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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1answer
25 views

QR decomposition subcases

Is the full QR decomposition the most general, which includes the reduced QR, i.e, is it alright to always compute the full QR Decomposition for a given matrix blindly? What's the point of having two ...
0
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0answers
31 views

Linearize Matrix Equation

I want to find a linearized formula for G in terms of A. $G = B^TC^{-1}T(I+BA)$ $G$ is 4x2 $B$ is a constant matrix 2x4 $A$ is a variable matrix 4x2 $C = I + A^TB^T + BA + BAA^TB^T$, so $C$ is ...
2
votes
1answer
29 views

Characterization of a square matrix.

I would like to see a proof to this fact. For a square matrix the following are equivalent: $A$ has a right inverse. $A$ has rank $n$, where $A$ is $n \times n$. $A$ is invertible.
5
votes
1answer
57 views

Rank of a lower triangular block matrix

For $$A= \begin{bmatrix}B&0\\C&D\end{bmatrix}$$ where $B, C, D$ are matrices that may be rectangular, is it true or false that $$\text{rank}(A)=\text{rank}(B)+\text{rank}(D)$$ I think that if ...
1
vote
1answer
31 views

Right inverse matrix

I know that if $A, B$ and $C$ are square matrices such that $$ AC=I \quad \mbox{and} \quad BA=I, $$ then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \...
1
vote
2answers
47 views

A proof of the Continuity of the inverse matrix function

I would like to see a proof to this fact. If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there ...
1
vote
1answer
65 views

How to prove that the vectors of the Krylov space of A are linearly independent if A is nonsingular.

$\mathbf{K}$ is a Krylov matrix. \begin{align} \mathbf{K}&= \left[ \begin{array}{ccccc} \mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots & \mathbf{A}^{N-1}\...
4
votes
4answers
130 views

Not understanding derivative of a matrix-matrix product.

I am trying to figure out a the derivative of a matrix-matrix multiplication, but to no avail. This document seems to show me the answer, but I am having a hard time parsing it and understanding it. ...
2
votes
2answers
52 views

Given a symmetric matrix $A$, find $P$ such that $P^T A P$ is a diagonal matrix

Given $$A =\begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0\end{pmatrix}$$ find a matrix $P$ such that $P^T A P$ orthogonally diagonalizes $A$. Verify that $P^TAP$ is ...
3
votes
1answer
95 views

Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...
0
votes
1answer
25 views

Change eigenvalues of correlation matrix and transform into original basis

I use the Random Matrix Theory to filter out the information from the correlation matrix that is associated with noise - Marcenko Pastur band. That is straight forward. Then I follow Rosenow, Bernd, ...
0
votes
0answers
49 views

Simplify Series composed by Noncommutative Matrices

Problem I need to find a simpler formula for the following series: S = $\sum_{a=1}^{\infty} \frac{1}{a} \sum_{b=1}^{a} X^{b-1}MX^{a-b} = \sum_{a=1}^{\infty} \frac{1}{a} X^{a-1} \sum_{b=0}^{a-1} X^...
0
votes
1answer
23 views

Doubt on Prasolov's notation: $A=||a_{ij}||_1^n$.

I'm reading Prasolov's: Problems and Theorems in Linear Algebra. He defines the following notation: $||a_{ij}||_p^n$ as a notation for a matrix, where $p\leq i, \;j\leq n$. And there is a problem: ...
2
votes
1answer
45 views

Diagonalizability of a given matrix

I must find out under which conditions the matrix $$A= \left[\begin{array}{ccc|cc}& & & c_0 &\\ & & &c_1&\ddots\\ & & &c_2 &\ddots& c_0\\ &...
2
votes
1answer
22 views

Mutually orthogonal vectors in a complex vector space?

Consider a Matrix $A \in \mathbb C^{m \times n}$, $m<n$ which is build by vectors like $$ A = \begin{pmatrix} | & | & & | \\ \vec a_1 & \vec a_2 & \cdots & \vec a_n \\ | &...
0
votes
1answer
51 views

What is the determinant of exp(matrix)? [duplicate]

Given a square matrix $A$, form the Lie series of it, which is defined by: $$ e^A = I + A + \frac{1}{2} A^2 + \frac{1}{3!} A^3 + \cdots + \frac{1}{n!} A^n = \sum_{k=0}^\infty \frac{1}{k!} A^k $$ Is ...
1
vote
1answer
33 views

How to find the derivative of the following matrix?

