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
107 views

If $F\subseteq\mathrm{Mat}_n(\mathbb{Q})$, then $[F:\mathbb Q]\leq n$?

Let $F$ be a field contained in the ring of $n\times n$ matrices over $\mathbb Q$. Prove that $[F:\mathbb Q]\leq n$. I have an idea to consider a degree $n$ extension $K$ of $\mathbb Q$ and left ...
3
votes
2answers
189 views

Question on cofactors

If the row sums of a symmetric matrix of size 4 by 4 are all $0$, then why are all the cofactors of the matrix equal? Thanks in advance for any helpful answers.
1
vote
1answer
535 views

Proof of adjoint(ab) = adjoint(b)adjoint(a)

So I'm trying to prove whether $\operatorname*{adjoint}(AB) = \operatorname*{adjoint}(B)\operatorname*{adjoint}(A)$. Here, for any matrix $C$, the matrix $\operatorname*{adjoint}(C)$ is defined as ...
1
vote
2answers
30 views

Finding column and row space without computing A.

I have the a question that asks that I find the column space and row space of: $$A = \begin{bmatrix}1&2 \\4&5 \\2&7\end{bmatrix} \begin{bmatrix}3&0&3 \\1&1&2 ...
1
vote
1answer
67 views

How does the Maximal Eigenvalue of the Prime Index Matrix behave?

I had a look at the eigenvalues of the matrix, I called it Prime Index Matrix (is there a better name?), constructed like the following: $$ P_{k,p_k}=P_{p_k,k}=1, $$ where $p_k$ is the $k$th prime. ...
5
votes
1answer
83 views

Eigenvalues appear when the dimension of the Prime Index Matrix is a prime-th prime. Why?

I had a look at the eigenvalues of the matrix, I called it Prime Index Matrix (is there a better name?), constructed like the following: $$ P_{k,p_k}=P_{p_k,k}=1, $$ where $p_k$ is the $k$th prime. ...
0
votes
4answers
96 views

Why is this not a 1-1 Function?

Linear mapping: \begin{align*} F: \mathbb R^3 &\to \mathbb R^2,\\ \begin{pmatrix} x\\y\\ z \end{pmatrix} &\mapsto \begin{pmatrix} x\\y\\ \end{pmatrix} \end{align*} I thought the check for ...
1
vote
1answer
69 views

How to prove this kernel is positive semi definite

How to prove $k(x_i,x_j)=e^{-(LR(x_i-x_j))^TLR(x_i-x_j)}$ is a valid kernel function or positive semi definite? $x=(\mu,\lambda)^T$ and R is a 2x2 rotation matrix, L is a 2x2 diagonal scaling matrix ...
7
votes
4answers
355 views

Largest singular value of non square matrix

Let $B$ be an $m\times n$ matrix with complex number as its element. Let $\sigma$ denotes the largest singular value of $B$ Prove that \begin{equation} \sigma = \max\limits_{\|u\|_2=1,\|v\|_2=1} ...
1
vote
1answer
47 views

matrix row/col mapping

Many square matrices are symmetric. i.e. $a_{i,j}=a_{j,i}$ For such matrices, we can only store the upper triangle elements, i.e. any $a_{i,j}$ for which $i<=j$. Assume a 10x10 matrix. Using this ...
1
vote
1answer
221 views

Column Space of Square Matrix

Assume that I have a $3 \times 3$ matrix $A$ with columns $A_1$, $A_2$, and $A_3$ that are linearly independent. Say that I want to find the column space of A. Isn't it possible for me to find some ...
0
votes
1answer
76 views

Proving inverse of transpose is transpose of inverse [duplicate]

I'm to prove that $(A^T)^{-1} = (A^{-1})^T$ but I'm not really getting anywhere. What I've got so far: $\frac{1}{detA} \cdot A^T = \left( \frac{1}{detA} \cdot A \right) ^T$ But that gets me ...
0
votes
0answers
54 views

Multiplying matrices of matrices

A question about multiplying the matrix below that is a variation Henderson's mixed model equations: |M1 M2| |M5 M6| |M3 M4| |M7 M8| whre M1-M8 are matrices. ...
1
vote
0answers
120 views

Solving a Large band system using Gauss-Seidel Iteration

Sorry for my english. I have to solve the following band system using Gauss-seidel iteration program in matlab. $$ \begin{array}{cccccc} 12x_1&-2x_2&+x_3&&&=&5\\ ...
0
votes
1answer
42 views

What constraints do we get on the matrices $A,B$ when we require $AV=VB$?

