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
68 views

Is there a non trivial ideal for the set of upper triangular matrices?

Is there a non trivial ideal for the set of upper triangular matrices? the zero matrix is a trivial ideal. Also, the set of upper triangular matrices is an ideal. Are there any other ideals?
1
vote
1answer
233 views

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a ...
2
votes
2answers
52 views

What is this good for - determinants

Ok, Using RRef and the identity matrix I can find the inverse matrix and the solution vector with out (directly) finding the determinant of a square matrix. But I have to believe, if this was the only ...
5
votes
2answers
273 views

Majorization relation between the absolute values squared of the entries of a matrix and the singular values squared

Let $A = [A_{ij}]$ be an $n\times n$ square matrix with complex entries, and let $\sigma_k$, $k=1,\ldots, n$ be its singular values. Suppose that the squared Frobenius norm satisfies $$ ...
1
vote
1answer
106 views

Equality involving norm on Cholesky/QR factorization

Let $B$ be symmetric and positive definite and $B=B^{\frac{1}{2}}B^{\frac{1}{2}}$ the Cholesky factorization. Having $A=QR$, why can we follow the last equality in the following? $$ || A ...
2
votes
1answer
29 views

Question on Matrix Transform Operations

Given $e = Y - XB$, where $ e = \begin{bmatrix} e_1 \\ \vdots \\ e_n \\ \end{bmatrix} $, $ Y= \begin{bmatrix} y_1 \\ \vdots \\ y_n \\ \end{bmatrix} $, $ X= \begin{bmatrix} 1 & ...
0
votes
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
130 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
42 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
129 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
1k 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
29 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
850 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
118 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
63 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
42 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
79 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
149 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
331 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
vote
0answers
97 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
52 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
166 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
82 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
0answers
88 views

Multiplication of two Symetric Matrices: Relation among their eigenvalues

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric nonsingular positive definite matrix, and $B\in\mathbb{R}^{n\times n}$ be a symmetric singular positive semi-definite matrix, this is ...
0
votes
1answer
61 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
127 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
91 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$^n$?
1
vote
1answer
335 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
vote
1answer
42 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
150 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
490 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
36 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
22 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
29 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
316 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
275 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
71 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
5k 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
41 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
208 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
36 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
75 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
137 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
37 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 $(I+cH^{-1})^{-1}$ ...
2
votes
1answer
585 views

formula for calculating determinant of the block matrix

I saw a formula on the wikipedia page about determinant that $\det\begin{bmatrix}A & B\\ C & D \end{bmatrix}$ = $\det(AD-BC)$, if $C$ and $D$ commute. Is this always true? Or is there a good ...
0
votes
2answers
63 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
198 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
564 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 ...