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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8 views

Eigenvalues of the subtraction of a gram matrix and a psd rank $1$ matrix.

If $V$ is $p \times r$ matrix ($r<p$) with rank $r$, then $G=V^TV$ is a $r \times r$ gram matrix with rank $r$. Let $x$ be a $r \times 1$ vector, and $xx^T$ is a psd rank $1$ matrix. It's already ...
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0answers
21 views

Attempt to solve a matrix (counterbalancing) problem computationally gives “spooky” result: why?

This question is posted on the mathematics section of stackexchange because my uneducated guess is that the answer involves some basic mathematical principles, possibly in the domain of linear ...
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0answers
17 views

If two linear systems are equivalent, they have the same size augmented matrix.

If two linear systems are equivalent, they have the same size augmented matrix? It is false but do any one know why for this?
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1answer
20 views

Multiplying inverse matrices easily.

Okay, this is more of a confirmation question than anything: I have been given two matrices $A^{-1}$ and $B^{-1}$. Then the inverses of these are: $A$ and $B$. I need to calculate $(AB^{T})^{-1}$. ...
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1answer
14 views

Alternative methods to solve DLP for $GL_{3}(\mathbb{F}_2)$

Is there (or rather what is) a more elegant/efficient way to solve the DLP for $g^x=h$ in $GL_3(\mathbb{F}_2)$ where $$g=\begin{pmatrix}0 &1 & 1 \\ 1 &1 &1 \\ 1&0&1 ...
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1answer
17 views

Can square matrices be represented as the union of vectors and some other set?

I believe all invertible matrices can be representable as $A = |A| \, \mathrm{adj}\left(A\right)^{-1}$ (a rotation part times a scaling part.) All invertible matrices can then be mapped to vectors ...
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3answers
26 views

Matrices and diagonalization.

I could verify that $P$ statement is false by just calculating the determinant but couldn't answer $Q$ statement. Any clue about $Q$??
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1answer
12 views

Row Equivalent Matrices

If I have a matrix $A$, where there are zeros everywhere apart from the first row, what are the matrices that are not row equivalent to $A$. I know that if two matrices are row equivalent, we can ...
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1answer
32 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
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votes
2answers
34 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
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2answers
2k views

Is the set of all invertible $n \times n$ matrices a vector space?

I'm studying Algebra and I'm asked to prove or disprove "Is the set of all invertible $n \times n$ matrices a vector space?" I assume with respect to the usual matrix-sum and scalar multiplication. I ...
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1answer
42 views

A question on boundary of set

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
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2answers
32 views

It is true that $rank(xy^T)=1$? [on hold]

Let $x,y\in \mathbb{C}^n$. It is true that $rank(xy^T)=1$?
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votes
2answers
48 views

Testing the diagonalizability of matrix $B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$

How to show that the matrix $$B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$$ is diagonalizable when $a\neq0$, when ...
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0answers
21 views

Gaussian Elimination vs matrix inversion [on hold]

Why Gaussian Elimination is better than matrix inversion in therms of FLOPS? Also how LU decomposition improves the shifted inverse power method?
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0answers
9 views

Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...
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0answers
18 views

Matrix transformation into block off-diagonal form

Consider the 4-by-4 matrix $\boldsymbol M = \boldsymbol M_0 + \boldsymbol M_1$, where $\boldsymbol M_0 = \alpha \left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 ...
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0answers
14 views

Eigen vectors and determinant of a block matrix

I have two questions regarding matrix $A$. The matrix $A$ can be partitioned into four tridiagonal matrices $A_1$, $A_2$, $A_3$ and $A_4$. $$A=\begin{pmatrix} A_1&A_2\\A_3&A_4 \end{pmatrix}$$ ...
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1answer
22 views

Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
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3answers
11k views

How is the column space of a matrix A orthogonal to its nullspace?

How do you show that the column space of a matrix A is orthogonal to its nullspace?
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1answer
37 views

inequality with a positive matrix

Let $$ A=\left[ \begin{array}{cc} a & b\\ \overline{b} & c\\ \end{array} \right]$$ be a positive semi-definite positive of $M_2(\mathbb{C})$. How prove the inequality $ac \geq ...
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2answers
23 views

Null space is an Invariant subspace

Let $\lambda$ be an eigenvalue of a square matrix $A$. Show the null space of $(A-\lambda I)^j$ is an A-Invariant subspace of $\mathbb{C}^n$ for all positive integers $j$. Proof without requiring ...
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1answer
29 views

What is meant by In-Place Matrix Inversion?

I come across the term "In Place Matrix Inversion" a lot in numerical libraries like NumPy and ND4J. What does it mean ? How is it different from the normal matrix inversion ? What are the advantages ...
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2answers
107 views

Conditions for the value of a determinant to be zero

The theory states that the value of a determinant will be zero if it contains a row or column full of zeroes or if it has two identical rows or two rows proportional to each other. Similarly can we ...
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0answers
27 views

SVD of a Matrix Product

Suppose we have a matrix $A$ with dimensions $m$ by $n$ and a column-wise permutation matrix $R$ (re-orders columns) with dimensions $n$ by $n$. Then we have a matrix $X$ which is constructed as $X ...
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votes
3answers
126 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
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0answers
119 views

Is there matrix representation of the line graph operator?

