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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0answers
51 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 ...
0
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1answer
41 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 ...
1
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0answers
58 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: ...
1
vote
2answers
239 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 ...
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1answer
55 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
19 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
21 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
46 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 ...
3
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1answer
771 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 ...
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2answers
49 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
57 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.
2
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3answers
39 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
1k 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}$ ...
1
vote
1answer
77 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
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2answers
36 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
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1answer
37 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 ...
2
votes
3answers
56 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
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0answers
45 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
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2answers
69 views

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

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
2
votes
1answer
608 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 $A$ ...
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0answers
70 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 ...
5
votes
1answer
74 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
70 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 ...
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0answers
56 views

Parallelogram with vertices $\mathbf{0}$,$\mathbf{Xa}$,$\mathbf{Xb}$,$\mathbf{Xa+Xb}$ ($\mathbf{X}$ matrix, $\mathbf{a}$ and $\mathbf{b}$ vectors)

There is a paralellogram with vertices $\mathbf{0}$, $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{a+b}$, whose area is $34$. What is the area of the parallelogram which has vertices $\mathbf{0}$, ...
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1answer
76 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
110 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
1answer
543 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
vote
4answers
78 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 ...
1
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1answer
45 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 ...
2
votes
2answers
54 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} ...
1
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1answer
161 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
votes
2answers
140 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
43 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\\ ...
1
vote
1answer
354 views

How to calculate Cartesian coordinates for an element after rotation has been applied?

I have a square on a Cartesian coordinate system with origin (0,0) on top left (yellow arrow from the picture). The initial coordinate of the square from the ...
0
votes
0answers
70 views

Matrix norm optimization problem : $\min_{\textit{ }x} \| A x B \|_4$, $x$ in the “unit” circle

Bonjour, Let $A$, $B$, $C$ and $D$ complex matrices. Is there a way to find a matrix $x$ (edit: non trivial) as: $\min_{\textit{ }x} \| A x B \|_4^4$ Or, more complicated, $x$ as $\min_{\textit{ ...
4
votes
1answer
102 views

Positive semidefinite matrix proof

Let $e_i$ the $i$-th column of the identity matrix. Is there an easy way to prove that the matrix $$\left[\matrix{\mathbb{I}_n & e_1e_2^T & \cdots & e_1e_n^T\\ e_2e_1^T & ...
0
votes
1answer
177 views

On matrices with trace value zero

I would like to ask you something regarding the trace of a matrix (the value of the diagonal after adding all its members, a value which is said to remain constant independently from base changes): ...
0
votes
2answers
72 views

What are some interesting properties of $A - 2I$ when the matrix $A= \tiny \begin{pmatrix} 1&*&* \\ 0&2&* \\ 0&0&3\\ \end{pmatrix}$

Without using the Jordan forms, What are some interesting properties of $A - 2I$ when the matrix $A= \begin{pmatrix} 1&*&* \\ 0&2&* \\ 0&0&3\\ \end{pmatrix}$. Attempt: $A - ...
1
vote
1answer
32 views

Which way to write the transition matrix arrow?

On a study guide my professor writes the problem: Let $B\:=\:\left\{\left(1,-1\right),\left(-2,1\right)\right\}$ and $B'=\left\{\left(-1,1\right),\left(1,2\right)\right\}$ be bases for $\mathbb{R}^2$ ...
0
votes
3answers
45 views

Find determinant value

\begin{vmatrix} 3 & 2 & 0 & 0 & . &. & . & . &0 &0 \\ 1 & 3 & 2 & 0 & . &. & . & . &0 &0 \\ 0 & 1 & 3 & 2 & . ...
3
votes
1answer
105 views

How can I project a matrix on the set of symmetric positive definite matrices with trace 1?

Given a square matrix $A \in \mathbb{R}^{n \times n}$, I need to compute $$ \min_{X \in \Omega} \lVert A - X\rVert^2$$ where $\Omega = \{X \in \mathbb{R}^{n \times n} |\, tr(X) = 1, X \text{ is ...
0
votes
2answers
38 views

A matrix $Q$ has orthonormal columns, but $QQ^T \neq I$

I have to find an example of a matrix $Q$ that has orthonormal columns, but $QQ^T \neq I$. If a matrix has orthonormal columns, it does not imply that the matrix is orthogonal, so that it is a ...
2
votes
0answers
34 views

How to solve the equation $Au+Bv=C$

How do I solve $Au+Bv=C$ Where $A$ and $B$ are constant known matrices that are nxn, $C$ is a constant known nx1 vector while $u$ and $v$ are unknown nx1 vectors with the condition given that $u_i = ...
0
votes
2answers
126 views

Direct sum of range(B) and kernel(B*)

Is it true that for some matrix $B$ of dimension $d\times d$, $ker(B) \oplus Range(B^*) = \mathbb{C}^d$???. I know that dim$(Range(B)) + $dim$(ker(B)) = $dim$(\mathbb{C}^d)$ but I don't know how to ...
3
votes
0answers
95 views

Does this have a name?

While messing around, I seem to have stumbled upon an interesting family of matrices: $$\mathbb{S} = \bigg\lbrace A\in\mathbb{M}_{n\times n}(\mathbb{R}) : A^{T}A=AA^{T}=\frac{1}{2} (A + ...
2
votes
1answer
53 views

algebraic equation with trace

I have a problem which I don't know how to attack. Actually I am not even sure there is a way to do it. Is it possible to solve an equation of this form $$A²-\frac{tr(A)²}{4}Id_{4\times 4}=B$$ where ...
0
votes
1answer
50 views

Find basis for kernel and matrix representation

Problem 4 from https://math.berkeley.edu/~ogus/Math_54-07/Exams/midsol1.pdf $\beta$ is a basis of $P_3$, the set of all polynomials of at most degree 3.$\beta = (x^0,x^1,x^2,x^3)$. Let $T$ be a ...
1
vote
0answers
38 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
0
votes
1answer
50 views

Find the matrix $A$

Let $A$ be a matrix such that $A\vec{x}=\begin{bmatrix}2 \\ 4 \\6 \end{bmatrix}$, where $\vec{x}=\begin{bmatrix}2 \\ 0 \\0 \end{bmatrix}+c\begin{bmatrix}1 \\ 1 \\0 \end{bmatrix}+d\begin{bmatrix}2 ...
2
votes
1answer
34 views

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the ...