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
52 views

Prove that $ A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$

I'd like to get some help So I need to prove that when $A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$ Linear Algebra, of course. Thanks
2
votes
1answer
150 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
0
votes
1answer
64 views

Induced Matrix Norm

I have trouble following a proof of the induced Norm $||\cdot||_1$ The proof can be found here: ...
3
votes
1answer
144 views

Rank one plus diagonal matrix approximation

Given $A \in R^{n \times n}$, $A$ symmetric. I'm trying to solve the following minimization problem: $\underset{u \in R^n, d \in R^n} \min \, \frac{1}{2} \|X - A\|_F^2$ subject to $X = u u^T + ...
1
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0answers
44 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
3
votes
3answers
48 views

Performing matrix chain multiplication by hand

I'm trying to gain intuition for writing a matrix chain multiplication algorithm by working through a few problems by hand. I see plenty of worked-through solutions on sets of three or four solutions, ...
4
votes
2answers
124 views

Product of positive matrices

I have two positive-definite matrices $A$ and $B$ (not necessarily symmetric), and we have $AB=BA$, is there any theorem that ensures that the product of $A$ and $B$, $AB$ is positive definite? Or ...
2
votes
1answer
100 views

Least squares problem with orthonormality constraints

Given $y_1,\ldots,y_n\in \mathbb{R}$,$w\in \mathbb{R}^d$, and $x_1,\ldots x_n\in \mathbb{R}^D$, how do we solve the following optimization problem \begin{align} \min_A \sum_{i=1}^n (y_i-w^TA^Tx_i)^2\\ ...
1
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2answers
51 views

Prove that if $B=P^{-1}AP$, then $q(B)=P^{-1}q(A)P$

Is it possible to prove this using a similarity invariant? For example showing that $$\det(q(B))=\det(q(A))$$
0
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1answer
62 views

properties of left-invertible matrix

When reading the notes on Left-invertible matrix, where $A$ is a matrix of dimension of $m\times n$, and $X A=I$. It is claimed that it $m$ must be larger than $n$, and $rank(A)=n$.How to get these ...
1
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1answer
305 views

how to find matrix A from complete solution to Ax=b

I am trying to solve a problem. I was stuck.Any help is appreciated. The complete solution to $Ax=\left[\begin{array}[c]{rr}1 \\3 \end{array}\right]$ is $ x= ...
1
vote
2answers
336 views

Inverse of a sum of positive definite matrices

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ ...
1
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1answer
44 views

get an element by finitely generated set

I want to know the method to get a element in a finitely generated group by its generated set, is there a general way to calculate? For example, $SL(2,\mathbb{Z})=<a,b|a=\begin{pmatrix}0 &1\\-1 ...
1
vote
1answer
30 views

the differences and relationship between linear independent and affinely independent

When learning optimization, I heard the two related concepts on linear algebra: linearly independent and affinely independent. ...
0
votes
2answers
412 views

Gauss Elimination - Diagonal dominant matrices don't need row changes

I was asked to prove the following statement: let $A$ be an $n$ by $n$ matrix with real entries such that $\forall k \in \mathbb N, k\leq n$: $$\sum_{i \neq k} |A_{i,k}| < |A_{kk}|$$ Show that if ...
2
votes
1answer
487 views

Sum of principal minors

Is there any formula for the sum of principal minors? (note: $i^{th}$ principal minor which results from omitting the $i^{th}$ row and $i^{th}$ column)
1
vote
0answers
94 views

Calculating MSE for two different size matrixes

I have two $2$-column matrixes, one of the has $467$ rows while the other one has $61468$ rows. Both them are trajectory paths of same robot, the big matrix is kind of raw data and the smaller one is ...
3
votes
1answer
88 views

Find rank of the matrix $a_{ij}=(i-j)^2$, $i,j=1,\dots, n$

Let is $A$ $n\times n$ matrix defined in following way $a_{ij} = (i-j)^2$. For example when $n=4$ $$ A= \begin{pmatrix} 0&1&4&9\\ 1&0&1&4\\ 4&1&0&1\\ ...
3
votes
1answer
51 views

