mathbf{ U U }^\dagger)$? Is it mathematically sound to say: \begin{equation} d\mathbf{U} ...
3
votes
3answers
379 views

Is $T$ singular or nonsingular for $T(A)=AB - BA$?

Let $B$ be a complex $n\times n$ matrix. Prove or disprove: The linear operator $T$ on the space of all $n\times n$ matrices defined by $ T(A) = AB - BA $ is singular.
0
votes
1answer
23 views

Equality with matrices

I need prove the following: $(\mu-\lambda)(D-\mu I)^{-1}(D-\lambda I)^{-1}=(D-\mu I)^{-1}-(D-\lambda I)^{-1}$ where $D$ is a matrix, $I$ is the indentity matrix and $\mu,\lambda$ reals. Thanks!
0
votes
1answer
57 views

what is the meaning of Det in the context of multiplication of two matrices

Does such a Determinant indicate a structural relationship between two variables for which matrices have been indicated.
1
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0answers
109 views

Finding the inverse and the solution for NxN system of equations in “one” step.

This is cool... at least if I have it right. Do I understand correctly that (using the gauss/Jordan method) Finding the RRef of |A|I||x| ( Matix A augmented by the Identity Matrix augmented by |x|) ...
1
vote
2answers
61 views

Solving for an $x$ in matrices, with condition $AB=BA$

I'm just starting to learn about matrices, and during one exercise I got a question to which I have no answer; Due to the fact that I haven't learned it yet... The question is as follows: Let $A ...
2
votes
0answers
206 views

Using Rref to find the inverse of a matrix.

Since, I can't divide vectors to deduce an inverse matrix I have dismissed that approach. I did find that if I multiply all of the matrix row operators It will yield the inverse. Since I did the logic ...
1
vote
0answers
85 views

derivative of determinat

For a lower trinagular, invertible but asymmetric matrix $X$, how to calculate the following: $$ \frac{\partial |XX^T|^{-1/2}}{\partial X} $$ I was doing the derivation, but not sure whether it was ...
0
votes
1answer
62 views

Aggregating a vector of $1\times K$ into a vector $1\times J$, such as $J<K$

I am stuck with a matrix algebra operation: how do I do (and mainly which notation to use) to aggregate the numbers of a vector $1\times K$ into a vector of $1\times J$, such as $J$ is of course lower ...
1
vote
1answer
145 views

Condition Number of a Product

Is this hypothesis true? $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where $\operatorname {cond}$ is the condition number. And is this true for rectangular matrices? ...
1
vote
1answer
106 views

If two matrices both multiplied by the same vector are equal are the matrices equal?

Assume A and B are n x n matrices. If Av$_k$ = Bv$_k$ then is A = B where v$_k$ is a vector in R$^n$?
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vote
1answer
356 views

Inverse complex matrix

I calculated the inverse of an complex matrix $C=A+iB$, where $A,B$ are real matrices and $i^2=-1$: $C^{-1}=(A+BA^{-1}B)^{-1}-iA^{-1}B(A+BA^{-1}B)^{-1}$ my question is: what assumptions must be met ...
1
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1answer
51 views

an estimate for condition number: $\kappa(C^{-1}A)\leq \kappa(C^{-1}B)\kappa(B^{-1}A)$

I'm currently reading through "Domain Decomposition Methods" by Tosseli and Widlund and in the appendix I found the following Theorem: Let A, B, C be symmetric positive definite matrices. Let ...
0
votes
1answer
192 views

Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
0
votes
0answers
667 views

Matrix with trig functions and Cramer's rule

Using Cramer's rule solve for $x'$ and $y'$ in term of $x$ and $y$ $x = x'\cos\theta - y'\sin\theta\\ y = x'\sin\theta + y'\cos\theta$ So what I have is this $\det\begin{bmatrix} \cos\theta& ...
0
votes
2answers
37 views

matrices equation

Let A, B, J 4x4 matrcies, such that: $\eqalign{ & {A^t}JA = J \cr & {B^t}JB = J \cr} $ Prove that: ${(AB)^t}J(AB) = J$ Any help will be appreciated.
0
votes
1answer
24 views

How to determine basis by Reducing a sets

How to do Reducing set of $(x_1,x_2,x_3,x_4)$ So that form basis for $\mathbb{R}^3$ for vectors: $\displaystyle x_1 = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}x_2=\begin{bmatrix} -3 \\ 2 \\ 1 ...
0
votes
2answers
32 views

How to determine Depedent and Span of matrices?

$ \displaystyle s= (2,4,6)^T ,(0,0,0)^T ,(0,1,1)^T \in R^3 $ Does S are dependent linear? Does S are Span of $R^3$ ?
2
votes
0answers
397 views

“A dominant eigenvalue of a non-negative matrix has a non-negative eigenvector”

I have the non-negative 3x3 matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 4 & 1 \\ 3 & 2 & 1 \end{bmatrix}$. I've calculated the eigenvalues of this matrix, ...
1
vote
1answer
390 views

Logarithm of singular matrix

How do we define logarithm of a singular matrix(Say it is real square symmetric and has distinct eigen values). I tried searching online but could not find much information(Something that someone as ...
1
vote
1answer
72 views

derivative of product of 2 inverse matrices

I was trying to differentiate the equation below: $$ \frac{\partial a^T X^{-T}X^{-1}a} {\partial X} $$ where X is invertible but not symmetric and $X^{-T}$ means transpose of inverse of X. In the ...
4
votes
1answer
7k views

Where can I calculate the exponential of a matrix online?