I had the need to calculate the adjacency matrix $L$ of the line graph of a certain planar $k$-regular graphs $G(n,e)$ ( $n$ vertices and $e=\frac k2 n$ edges) given its adjacency matrix $A_G$. Here I ...
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votes
1answer
41 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
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1answer
389 views

changing bases/rotating axes to find reflection across y=2x

Find the (exact) reflection of the vector v = (5, 1) across the line: y = 2x. Hint: A sketch of v and the line may suggest an approach. I found the matrix -3/5 6/5 4/5 2/5 which seems like it gives ...
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1answer
33 views

Is matrix $A^i A^j = A^j A^i$

I want to know if $$A^i A^j = A^j A^i$$ holds or not. It seems like an obvious, but I am wondering if there is a more formal proof
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2answers
51 views

Does $Ax=b$ have a solution for every vector $b$ in $\mathbb{R}^3$

Let $A$= $\begin{bmatrix}3 & 1 & -1\\0 & 4 &0\\6 &3&-2\end{bmatrix}$ and $x= \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$ and $b=\begin{bmatrix}0\\4\\1 \end{bmatrix}$ Does $Ax ...
0
votes
2answers
45 views

How can I show that for matrix $A$ , $A^t A $ is not equal to $ A A^t $ in general?

How can I show that for matrix $A$ , $A^t A \neq A A^t $ $A^t$ means the transpose of $A$. That is the entire question and I have no idea how to begin... please help!
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1answer
47 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb ...
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2answers
34 views

How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans?

Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$ Apparently $\dim W_1=\dim W_2=2$. For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as ...
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0answers
24 views

Looking for mathematical/combinatorial and computational explanation regarding adding values in a $5 \times 4$ (matrix?) with a constraint.

Given the following matrix (not sure if I should call it that): Matrix $5 \times 4$ I want to add all possible combinations of values such that each Horse gets but one value from each Bookie. What I ...
4
votes
1answer
3k views

How to construct the graph from an adjacency matrix?

I have the following adjacency matrix: a b c d a [0, 0, 1, 1] b [0, 0, 1, 0] c [1, 1, 0, 1] d [1, 1, 1, 0] How do I draw the graph, given its adjacency ...
0
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1answer
403 views

Simple Probability Matrix

Question: Consider a simple model that predicts whether you pass your next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will ...
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0answers
43 views

maximum frequencies of numbers in a matrix

I have a matrix A of size n*n.Consider a new matric M : M[i][j]=max of frequencies of numbers occuring in ith row and jth column(A[i][j]) counted once. I have a ...
0
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1answer
29 views

Dimension of the image of a matrix

So the question asks: Verify if the image of the linear map $T : \mathbb{R}^6 \to \mathbb{R}^3$ given by left multiplication by A= $$\begin{bmatrix}6 & 0 &2 & 2& 3& 4\\0 & -1 ...
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1answer
22 views

choosing a square matrix to have a product with one 1 und other 0's

Let $A$ be a $m\times n$ real matrix with maximal rank. Let $i\in\{1,\dots,m\}$, $j\in\{1,\dots,n\}$. I'm curious if it is possible (for any choice of $i,j$) to find a square matrix $B$ such that ...
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1answer
52 views

Compare analytic model with numerical, mass spring system.

So I'm trying to solve a problem here and I have been working on it all day, clearly i'm in need of some guidance. I have a rod of length $L$ and cross section area $A$, Young's modulus $E$ and ...
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1answer
35 views

Matrix equation implies invertibility

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix with positive entries $\lambda_i > 0$ (some of them might coincide). If we have the matrix equation $A D A^t = ...
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2answers
39 views

Unit vectors with imaginary numbers

I'm trying to determine if the matrix: \begin{bmatrix} 0 & i \\ 1 & 0 \end{bmatrix} is a unitary matrix. Therefore, the first step I'm taking is to figure out if both $\langle 0, 1\rangle$ ...
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0answers
30 views

What is the maximum value of coefficient $f_v$ with the constraint that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
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0answers
9 views

Entry Expansion of Power Matrix

Suppose $A:=\{a_{i,j}\}, 1\le i,j, \le n$ is a $n\times n$ matrix with real positive entries. Now replace the constant $a_{1,1}$ with a real variable $x$. Denote by $A_x$ the resulting variable-Matrix ...
3
votes
1answer
40 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
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0answers
29 views

Linear transforms and their corresponding invertible matrix.

Let $(1,x,x^2,x^3)$ be a basis for $\mathscr{P_3}(\mathbb{R})$ and let $(1,x,x^2,x^3,x^4)$ be a basis for $\mathscr{P_4}(\mathbb{R})$. Suppose $R \in ...
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0answers
23 views

Matrix Inverse as Series

I am looking for different representations of the inverse of a matrix as a power series. One obvious candidate is the Von Neumann series which is given $$A^{-1} = \sum_{k=0}^{\infty} (I-A)^k$$ ...
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0answers
12 views

Is it always possible to find the Reduced Row Echelon form of a matrix, given the basis of its null space? [on hold]

I tried starting with multiple bases of the null space and each time I was able to write the RREF form of the matrix. However, I have not been able to prove that this is true for all possible bases.
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1answer
36 views

When is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible?

The question is quite simple: for a $N \times p$ matrix $\mathbf{X}$ with real entries, when is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible (where $\mathbf{I}$ is the $p \times p$ identity ...