Diagonalisability…without the characteristic polynomial

Let us consider an $n\times n$ matrix $A$ defined as follows $$ A=\begin{pmatrix} 1+a&1&\cdots &1\\ 1&1+a&\ddots&\vdots\\ \vdots&\ddots&\ddots&1\\ ...
2
votes
1answer
42 views

Showing the existence of an eigenvalue whose real part is positive

$$M = \left(\begin{array}{cc|cc|cc|cc|cc} -b_1 &0 &b_2 &0 &0 &0 &\ldots &\ldots &0 &0\\ 0 &-a_1 &0 &a_2 &0 &0 &\ldots &\ldots &0 ...
0
votes
1answer
38 views

Symmetric Matrix Algebra

This should be a straightforward answer but my matrix algebra skills are weak. If I have a symmetric matrix $X$, is $A^TX^{-1}A$ symmetric for any matrix $A$? I know the inverse of a symmetric ...
1
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2answers
2k views

How to test if a graph is fully connected and finding isolated graphs from an adjacency matrix

I have large sparse adjacency matrices that may or maybe not be fully connected. I would like to find out if a matrix is fully connected or not and if it is, which groups of nodes belong to a ...
13
votes
1answer
224 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
1
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1answer
68 views

How do I rewrite vectors in other basis' given change of coordinate matrices?

$\displaystyle β= \begin{bmatrix}2\\2\\\end{bmatrix}$,$\displaystyle \begin{bmatrix}4\\-1\\\end{bmatrix}$ $\displaystyle C= \begin{bmatrix}1\\3\\\end{bmatrix}$,$\displaystyle ...
0
votes
0answers
63 views

Matrix Induction proofs using $\rm mod$.

Please could someone advise me on where to even start with this question. Thanks in advance Let $A$ be an $n\times n$ matrix, for some $n\geqslant 1$. Given any prime number $p\gt1$, write $A_p$ for ...
1
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1answer
69 views

matrix representation of linear transformation

For a set $N$ let $id_N:N \rightarrow N$ be the identical transformation. Be $V:=\mathbb{R}[t]_{\le d}$. Determine the matrix representation $A:=M_B^A(id_V)$ of $id_V$ regarding to the basis ...
0
votes
0answers
58 views

Notation in Linear Algebra

What does $(A\mid b)$ denote in Linear Algebra? Specifically in the context of the following question: "If $(A\mid b)$ is in reduced row echelon form, prove that A is also in reduced row echelon ...
9
votes
2answers
136 views

How many Matrices exist with increasing row and increasing column condition?

Given $N$, I would like to know the number of matrix constructed from $1$ to $N$ which satisfies the following condition: 1. The each row entries should be in increasing order. 2. The each column ...
0
votes
1answer
418 views

Eigenvector when all terms in that column are zero?

so I have this matrix: $$ \begin{matrix} 0.7 & 0 & 0 \\ 0.1 & 0.6 & 0 \\ 0 & 0.2 & 0.8 \\ \end{matrix} $$ I managed to solve ...
0
votes
1answer
22 views

Multiplying matrices, need some clarification simple dot product

I understand that when you want to multiply two matrices that the number of rows in the left matrix have to be equal to the number of columns in the right, otherwise the result of the multiplication ...
3
votes
2answers
164 views

Eigenvectors of the Zero Matrix

Given the following matrix: $ \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix} $. I have to calculate the eigenvalues and eigenvectors for this matrix, and I have calculated that ...
1
vote
2answers
146 views

Find a matrix such that $Ax=0$

Let $$W = span\left\{ {\left( {\matrix{ 1 \cr 0 \cr 0 \cr 1 \cr } } \right),\left( {\matrix{ 0 \cr 2 \cr 1 \cr { - 1} \cr } } \right)} \right\}$$ I was asked ...
0
votes
3answers
111 views

What is the dot product of two or three vectors graphically or visually?