Where can I exponentiate a $3\times 3$ matrix like $\left[\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]$ online? Is there some website where this ...
0
votes
1answer
46 views

In a boolean matrix, what does the $n$ in $M_{R^n}$ represent?

I'm now learning about binary relations. I stumbled upon this question in the book: Given $A = \{1,3,5,6\}$ and $R$ is a relation over $A$, whose matrix is defined by $$\begin{pmatrix} 0 ...
0
votes
2answers
430 views

Possibility of making diagonal elements of a square matrix 1,if matrix has only 0 or 1

Let $M$ be an $n \times n$ matrix with each entry equal to either $0$ or $1$. Let $m_{i,j}$ denote the entry in row $i$ and column $j$. A diagonal entry is one of the form $m_{i,i}$ for some $i$. ...
1
vote
1answer
37 views

matrix differentials for the product of three functions of matrixes

Can anyone give me an expression for following differential problem: $$ \frac{\partial f_1(X)^Tf_2(X)f_3(X)}{\partial X} = ?$$ where $ X $ is a matrix, $ f_1(X) $ is a vector, $f_2(X)$ is a matrix, ...
4
votes
1answer
76 views

Inequality with determinants problem

Let $A,B \in M_{2}(\mathbb{R})$ with $AB=BA.$ Prove that: $$\det(A^{2}+AB+B^{2})\geq (\det(A)-\det(B))^{2}$$
3
votes
1answer
143 views

Matrix with eigenvalue pairs $\pm\lambda$

Consider a real differentiable function $f:\mathbb{R}\to\mathbb{R}^N$ and define a matrix $A_{ij}=\mathbb{E}[\frac{\partial}{\partial x}[f_i(x) f_j(x)]]$ where the expectation is with respect to some ...
0
votes
1answer
41 views

About the inverse matrix of the form $(I+cH^{-1})^{-1}$.

Given $(I+cH^{-1})^{-1}$, where $c$ is a constant and $H$ is a $\mathbb{R}^{n\times n}$ matrix. Suppose $(I+cH^{-1})^{-1}$ has a inverse matrix. Is there any way to calculate $(I+cH^{-1})^{-1}$ ...
0
votes
2answers
65 views

Bi-linear function

Prove that bi-linear function $f:M_n(\mathbb{C})\times M_n(\mathbb{C}) \rightarrow \mathbb{C}$ defined as $f(A,B)=Tr(A^t\overline{B})$ is non-singular. I don't quite know where to start. Thanks!
1
vote
1answer
255 views

condition number of sum of matrices

To my knowledge, there are no explicit formulas linking the singular values of a matrix sum to the singular values of the summand matrices, i.e. it is hard to guess the singular values of matrix ...
7
votes
1answer
599 views

When does the adjacency or incidence matrix of a graph have consecutive ones property?

Given a graph, what are some sufficient (and necessary) conditions to tell if its adjacency matrix has the consecutive ones property? Similar question for its incidence matrix? Note that a ...
1
vote
0answers
256 views

Question about proof of bounded real lemma

My question is: it is possible to proof the bounded real lemma for $H_\infty$ performance with the following procedure? The $H_\infty$ performance is defined as: \begin{align} \parallel ...
0
votes
2answers
131 views

Multivariate normal distribution from invertable covariance matrix

I want to generate a random vector with $\mathcal{N}(0, C)$ distribution, i.e. normal distribution with $0$ mean and given covariance matrix $C$. $C$ is not invertible (singular). Here it's written: ...
3
votes
3answers
623 views

Determinant from matrix entirely composed of variables

I don't want the answer, but I'd love to kick in the right direction. I'm really not sure how to approach this question. $$\begin{align} & -6 = det\begin{bmatrix} a & b & c \\ d & e ...
0
votes
1answer
36 views

How to find matrix $\left(\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{matrix}\right)$.

I need to find all $a_{11}$ to $a_{22}$ can anyone help me pls. $2a_{11}+3a_{21}= 3$ $2a_{12}+3a_{22}= 0$ $1a_{11}+4a_{21}= 1$ $1a_{12}+4a_{22}= 2$ I am stuck here don't know how to calculate ...
0
votes
1answer
462 views

Find a basis for ker(T) and range(T) for the given transformation and compute T(5x-4)

I am not really having trouble with $a)$, $c)$, $d)$ or $e)$. For $a)$ I put that $B$ is a basis for $\mathbb{P}^1$ because it has ${\rm dim} = 2$ and the highest degree is 1, and for $B'$ it has ...
3
votes
2answers
364 views

Solutions of Matrix Equation $XAX^{T}=A$ for unknown $X$

Given a square, symmetric positive semidefinite matrix $A$, I am looking for solutions to the matrix equation $XAX^{T}=A$ for unknown $X$ (not necessarily symmetric). Clearly, setting $X$ equal to ...
0
votes
2answers
79 views

Non-elementwise Matrix Derivatives

Let A,B,C,D,X be matrices. I'd like to perform a Gradient Descent minimization to the loss functin $$ tr[(AXBX^TC-D)^T(AXBX^TC-D)] $$ My question is, how to take the gradient efficiently w.r.t. $B$? ...
0
votes
1answer
35 views

inverse of subelliptic matrix valued function

Let $M:\mathbb{R}^n\rightarrow GL(n,\mathbb{R})$ be such that for some $C>0$ $\frac{1}{C}\vert x\vert^2\leq x^TM(.)x\leq C\vert x\vert^2$ for all $x\in\mathbb{R}^n$. Does the map $x\mapsto ...