Let $V$ be $n$ by $m$ matrix and let $x$ be $m$ by $1$ vector, i.e., $$V = \left[\begin{array}{cccc} V_{11}&V_{12}&\cdots&V_{1m}\\ V_{21}&V_{22}&\cdots&V_{2m}\\ \vdots&\...
0
votes
0answers
14 views

Generate a class of matrices via optimization

I want to generate a matrix (using Matlab) with the following properties: (1) $A = (a_{ij}) \in \mathbb{R}^{n \times n}$; (2) $a_{ij} \in \{0,1\}$ and $a_{ii} = 0$ for all $i\in\{1,2,\cdots, n\}$; (...
0
votes
0answers
18 views

How do I calculate similar matrix with arbitrary change of base matrix

$P=[\begin{matrix}\overrightarrow v_1 & \overrightarrow v_2 & \overrightarrow v_3\end{matrix}]$ with $\overrightarrow v_1$, $\overrightarrow v_2$, $\overrightarrow v_3 \in \Bbb R^3$ and $A= \...
0
votes
0answers
21 views

Is there an algorithm for finding the largest possible linear subspace of a given vector space having this specific property?

Let $G_1,G_2,\dots,G_k$ be $n\times n$ real matrices, and let $\mathcal{G} = \operatorname{span}\left\{ G_k\right\}$. Let $\mathcal{V}$ be a linear subspace of $\mathcal{G}$, i.e. $\mathcal{V} \...
2
votes
2answers
58 views

What are the constraints on $\alpha$ so that $AX=B$ has a solution?

I found the following problem and I'm a little confused. Consider $$A= \left( \begin{array}{ccc} 3 & 2 & -1 & 5 \\ 1 & -1 & 2 & 2\\ 0 & 5 & 7 & \alpha \end{...
0
votes
0answers
34 views

Matrix that changes basis

Does change of basis matrix we use in linear transformations change both the domain and range of the transformation matrix? By the way, I have a hard time calculating the change of basis matrix. I've ...
2
votes
4answers
46 views

Show that the diagonal elements are not all $0$

If the rank of a real symmetric matrix be $1$, show that the diagonal elements of the matrix can not be all zero. Since the rank is $1$, the determinant of the entire matrix is $0$, so it is ...
0
votes
1answer
16 views

Embedding A Matrix

Okay, I have a matrix $A \in M_k(\mathbb{C})$ that I want to view it as embedded in some larger matrix in $M_n(\mathbb{C})$, which means $k < n$, with zeros filling in the rest of the entries so as ...
0
votes
0answers
43 views

represent an image in linear algebra

Can we represent a grayscale image as a matrix of values, and then apply all our linear algebra techniques to that? Like finding the column space and null space, reason about the matrix structure. ...
2
votes
1answer
60 views

How can I divide a vector by a matrix?

I am trying to go backwards through a neural network. I have an output and I want to see what input would lead to that output. To go forwards I start with a vector and multiply by a matrix and then ...
1
vote
0answers
27 views

Is there a non-trivial special orthogonal transform which preserves the diagonal elements of a symmetric matrix with positive entries?

This problem is at the interface of matrix algebra and spectral graph theory. Let $\mathbf{S}$ be a symmetric $n\times n$ matrix, with positive entries $S_{ij}\geq 0$, and $\mathbf{D} = \mathrm{diag}(...
2
votes
2answers
33 views

Matrix decomposition in unipotent matrices

Consider the positive definite and symmetric matrix $$A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 6 & -1 \\ 0 & -1 & 1 \end{pmatrix}$$ Find a decomposition with unipotent $U ...
0
votes
0answers
21 views

Nonnegative solutions of linear equations

I am wondering whether the follow proposition holds or not: Let $A$ be an $n\times k$ matrix with real entries where $k<n$. Suppose that there is some $y\geq 0$ and $y\neq 0$ with $yA^{\...
2
votes
0answers
22 views

Cholesky decomposition of $I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$

I need to compute the Cholesky decomposition of the following matrix: $\varPi=I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$ Here $n$ is the dimension of the matrix and $x>0$. $\iota_{n}$ is ...
1
vote
2answers
24 views

Projectors solution of a matrix equation

If I have two $n \times n$ complex matrices $A$ and $B$, where $A$ and $B$ are both projectors, i.e., $A^2=A$ and $B^2=B$. If $A B = A$ and $A \neq I$, clearly if $B=A$, or $B=I$ then the equality ...
1
vote
2answers
46 views

How do I write $B = \left\{\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \in \boxed{?}| \ldots\right\}$ with proper notation

Let $x \in X \subset \mathbb{R}^n$, then I define a set: $$A = \{x \in X| 1^Tx = 0\}$$ Now supose I have another element $y \in Y \subset \mathbb{R}^n_{+}$ I concatenate $x,y$ in to a single vector ...
1
vote
1answer
25 views

What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible

I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is ...
3
votes
1answer
42 views

Lower bounding the trace of $A^2$ using the trace of $A^T A$

$\DeclareMathOperator{\tr}{tr}$For a real, square matrix $A$, I believe that one has a simple upper bound on the (absolute value of the) trace of its square in terms of the trace of its Gramian-type ...
0
votes
1answer
34 views

What if 1st pivot is missing but the 2nd one is there?