I don't understand what the dot product actually is. I understand when and where to use it, but when it comes to proving things with it, I don't really grasp what it actually is making it difficult ...
1
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0answers
39 views

Product of two matrices with simple spectrum

We are given two square matrices $A$ and $B$ of the same size over the field of complex numbers and $\epsilon > 0$. Then it can be shown that there exist non-singular (even diagonalizable) ...
0
votes
1answer
31 views

For how many values of $a,b,c\in(1,2\ldots,p-1)$ does $p$ | $({a^2}-bc)$ where $p$ is an odd prime number

In a mock test for an entrance exam I am preparing for came the following question: Let $p$ be an odd prime number and $T_p$ be the following set of matrices $$ T_p= \left( ...
0
votes
1answer
24 views

question related to eigen value of matrix

Let $A$ belong to $M_2(R)$ be a matrix which is not a diagonal matrix. If $A^3 = I$, then why is $\operatorname{tr}(A) = -1$ and $\det(A) = 1$? I am trying to solve it as follows: let $x$ be an ...
0
votes
1answer
324 views

Finding the representing matrix with respect to the standard basis.

Let $B=[(1,0,0),(1,2,0),(1,2,3)]$ be the basis for $\mathbb{R}^3$. Let $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear transformation, such that its representing matrix with respect to basis $B$ ...
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0answers
107 views
0
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1answer
194 views

Regular matrix and regular stochastic matrix

We know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular if at a certain power all elements are positive. Question is how can I make the link between the ...
2
votes
1answer
43 views

is some of matrice with it's transpose positive definite? when eigenvalues of matrix is positive

Suppose M = A+ A^T , and we know that all of eigenvalues of A are real and positive, is M positive definite? or semi positive definite?
0
votes
1answer
29 views

what does this question about a matrix mean?

here is a question says : what does that mean ? I did my best to solve this question myself but i didn't find a way to solve it is this question possible or there is something else that i don't ...
0
votes
1answer
44 views

find minimal polynomial of $T(p)=p'+p$

I'm trying to solve the following question: let $T: \mathbb C_n[x] \to \mathbb C_n[x]$, $T(p)=p'+p$ find the characteristic and minimal polynomial of $T$. What I'm trying to do is the following: I ...
0
votes
1answer
39 views

Is this rank $1$ matrix is semidefinite?

I have a matrix, $X = xx^T$, where $x \in \mathbb{R}^n$. Is the matrix $X$ semidefinite?
1
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2answers
63 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
2
votes
4answers
98 views

how to prove that this determinant is a polynomial? [closed]

Given two square matrices $A$ and $B$ of size $n\times n$. I am wondering how to prove that $\det(A+xB)$ is a polynomial function of $x$ ? does anyone has a (simple if possible) proof of this fact? ...
2
votes
3answers
71 views

Is there any way to check wheter the determinant of a matrix $A$ with $|\text{det }A|=1$ is positive or negative?

Let $A\in\text{GL}(n,\mathbb{R})$ with $|\text{det }A|=1$. Is there any way to check wheter $\text{det }A$ is positive or negative without computing it?
0
votes
1answer
17 views

Find which points are unchanged under this reflection?

I am doing revision and I can't understand how to get the points here. The Question is The matrix $F = \pmatrix{0& -1& 0& 10 \\ -1& 0& 0& 10& \\ 0& 0& 1& ...
0
votes
1answer
53 views

Differences of skew symmetric matrices

Let $A$ be an invertible real skew-symmetric matrix, and consider the difference $A_R:=RAR^{-1}-A$, for orthogonal $R$. Is it true that $A_R$ is either zero or invertible? Does the answer depend on ...
1
vote
1answer
29 views

Matrix equality has a certain solution

I am wondering about the following matrix equality $$ \begin{pmatrix} 0 \\ 1 & \lambda_{1} \\ & 1 & \lambda_{2} \\ && \ddots & \ddots \\ &&& 1 & \lambda_{k} \\ ...
1
vote
2answers
87 views

b such that Ax = b has no solution having found column space

$A:=\begin{bmatrix} 2 & 6 & 0 \\ 3 & 1 & 3 \\ 1 & 0 & 0 \\ 4 & 8 & 1 \end{bmatrix}$ I've found the basis for the column space by doing row reduction (i.e. ...