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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29 views

Why $P D P^{'T}$ is positive semidefinite matrix

I am reading a paper about 3D mesh deformation. One statement makes me confused $ S = \sum_{i = 0}^N w_ie_i (e_i^\prime)^T = PD(P^\prime)^T $ where $e$ and $e^\prime$ are $3\times 1$ vectors ...
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0answers
16 views

Problems with the inverse of a banded matrix: not invertible?

I am creating with a software a banded matrix, which is also symmetric. In fact, its definition comes from an array, Array[q], whose length is ...
1
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2answers
31 views

Question about matrix multiplication notation

I have the following matrices: $A=\begin{pmatrix} -\frac{2}{3} & \frac{1}{3} & 0 \\ \frac{1}{6} & -\frac{1}{3} & \frac{1}{2} \\ \frac{1}{6} & \frac{1}{3} ...
0
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0answers
16 views

Setting up a matrix using logical constraints?

Hello all at Stack Exchange! This is my first post! It took me a while to learn MathJax, but a buddy who referred me said people heavily prefer this format, so I thought I'd just follow the rules of ...
0
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0answers
31 views

Algebraic multiplicity of an eigenvalue $λ$

I was going through a question posed on the expression for algebraic multiplicity of an eigen value $\lambda$ on this page : Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is ...
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0answers
12 views

How to get the projection matrix from coordinate/transformation?

I would like to compare my results with the groundtruth provided by a dataset. For each frame (image) in the groundthruth, I have a projection matrix. For example (for the 0th frame): ...
0
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1answer
33 views

What are the names of these variations on the transpose of a matrix and symmetric matrices?

Is there a name for the operator that reflects a matrix over the diagonal running from the top-right to the bottom-left? For the moment, define this reflection of a matrix $A$ as $A^*$. Is there a ...
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0answers
30 views

Confusion about finding matrix of Linear Transform w.r.t to different bases

I have come across two questions about matrices and changes of bases. They seem to be the same question, but require different approaches. I can't figure out why. First question can be found at: ...
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0answers
18 views

Shortest distance and Cross Product [on hold]

Show that the shortest distance from a point P to the line through Po with direction vector d is $$ ||P_oP \times d||/||d||$$. I need help writing the proof for this. So far I have: let $ ...
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0answers
6 views

On the interval minor extremal function of a j × k matrice.

I was going through papers by Marcus/Tardos and Fox and I have this small doubt. If L is a j×k matrix which has every entry equal to 1, what is the interval minor extremal function of L? Can someone ...
1
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2answers
23 views

Row Reduce Echelon Form on 3x4 Matrix

I understand the rules for RREF are: 1) Each leading entry must be a 1 in each row 2) Each leading entry's column must be 0's other than the leading entry 3) In stair case order, the next element of ...
0
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1answer
35 views

What is known about optimization of spectral properties of matrices over finite fields?

[I am solving the characteristic polynomial over complex numbers but since the matrices are symmetric all eigenvalues are real] Like for symmetric $d-$regular matrices over 0/1 or 0/1/-1 what are ...
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0answers
12 views

What is or how do you get the rotational matrix of 4-D vector onto the xyz-space?

which would make the 4-D component 0. To be honest I'm not really sure how 4-D rotations work. I know about the simple rotations but not the mechanism in how it rotates, and I'm not sure whether to ...
0
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1answer
17 views

Explain $(\|Mx\|_2)^2 = (M^Tx)^T(M^Tx) $ (positive definite, positive semi definite)

Would really appreciate if someone can explain: $$ (\|Mx\|_2)^2 = (M^Tx)^T(M^Tx) $$ can't get my head round with this.
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0answers
17 views

Find rank of parametered matrix

Today, I've got task: Find all values of parameter a, when rank of matrix M equals 2, where matrix M is 3x3 and has some dependencies on a, for example: $$\begin{pmatrix} 1 & a & a^2-1\\ 1-a ...
2
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2answers
58 views

Volume of a parallelepiped, given 8 vertices

Given the eight vertices $(0,0,0)$, $(3,0,0)$, $(0,5,1)$, $(3,5,1)$, $(2,0,5)$, $(5,0,5)$, $(2,5,6)$, and $(5,5,6)$, find the volume of the parallelepiped. I'm having trouble finding the 1 vertex ...
0
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2answers
43 views

Inverting matrix multiplication “and” representing with a smaller sized matrix

Consider I have a vector $A=[a_0 \ \ a_1]$ and a random binary matrix $B$ which is $2\times 2$. I compute $C=A\cdot B$. My question is: " Can one compute $B$ Given $C$ and $A$? " Note: By binary ...
2
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1answer
42 views

Matrix multiplication memorisation

So I'm writing an exam about matrices in a few weeks time, and I'd like to know if anybody has any tips about multiplying matrices.
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3answers
31 views

$n \times m$ matrix conversion?

