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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1
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1answer
411 views

Rotate a vector in higher dimension

I am trying to figure out way to rotate a vector X=[x1,x2.........,xn] taking another point p as a reference, where the vector is in higher dimension with n>3 something like n=30. At this point, I am ...
4
votes
1answer
74 views

Find a lower bound

Let $M$ be an $N\times N$ symmetric real matrix, and let $J$ be a permutation of the integers from 1 to $N$, with the following properties: $J:\{1,...,N\}\rightarrow\{1,...,N\}$ is one-to-one. $J$ ...
2
votes
3answers
125 views

symmetric positive definite matrices

Why must a symmetric positive definite matrix must be invertible? I'm reading a proof of the Levi-Civita theorem in differential geometry but the author states this without proof and I haven't been ...
6
votes
2answers
230 views

Besides Vandermonde matrix, is there any other $m$ by $n (m>n)$ matrix in which any $n$ rows has a full rank?

I want to find some $m \times n(m>n)$ matrices that have the property that any $n$ rows has a full rank. Vandermonde matrix and Cauchy matrix are the only two matrices I know, can you guys give me ...
1
vote
2answers
94 views

Revisited: $[T]^{\gamma}_{\beta}$ is diagonal?

Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that ...
1
vote
1answer
132 views

How to show by induction that, for $0<\theta<\pi$, $\det A_n=\frac{\sin (n+1)\theta}{\sin \theta}.$

I need help with the underlined part. Thanks in advance Let $A_n$ be the $n\times n$ matrix given by $$a_{ij}= \begin{cases} 0 & \text{if }|i-j|>1, \\ 1 & \text{if }|i-j|=1, ...
-1
votes
1answer
75 views

Matrix Polynomial Question

Suppose $A$ is a matrix with complex coefficients. Suppose $f(x)$ is a polynomial of minimal positive degree with property that $f(A)=0$. Let $P_A(x)$ be characteristic polynomial of $A$. Prove that ...
-1
votes
1answer
61 views

Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$

Theorem to prove: Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
4
votes
1answer
161 views

Solving $Ax = b$ when $A$ is singular

I have a system of equations, expressed as $\mathbf{A} \begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ x_4 \end{pmatrix} = \begin{pmatrix} 0 \\ i (\frac{1}{2} + C - a) \\ i(\frac{1}{2} - C - a) \frac{m ...
1
vote
0answers
78 views

Does this matrix have any properties?

The matrix is: $\left( \begin{array}{cc} \sin{\theta} & \cos{\theta} \\ \cos{\theta} & \sin{\theta} \\ \end{array} \right) $ I'm interested in its effect on points in the first quadrant, ...
8
votes
6answers
2k views

If $A^2$ is invertible, then $A$ is also invertible?

True or False: If $A^2$ is invertible, then $A$ is also invertible. ($A$ is a matrix here.) The answer is true. I was trying to come up with an example that makes this false. But I couldn't. ...
0
votes
2answers
77 views

Linear Algebra: how do I know this is linear transformation?

T is the transformation from $\mathbb{R}^2$ to $\mathbb{R}^3$ $$T \left(\begin{array}{cc} x_{1}\\x_{2}\\\end{array}\right) = x_{1}\left(\begin{array}{cc} 1\\2\\3\end{array}\right) + ...
0
votes
2answers
142 views

Linear Algebra : find the kernel of this transformation.

Q. I think I find the kernel but several... which is correct? Seems like depending on which variable I put as kernel, I can get several kernels. Correct? T is the transformation from $\mathbb{R}^2$ ...
0
votes
1answer
78 views

Linear approximation of matrix norm

Given a square matrix $X=[x_1...x_N]$, and can be vectorized by $y=vec(X)=[x_1^T ... x_N^T]^T$ Is there any linear function can approximate $|| X ||$ (any matrix norm is okay) by using $y$?
0
votes
1answer
67 views

How to write this proof formally?

I have to prove that if $V$ is an unitary vector space and $W,U$ are subspaces of $V$, then $W^\bot \cap U^\bot=(W\land U)^\bot$ where $\bot$ means orthogonal complement and $\land$ is conjunction of ...
3
votes
2answers
66 views

Linear algebra, eigenvectors problem

Suppose you know that A is $2x2$ and symmetric. Assume the eigenvalues are $4$ and $7$. An eigenvector for $4$ is the vector $(3, -4)$. What is an eigenvector for $7$? So first we let ...
0
votes
2answers
380 views

Transpose of matrix inverse: $(AA^T)^{-1}A^Tb \stackrel{?}{=} (A^TA)^{-1}A^Tb$

Given the matrix equation: $$ x^TA^TA = b^TA $$ I'm trying to find the least squares solution (i.e.; trying to minimize $r=||Ax-b||$). The matrix $A$ is not necessarily symmetric. When I solve it ...
0
votes
1answer
110 views