The matrices $A$ and $B$ are, a priori, general unitary $3\times3$ matrices and $V$ is some fixed unitary $3\times3$ matrix. When I impose the following requirement on $A$ and $B$: \begin{equation} AV ...
2
votes
2answers
39 views

Merging Linear Regression

If I have built two linear regression models over sets $A$ and $B$, and now want a linear regression over set $A\cup{}B$. Is there a way to reuse what I already have?
1
vote
0answers
85 views

Algorithm to determine matrix equivalence

I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of. I have two real rectangular matrices $A_{mxn} ...
0
votes
2answers
46 views

How to prove this implication?

Let $A$ be a matrix $n \times n$ and $b_1....b_k$ are $k$ vectors in $\mathbb{R}^{n}$. Does anyone know how to prove the following implication $Ab_1, ..., Ab_{k}$ is a spanning set of ...
1
vote
1answer
42 views

Basis of this vector space.?

Let F denotes the set of sequences such that $u_{n}+u_{n+1}-u_{n+2}=0$. How to find a basis of this vector space? Thanks.
0
votes
2answers
32 views

Line in vector form?

Given the line y=3x my book states it is $\left(\begin{array}{c}1 \\ 3\\\end{array}\right)$ as a matrix. Why is it not $\left(\begin{array}{c}3 \\ 1\\\end{array}\right)$, I thought the upper number ...
3
votes
2answers
98 views

Basis of Kernel of a matrix

Given $\theta>0$. Let $H$ be $5 \times 6$ matrix $$\left[\begin{matrix} 1 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & -1 ...
2
votes
1answer
72 views

Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ? To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
4
votes
1answer
157 views

Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
1
vote
1answer
75 views

Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
0
votes
0answers
69 views

Summing the product of combinations of matrix elements

I have a situation where I have an $NxN$ matrix $A$ where each element $a_{i,j}\in\mathbb{R}_{\leq 0}$. I would like to consider the set of all collections of elements such that each collection of $N$ ...
2
votes
3answers
138 views

to prove $f(P^{-1}AP)=P^{-1}f(A)P$ for an $n\times{n}$ square matrix?

let $f(X)$ be a polynomial and let $A$ be $n\times n$ matrix.We have to show that for any $n\times n$ invertible matrix $P$, $f(P^{-1}AP)=P^{-1}f(A)P$ and that there exist a unitary matrix $U$ such ...
1
vote
2answers
59 views

Non-nilpotent matrix with $0$ determinant

I know that any nilpotent matrix $M$ has $\det(M)=0$, because $M^k=0$ and thus $\det(M^k)=0$. Are there any simple examples of matrices $A$ that do have $\det(A)=0$ that are not nilpotent? I've tried ...
0
votes
3answers
231 views

how to find a matrix A given the solution?

if we need,for example, to find a nonzero 3x3 matrix A such that we are given a 3x1 vector as a solution to Ax = 0. What is the general procedure we can follow to obtain such Matrix A? Thank you :)
1
vote
1answer
64 views

One definition of the determinant of a matrix

Suppose you define as follows : for $(a,b,c,d)\in \mathbb{R}^4$, $\det \begin{pmatrix} a & b \\ c & d\end{pmatrix} = ad-bc$. for $A$ a square matrix of size $n$, you define $\det A$ ...
0
votes
1answer
61 views

Position of 20096 in triangular array of all natural numbers

Write the set of all natural numbers in a triangular array as Find the row number and column number where $20096$ occurs. For example, $8$ occurs on row: $3$, column: $2$ Now, the upper row is ...
0
votes
1answer
40 views

A problem about the matrix equation $x^{\sf T}Ax+x^{\sf T}By+y^{\sf T}Cy=0$.

Let $k\in\mathbb{N}, n\in\mathbb{N}, p\in\mathbb{N}, A\in\mathbb{R}^{k\times k}, B\in\mathbb{R}^{k\times n},$ and $C\in\mathbb{R}^{n\times n}$. If $A^{\sf T}+A=0_{k\times k}, B=0_{k\times n},$ and ...
0
votes
2answers
73 views

Is the space isomorphic?

$\mathcal{P}_5$ and $\mathbb{R}^5$. So $\mathbb{R}^5$ has a dimension of 5, but how do you determine the dimensions of $\mathcal{P}_5$? Any element of $\mathcal{P}_5$ is of the form ...
0
votes
1answer
171 views

Matlab - Generate square convex function with positive definite Hessian Matrix

So, I have to generate a square convex function in Matlab and it's Hessian Matrix must be positive definite but I can't find any function that can help me do that. Is there anything I should search ...
2
votes
1answer
143 views

The Matrix Inversion Lemma: the General Case

I find it is hard to understand the application senario of the Matrix Inversion Lemma in non-special cases. Suppose I already computed $A^{-1}$ and want to find $\left(A+UCV \right)^{-1}$. The Matrix ...
1
vote
1answer
86 views