I have the following matrix : $$A= \begin{bmatrix} 0 &1 &2 &3 &4 \\ 0 &0 &0 &1 &2\\ 0 &0 &0 &0 &0\\ \end{bmatrix} $$ So here the 1st pivot is missing ...
0
votes
0answers
14 views

CUR decomposition

In SVD, reconstruction of matrix X from its rank-2 approximation X2 ( i.e. using two PCs) is as follow: X_reconstructed = U(:,1:2) * S(1:2,1:2) * V(:,1:2)' How to reconstruct matrix X from its ...
0
votes
2answers
30 views

Finding the matrix representation of a transformation

Question is : The vectors $(2,1)$ and $(1,1)$ form a basis for $R^2$. Let $T$ be a linear transformation satisfying $T(2,1)=(-2,6)$ and $T(1,1)=(0,5)$. Find the matrix of $T$ with respect to the ...
3
votes
1answer
52 views

Matrices that represent rotations

So the question is What 3 by 3 matrices represent the transformations that a) rotate the x-y plane, then x-z, then y-z through 90? I believe this is the matrix that rotates the xy plane \begin{...
0
votes
0answers
7 views

Minimize the inner product of this tensor function

Minimize the following function: $ f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
1
vote
2answers
29 views

Calculate the spectral norm

Consider the four vectors $v_1, v_2, u_1, u_2 \in \mathbb{C}^2$ with $$v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad v_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \qquad u_1 = \begin{pmatrix} ...
0
votes
0answers
19 views

Positive values of Quadratic Form with nonnegative vectors

I hope somebody can give me a hint on the following problem: Consider a real valued $n \times n$ matrix $A$. Let $x$ be a real-valued "non-negative" vector, i.e. a vector with nonnegative components,...
1
vote
1answer
39 views

$x=x_p+x_n$ is given, asked to find the matrix

The question is : Find a 2 by 3 system $Ax=b$ whose complete solution is : $$ x=\begin{bmatrix} 1 \\ 2 \\ 0 \\ \end{bmatrix}+w \begin{bmatrix} 1 \\ 3\\ 0\\ \end{bmatrix} $$ So I treated this as $x=...
5
votes
1answer
34 views

a property of infinite matrices

An infinite matrix $[a_{ij}]_{i,j\in\mathbb{N}}$ is called invertible, if for any convergent sequence $(y_m)$ there exists exactly one sequence $(x_m)$ such that $y_m=\sum_{n\ge 1}a_{mn}x_n$ for all $...
1
vote
1answer
29 views

matrix multiplied by rotation matrix on right side and transpose(rotation) on left side

Would a matrix remain un-rotated if it is multiplied by an orthonormal rotation matrix on right side and transpose of same rotation matrix on the left side?
0
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0answers
45 views

Attempt to represent gaussian integers with matrices over ${\mathbb Z_+}^{4\times4}$

Let us first consider the generating element for $C_2$ : $$M_1 = \left[\begin{array}{cc}0&1\\1&0\end{array}\right], \text{ and } P_1 = ({M_1})^2 = I_2 = \left[\begin{array}{cc}1&0\\0&1\...
-1
votes
1answer
43 views

Linear Algebra Eigenvalues and Eigenvectors [closed]

So I have a 2x2 matrix where equation 1(EQN1) is 1 and 2; equation2(EQN2) 2: 4 and 3 The determinant is det(A-λI)=0 When I first solve the eigenvalues I get λ=5, λ=-1 Now this is where I am lost,...
0
votes
0answers
23 views

About the distributive property of matrices

So We all know that matrix operations are distributive, so here is my question.$A^2+AB\\$ and $BA+B^2$ is two matrix operations I have, I know we can do $A(A+B)$ in the first operation but I'm not ...
4
votes
2answers
122 views

Matrix decomposition into square positive integer matrices

This is an attempt at an analogy with prime numbers. Let's consider only square matrices with positive integer entries. Which of them are 'prime' and how to decompose such a matrix in general? To ...
0
votes
1answer
34 views

exponent of a matrix, equivalent conditions

Let $A=[a_{ij}]$ be a real $n\times n$ matrix. Prove that the following conditions are equivalent: $(1)$ for every $t\ge 0$, all elements of the matrix $\exp (tA)$ are nonnegative $(2)$ $a_{ij}\ge ...
0
votes
1answer
24 views

How to find a scalar given a matrix equation with an unknown matrix?

I am not expert in linear algebra. I couldn't come up with any solution for my problem with my limited knowledge. So the question may be even silly or have no solution, I don't know. But I appreciate ...