Derivative of matrix inner product

Let $A$ be a matrix with no restrictions, then I can compute, $$\nabla_x (x^t Ax) = \nabla_x (\sum_{i,j}x_i A_{ij} x_j) = Ax+A_Nx$$ Where $A_N$ is equal to $A$ on its diagonal entries and zero ...
1
vote
0answers
76 views

An interesting variant of Rayleigh quotient

Edit: The norm constraint in the optimization problem in the below question was not there earlier. I apologize to the answerer user1551 who had to put his time and effort for my mistake. Let $A$ ...
0
votes
1answer
309 views

Linear Transformation Reflection Equation

Linear Transformation $$T(\vec{x}) =\left(\begin{array}{cccc} 0.6&0.8\\0.8&-0.6\\ \end{array}\;\begin{array}{c}\end{array}\right)\vec{x}$$ is a reflection about a line L. I need to find the ...
1
vote
2answers
78 views

Matrix with same image and kernel

Does exists a matrix A for which kernel of A is the same as the image of A? Answer is True. But I couldn't find the example. I think I saw it from somewhere but I can't find it. It was 2 by 2 ...
0
votes
2answers
115 views

Linear Equations using matrix and variables on the line

Q. I need a find a system of linear equations with three unknown variables whose solutions are the points on the line through (1,1,1) and (3,5,0). $ \frac{x-1}{2} = \frac{y-1}{4} = \frac{z-1}{-1}$ ...
1
vote
1answer
56 views

Linear Algebra True False: Inconsistency

Q. The system $A\vec{x} = \vec{b}$ is inconsistent if and only if $rref(A)$ contains a row of zeros. Answer is False. But my answer was True because I was thinking of the following example. ...
1
vote
1answer
71 views

Matrix and Linear Algebra True or False problem.

Q. True or False: If matrix A is a reduced row-echelon form, then at least one of the entries in each column must be 1. It comes down to this question. Can I have the following as Reduced ...
2
votes
1answer
73 views

Linear Equation unknown variables and number of equations

Q. True of False The four linear equations with Three unknown variables is always inconsistent? Is it true or false? I thought of this example $$\left(\begin{array}{ccc} ...
8
votes
1answer
84 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
0
votes
1answer
828 views

Rank normal form of a matrix

There is a standard result in matrix theory that goes like this: Suppose $A$ is an $m\times n$ matrix of rank $r$, then there exist two non-singular matrices $E$ (of size $m\times m$) and $F$ (of size ...
2
votes
2answers
62 views

A commutator problem

Let us consider $N \times N$ complex matrices, with $N >2$. Let D be a diagonal matrix, with $$D_{kk} = \sin \left(\frac{2\pi k}{N}\right), \space k = 0,..N - 1$$ I am looking for two ...
5
votes
1answer
100 views

What is the isomorphism function in $M_m(M_n(\mathbb R))\cong M_{mn}(\mathbb R)$?

What is the isomorphism function in $M_m(M_n(\mathbb R))\cong M_{mn}(\mathbb R)$. I tried this $[[a_{ij}]_{kl}]\mapsto[a_{ijkl}]$ , but I couldn't prove all steps.
2
votes
1answer
49 views

how to prove $2^{n-1}|\det(A)$ where $A=[a_{ij}]\in M_n(\mathbb R)$ and $a_{ij}\in\{-1,1\} $

let $A=[a_{ij}]\in M_n(\mathbb R)$ such that $a_{ij}\in\{-1,1\} $ then how prove $$2^{n-1}|\ \det(A)$$ Thanks in advance
-1
votes
1answer
112 views

by finding the eigenvalues and eigenvectors the evaluate the following.

so the question is : by finding the eigenvalues and eigenvectors of the matrix $$ P=\begin{bmatrix}1&6\\0&-2\end{bmatrix}\qquad\text{evaluate }P^{20}\begin{bmatrix}-2\\1\end{bmatrix} $$ I ...
2
votes
1answer
55 views

are normal subgroups of $SL(2,\mathbb{Z})$ also normal under the action of integer matrices in $GL(2, \mathbb{Q})$?

Ie, if $\Gamma\subseteq\text{SL}_2(\mathbb{Z})$ is a normal subgroup, and $\alpha\in\text{GL}_2(\mathbb{Q})\cap M_2(\mathbb{Z})$, then is $\alpha\Gamma = \Gamma\alpha$? (if necessary we can assume ...
3
votes
1answer
542 views

what does it mean for a matrix to be greater than another?