Proving group properties of $G$, a set of $2 \times 2$ matrices with rational entries

Let $G$ be the set of all $2 \times 2$ matrices whose entries are rational numbers and whose determinant is equal to $3^n$ where $n$ is a nonnegative integer. Prove that $G$ is a group with respect ...
1
vote
0answers
85 views

find the standard basis of a linear transformation

Let $\beta =\{b_1,b_2,b_3\}$ be a basis for the vector space V and let $T:V\to R^2$ be a transformation with the property that $$T(x_1b_1 + x_2b_x + x_3b_3) = \begin{bmatrix} 2x_1-3x_2+x_3\\ ...
3
votes
3answers
80 views

How to construct a matrix $A$

Construct a matrix $A$ such that $A^2\ne 0$ but $A^3=0$. I need your help to find $A$. Please help. Thanks in advance.
2
votes
2answers
83 views

Partial Ordering of Positive Definite Matrices

$\mathbf{A}, \mathbf{B}, \left(\mathbf{A-B}\right)$ are all positive definite Hermitian symmetric matrices of same dimensions. Prove that $\left(\mathbf{B}^{-1}-\mathbf{A}^{-1}\right)$ is positive ...
0
votes
0answers
26 views

Rank of Rectangular Matrix

How I will be able to found: How many matrix, over $\mathbb{F}_q$, of dimensions $r\times m$ with $r$ L.I columns between the positions $r+1$ and $m$ exist?
0
votes
1answer
162 views

arrow structure matrices and Sherman-Morrison-Woodbury

I have two questions regarding "arrow structured" matrices and I'll be grateful if you can give more insights about them: 1- If A is an n-by-n SPD and has the arrow structure, e.g. A=[x x x x;x x 0 ...
1
vote
3answers
197 views

Help complete an exercise - Computing Determinant/Solution to 3$\times$3 matrix

The following is an excerpt from a book I am using: So, now I would like to compute $x_2$ and $x_3$ using the same approach that the author used to compute $x_1$. Here's my attempt for $x_2$: ...
2
votes
1answer
82 views

N-th power of matrix

Find the formula for the n-th power of this matrix. $$ \pmatrix{1&1\\1&0} $$ Well $f^2 = \pmatrix{2&1\\1&1}$ and $f^3 = \pmatrix{3&2\\2&1}$ and $f^4 = ...
0
votes
0answers
67 views

Why can I not generalize O(n^log5) for squaring matrice of size n

I have a question that is bugging me for around a 3 days, I first asked this question in stackoverflow but no one could answer it reasonably though they tried to help, so finally I found here as a ...
0
votes
2answers
40 views

What does this map represent? Matrix multiplication

Represent the derivative map on $Pn$ with respect to $B$, $B$ where $B$ is the natural basis $(⟨1,x,...,x^n⟩)$ how that the product of this matrix with itself is defined; what map does it represent? ...
0
votes
1answer
48 views

How to judge the covariance matrix?

As for a numeric matrix, how to judge that it is a covariance matrix? For example, the following $3\times3$ symmetric positive matrix is not a covariance matrix. $$ \begin{pmatrix} 2 ...
2
votes
1answer
57 views

Best algorithm for computing eigenvalue decomposition of a $3 \times 3$ symmetrix matrix

In one of my applications, I need to compute the eigenvalue decomposition of a $3 \times 3$ symmetric matrix. What algorithms can I use? Which is the most efficient one? More specifically, the ...
-1
votes
1answer
44 views

For each vector space of functions of one real variable, represent the derivative transformation with respect to B,B.

{acosx +bsinx | a,b ∈ R}, B = < cosx, sinx > My work is the following: {{1},{0}} h-> cos x {{0},{1}} h-> sin x RepB(cos x) = {{1},{0}} since cos x = Acosx +Bsinx RepB(sin x) = {{0},{1}} My ...
1
vote
2answers
54 views

Is a matrix equivalent with its row reduced one?

If I manage to reduce a matrix A to the identity one, does that mean that I can actually use it on any given equation instead? $$AB = 0 \implies IB = 0\;\;?$$
1
vote
1answer
57 views

bound on matrix inverse with different elements

I'm hoping that someone can point me to some literature on the following. Is there a way to bound the inverse of a matrix if I change the value of 1 element in that matrix. Let's say I have a matrix ...
0
votes
2answers
45 views

Relation between the weighted matrix norm and the weights

For a nonsingular matrix $W \in \mathbb{C}^{m\times{}m}$, the weighted vector norm is defined as $||\overrightarrow{x}||_W = ||W\overrightarrow{x}||$. Let $||A||$ denote the induced matrix norm by the ...