Is it possible to convert an $n\times m$ matrix $A$ such that $$ A=CB $$ where $B$ is a $1\times m$ matrix which contains all elements of $A$, and $C$ is a $n\times 1$ matrix. I'm assuming no since ...
1
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1answer
78 views

If A is a square matrix, and A^2 = 0 then A=0 Prove true or provide a counter example?

This is a proof question and I am not sure how to prove it. It is obviously true if you start with A = 0 and square it. I was thinking: If $ A^2 = 0 $ then $ A A = 0 $ $ A A A^{-1} = 0 A^{-1}$ ...
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0answers
18 views

Why matrix can have only two possible Jordan Canonical Forms

If matrix $A = \left(\begin{array}{ccc}1 & 1 & 1 \\0 & 1 & 0 \\0 & 0 & 3\end{array}\right)$. Why is the case that the following two matrices, $B,C$ are the only two possible ...
1
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1answer
35 views

Are rotations the result of composing two reflections?(Linear Algebra)

I mean, is it true that every rotation matrix is the result of multypling one reflection matrix by another? If the answer is yes, how do I prove it? And what are the reflection matrices I can use to ...
0
votes
1answer
24 views

How do I find this basis given matrix representations?

Here is the question: Consider the multiplication operator $L_A:{\mathbb R}^2\to {\mathbb R}^2$ defined by $L_A(x)=Ax$ where $A=\left[\begin{array}{cc}2 &0\cr1 &-1\end{array}\right]$. Find an ...
0
votes
2answers
25 views

Decomposition of a diagonal matrix into a product of particular matrices.

Could you tell me please, if it's possible to find a decomposition of a diagonal matrix $ 3 \times 3 $ : $ D ( \lambda_1 , \lambda_2 , \lambda_3 ) = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 ...
0
votes
1answer
30 views

Is this function of matrix F convex?

A scalar function $f(F)=vec(F)^{H}\Big(\Omega^{H}\big(M\otimes F(I+F^{H}F)^{-1}F^{H}\big)\Omega\Big)^{-1}vec(F)$ convex? $\Omega$ is arbitrary matrix and $M$ is positive definite. F can be reviewed ...
0
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0answers
6 views

Is it possible to obtain a Hermitian, positive semidefinite matrix as some sum of non-commuting matrices?

I am working with generalized Pauli matrices given by $X \vert j \rangle = \vert (j+1)mod~p \rangle$, where $p$ is a prime number. $Z = \vert j \rangle = \omega \vert j \rangle$, where $\omega = ...
2
votes
3answers
34 views

How to explain the calculation of the determinant of a $4\times4$ matrix

In my linear algebra lecture notes, I am studying an example which concerns the calculation of the determinant of a $4 \times 4$ matrix, by first reducing the matrix to upper triangular form. (See ...
2
votes
0answers
31 views

Does this matrix normal form have a name and has it been used?

In a research paper in Theoretical Computer Science, we are using a certain matrix normal form, which I was not able to find in the literature (I have to admit that my Linear Algebra got a bit rusty, ...
0
votes
2answers
40 views

If $A$ is idempotent and $B=(I-A)$, then $BA'=I$ [on hold]

Given that $A$ is idempotent and $B=(I-A)$, then prove that $BA'=I$. I try this by taking two idempotent matrices..but i am confused
1
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1answer
42 views

The number of linearly independent solution of the homogeneous system of linear equations $AX=0$

I came across the following multiple choice question: The number of linearly independent solution of the homogeneous system of linear equations $AX=0$, where $X$ consists of $n$ unknowns and $A$ ...
0
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0answers
40 views

Writing an abelian group as a direct sum of cyclic groups

I have this problem Determine an isomorphic direct sum of cyclic groups, where $V$ is an abelian group generated by $x,y,z$ and subject to:: $x+y=0, 2x=0, 4x+2z=0, 4x+2y+2z=0$ So I wrote ...
0
votes
1answer
21 views

How to use the crank-nicolson method

I'm going over my study questions for an exam I have tomorrow in Applied Numerical Methods and I know everything except for one thing. There's a sample question about using the Crank-Nicolson method, ...
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1answer
13 views
4
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1answer
27 views

Representing Matrix Subgroups

Suppose I wanted to describe the subgroup of $GL_n(\mathbb{R})$ of matrices of the form $$ \left [ \begin{array}{cc} A & B \\ 0 & C \\ \end{array} \right ] $$ where $A \in ...
1
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1answer
21 views