I am reading these notes on viscosity solutions, here is a theorem: Let us assume $u\in C^2$ is a classical solution of $F(x,u,Du,D^2u)=0$, $x\in \Omega$ then $u$ is a viscosity solution whenever ...
1
vote
1answer
221 views

How to deduce the psition mapping of entries of a matrix?

I would be thankful if any peer shed light on me. Assume that the mapping of a set is unknown. By knowing n number of E element sets and the transformed sets with positioned elements, How can I ...
1
vote
1answer
202 views

How do I write this $t^2$ as a linear combination of polynomials in the basis?

I have this homework problem that says "In $\mathbb P_2$ find the change of coordinates matrix from the basis $\mathcal B=\{1-3t^2, 2+t-5t^2,1+2t\}$ to the standard basis. Then write $t^2$ as a linear ...
0
votes
1answer
60 views

$[T]^{\gamma}_{\beta}=[v]_{\gamma}$ with $\beta=\{1\}$ a basis for $F$

Let $V$ be a finite-dimensional vector space over $F$ with basis $\gamma$ and let $v\in V$. Find a linear map $T:F \rightarrow V$ such that $[T]^{\gamma}_{\beta}=[v]_{\gamma}$, where $\beta = \{1\}$ ...
2
votes
1answer
90 views

elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
0
votes
1answer
37 views

Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$

I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
1
vote
2answers
63 views

Number of Solutions in Linear System

$$\left(\begin{array}{cccc} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0 \end{array}\;\middle\vert\;\begin{array}{c}2\\3\\4\\0\end{array}\right)$$ This is a $4$ ...
1
vote
0answers
121 views

Extracting hitting times from the pseudoinverse of a Laplacian matrix for an undirected graph

Provided a pseudoinverted Laplacian matrix for an undirected graph $G$, how can I extract first passage and commute times between vertex pairs in $G$?
2
votes
3answers
109 views

Revisited: How is $\phi:{\cal{L}}(V,W)\rightarrow M_{m\times n}(F)$ an isomophism of vector spaces?

I'm told in lecture that if $V,W$ are vector spaces over $F$ and ${\cal{L}}(V,W)$ is the vector space of all linear maps $V\rightarrow W$ and ${\scr{B}}$ and ${\scr{C}}$ are bases for $V$ and $W$ ...
2
votes
1answer
452 views

Commuting in Matrix Exponential

Let $A$, and $B$ be commuting $n\times n$ matrices, i.e. $A.B = B.A$. Let \begin{equation} \exp(A) = \sum_{i=0}^\infty\frac{1}{i!} A^i \end{equation} show that $\exp(A+B) = \exp(A).\exp(B)$.
1
vote
1answer
181 views

Question related to diagonally dominant matrix

A matrix is said to be positive if each entry in the matrix is positive. If $A$ is real, irreducible, diagonally dominant (or strictly dominant matrix) and has positive diagonal and non-positive ...
2
votes
1answer
455 views

Monotone matrix

A real matrix A is called monotone if Ax≥0 implies x≥0. If inverse of A exists and is real, then prove that A is monotone if and only if inverse of A≥0. (x≥0 means x is a column vector whose all ...
0
votes
1answer
52 views

Stability of a matrix

Suppose the hermitian part $H$ of a complex matrix $A$ be defined by $H=\frac{A+A^\ast}{2}$ and the skew hermitian part $S$ by $S=\frac{A-A^\ast}{2}$. If the hermitian part $H$ of $A$ is negative ...
1
vote
1answer
41 views

Combining elimination matrices

I am trying to combine several elimination steps into one matrix: more specifically I try to come up with a 3 by 3 matrix that first subtracts row 1 from row 2, subtract row 1 from row 3 and then ...
0
votes
1answer
148 views

Augmented Matrix with a constant in 'A'

I have an augmented matrix defined: $$\left[\begin{array}{ccc|c} 1& 0& 2& 1\\ 0& 1& -1& 2\\ 1& -2& k+4& 5 \end{array}\right]$$ ...
0
votes
1answer
49 views

Question related to non-negative matrix

I am facing a problem in this question. Let $y$ be fixed. Prove that $x \geq y$ implies $Ax \geq y$ for all $x$ if and only if $A \geq 0$ and $Ay \geq y$. A≥0 implies each entry in the matrix A ...
0
votes
2answers
153 views

Gram-Schmidt verifying orthonormal basis

Gram-Schmidt If I have an orthonormal basis, how do I verify that they are indeed orthonormal? I have Q, R and A is it enough to times Q` by Q to give me I? or A=QR? Edit: Let's say I ...
1
vote
0answers
581 views

Does a Symmetric Matrix with main diagonal zero is classified into a separate type of its own? And does it have a particular name?

For example, I have a Matrix as shown below. Does this Matrix belong to a particular type. I am CS student and not familiar with types of Matrices. I am researching to know the particular Matrix type ...