Possibilities of minimal polynomial for a matrix

I came across a problem recently in my linear algebra studies that went something like this: Let $A$ be a linear transformation on a finite-dimensional space $V$ with characteristic polynomial $(x ...
0
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0answers
11 views

Magnitude of linear transformation map

In a 2D situation: given a 2x2 matrix A and a vector $\vec u$ , does the magnitude $|A \vec u|=|\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \begin{pmatrix} ...
0
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0answers
17 views

Parallelogram with vertices 0, Xa, Xb, Xa+Xb (X is matrix, a and b are vectors)

There is a paralellogram with vertices 0, a, b, and a+b, whose area is $34$. What is the area of the parallelogram which has vertices 0, Xa, Xb, and Xa+ Xb, where X = \begin{pmatrix} 3 & -5 \\ -1 ...
0
votes
1answer
25 views

Find transformation matrix $T$ relative to new bases

T is a linear transformation represented as $\left(\begin{array}{ccc}1 & 1 & 0 \\0 & 2 & 0 \\3 & 1 & 0 \\0 & 1 & 1\end{array}\right)$ w.r.t the standard basis. Now ...
0
votes
2answers
55 views

What is the simplest way to find an inverse matrix?

let $A = \left( \begin{array}{cccc} 1 & -1 & 2 & -1\\ -1 & 2 & -3 & -2 \\ 2 & -3 & 7 & 5 \\ 3& -2 & 6 & -3\end{array} \right)$ I want to find the ...
0
votes
0answers
18 views

Inverse of a rigid transformation

I would be grateful for any help with the steps required to complete this calculation. You may assume that I have some experience with matrices from before, but I am obviously no master! So we have ...
1
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4answers
64 views

Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$

I was working on this problem Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$. My attempt:- I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should ...
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0answers
29 views

Relation between Eigenvectors and the commutivity between square matrices?

I'm stuck on this in a proof, any help is greatly appreciated! Thanks so much in advance! [EDIT] Okay, so basically I'm stuck on a proof, I need to proof that all eigenvectors of B are unique, ...
0
votes
1answer
32 views

About the special linear group $SL(n,\,\mathbb{Z})$

I found the following relation: $$SL_n(\mathbb{Z})=\langle e+e_{12},\, e_{12}+\dots+e_{n-1, n}+(-1)^{n-1}e_{n1}\rangle$$ where $e_{ij}$ is the matrix with its $(i,\, j)$th entry $1$, and all other ...
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0answers
86 views

If $AB$ has zero trace for every matrix $B$ of zero trace, then $A$ is a scalar matrix

Let $A$ be an $n \times n$ real matrix such that $\operatorname{Trace}(AB)=0$ whenever $\operatorname{Trace}(B)=0$. Show that $A=cI$ for some $c \in \mathbb{R}$. My attempt Let $U$ be the subspace ...
2
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2answers
29 views

Relation Between Eigenvalues of Block Matrices

Is there any relation between eigenvalues, or spectral radii, of $M$, $M_1$, and $M_2$ block matrices? \begin{equation} M= \begin{bmatrix} A&B\\B^T&C \end{bmatrix} \end{equation} ...
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0answers
11 views

How do you calculate the kernel of a matrix

How to find a kernel of a matrix. The teaching slides wasn’t very helpful. Can someone show me how you would find a kernel. 3...0... 0...-3 2...0...-1...1 3...-3...3...13 1...-1...1...4 If anyone ...
1
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0answers
44 views

Integration.Matrix.Determinant.Inverse.Trace.

Given $$ I_n=\int_0^1\frac{x^n}{x^{2012}-1}{\rm d}x\text{ and }J_n=\int_0^1\frac{x^n}{x^{2013}+1}{\rm d}x\quad\forall n>2012, n\in\mathbb N$$ If the matrix $$\rm A=[a_{ij}]_{3\times3}\text{ where ...
3
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0answers
16 views

How to use a Matrix Pencil to decompose Exponentials

I'm an Engineering wanting to do some analysis on decomposing exponentials. My problem is that in order to get the highest resolution results I need to decompose a signal that exponentially decays in ...
3
votes
2answers
106 views

Is this matrix positive semidefinite for all $n$?

This is an extension of my previous question (see here). In this follow-up problem extra ones have been added in the non-diagonal matrix elements. We want to prove the positive semi-definiteness of ...
1
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0answers
34 views

Eigenvector and eigenvalue of an infinte, symmetrical matrix

How to get eigenvectors and eigenvalue of an infinite matrix like $$ A= \begin{pmatrix} 1&0&1&0&\dots\\ 0&1&0&1&\dots\\ 1&0&1&0&\dots\\ ...