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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2answers
44 views

Matrix question help.

Consider $$X = \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}.$$ Find a real matrix $A$ for which $A^2 = X$. I don't know how to answer this or where to start. ...
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2answers
53 views

How many $3 \times 3$ matrices are singluar?

How many $3 \times 3$ matrices are singluar? Describe the methodology used to achieve the result.
1
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0answers
50 views

What does it mean to compute a normal to a triangle in a “clockwise direction”

I am trying to understand how this works. I am given 3 points, each representing a vertex of a triangle. I must then "organise" the points and calculate the normal of the resulting triangle in a ...
0
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1answer
36 views

Problem about M-matrix

Is a symmetric M-matrix positive definite? I intuitively think this is not correct. Can someone prove this or provide a counter-exmaple? I really appreciate it.
2
votes
1answer
60 views

Commutative matrix proof

I've got the following question: $A \in M_{nn}(\mathbb{K})$ is a matrix and $AB=BA \forall B \in M_{nn}$. Proof that $ A=aI_n \forall a \in \mathbb{K}$. and one given solution starts with: $ % ...
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0answers
18 views

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth.

For a matrix group $G$ of dimension $d$, I am trying to prove that the function $Ad : G \rightarrow GL_d(\mathbb{R})$ is smooth. So where I am starting is by extending $Ad : G \rightarrow ...
7
votes
1answer
260 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
2
votes
1answer
145 views

Converting a PDE to a matrix form

I have to solve the problem $U_{xx}-U_{xy}+2U_y+Uyy-3U_{yx}+4U=0$ using diagonal matrix as described in this article page 44 section 3.2 But my problem is there the matrix A is symmetric matrix ...
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1answer
32 views

Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
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3answers
60 views

For which values of $k$, we have $A = A^{-1}$?

I got this question in hw. Can anyone help me solve it? Let $ A = \left( \begin{array}{ccc} k & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & k \end{array} \right) $ For which values of ...
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1answer
60 views

Differential equation of the form $y'=Ay+b(x)$ with $b(x)=(\sin{(\omega x)},0)$

I have a question regarding the following specific differential equation. $$y'=\left(\begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix}\right)y+\left(\begin{matrix} ...
1
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1answer
27 views

Positivity of certain matrix

Let $A=[[a_{ij}]]$ and $B=[[b_{ij}]]$ be two positive semi-definite matrices of same dimensions. Further they have a property that, if $a_{ij}=0$ then $b_{ij}=0$ (i.e. the nonzero entries appear in ...
1
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1answer
98 views

Nonsingular block matrix

Let us consider a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ and the block partitioning $$ \mathbf{A} = \begin{pmatrix} \mathbf{A}_{11} & \mathbf{A}_{12} \\ \mathbf{A}_{21} & ...
0
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0answers
22 views

How to nullify one of the axises rotation in a rotation matrix?

Let's say that I've got a matrix with some rotation stored. Now I wan't to somehow make an Y rotation equals 0 (or rather make it equal to the starting moment without rotation). How would I do it? I ...
0
votes
1answer
54 views

Matrix multiplication and rank reduction? - What is the minimal polynomial?

given is a matrix A with $\begin{pmatrix} a & 1 & 0 & \cdots & 0 \\ 0 & a & 1 & \cdots & 0 \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0 & 0 ...
0
votes
2answers
36 views

How to calculate the determinant when the diagonal is in terms of $k$?

Determine the value of $k$ so that the columns in this matrix are linearly dependent: $$\begin{bmatrix} k & -1/2 & -1/2\\ -1/2 & k & -1/2\\ -1/2 & -1/2 & k ...
3
votes
0answers
36 views

Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...
0
votes
2answers
31 views

How to determine column dependency without calculating the determinant?

Determine whether this matrix' columns are linearly dependent or not. $$\begin{bmatrix} 1 & 0 & 2 \\ 0 & -1 & -2 \\ 2 & -2 & 0 \end{bmatrix}$$ The determinant is $0$ ...
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1answer
28 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
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votes
0answers
374 views

How to calculate a linear combination for a matrix' column?

I have a very weak understanding of linear dependency and linear combination, so I figured I'd check out some exercise about it: $$A = \begin{bmatrix} 4 & 0 & 1\\ 2 & 3 & 6\\ 6 ...
2
votes
2answers
3k views

why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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0answers
31 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
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0answers
35 views

three dimensional subspace question

If a vector is in $\mathbb{R}^5$, does this mean that the projection of this vector onto $S$ is in $\mathbb{R}^3$, where $S$ is some 3-dim subspace of $\mathbb{R}^5$?
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1answer
19 views

Going from $X^tAB - I = X^t$ to $X^t(AB-I) = I$ in matrix algebra.

Finding the value of the matrix $X$: $$X^tAB - I = X^t$$ I noticed that the next step chosen by my book is $$X^t(AB-I) = I$$ It's not clear to me how did they reach that. How did they go from one ...
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1answer
47 views

Proving vector projection

For nonzero vectors, how do you prove the following? $$\|u\|^2 = \|\text{projection of $u$ onto $v$}\|^2 + \|u - \text{projection of $u$ onto $v$}\|^2$$ I think what we need to do is split up the ...
0
votes
1answer
70 views

Solution of matrix equations

$$A=\begin{bmatrix} 3 & -2 & -1 \\ 1 & 2 & 1 \\ -1 & 1 & 1 \end{bmatrix}, X= \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 1 \\ 7 \\ 2\end{bmatrix}$$ ...
0
votes
1answer
31 views

Calculating matrix determinants based on another's.

$$A = \begin {bmatrix} a & b & c \\ 4 & 0 & 2 \\ 1 & 1 & 1 \end {bmatrix} \ \ , \ \ \left| \ A \ \right| = 3$$ Knowing only this, how does someone calculate the determinant ...
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vote
1answer
25 views

Can we list down all order 4 integer valued 3 x 3 matrices

Can we list down all integer 3 x 3 matrices($A$) whose are order 4 i.e $A^4= I$? or atleast get some examples? What should be the method for such thing?
2
votes
1answer
69 views

Is there a quicker way to show that a set of vectors is a spanning set?

Let's say set $S = \{(2,1,4), (1,-1,1), (3,2,5)\}$ and the vector space V is $R^3$. In this case, the number of independent vectors is equal to the dimension of the vector space. I know that one way ...
0
votes
2answers
43 views

How do I calculate the inverse of these matrices?

In learning how to rotate vertices about an arbitrary axis in 3D space, I came across the following matrices, which I need to calculate the inverse of to properly "undo" any rotation caused by them: ...
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2answers
88 views

What does it mean to multiply a real matrix by a complex scalar?

In this answer http://math.stackexchange.com/a/219508/27609 it is noted, that multiplying a matrix $A$ by a scalar $s$ is the same as multiplying a matrix $A$ by a diagonal matrix ${\rm ...
0
votes
1answer
132 views

rotation matrix to axis angle

from wikipedia the above rotation matrix has a rotation of -74 degrees. What does it mean "around the axis (−1⁄3,2⁄3,2⁄3)"? How can I determine how many degrees is rotated on X axis, Y axis and Z ...
2
votes
2answers
749 views

Finding the unknown matrix in an equation?

so I was wondering how can I find the unknown matrix from an equation, I need to find X [-1 2] X [1 0] [-2 -12] [ 0 1] [2 4] = [1 - 4] so I ...
1
vote
1answer
299 views

How to divide polynomial matrices

If I am given two $2\times2$ polynomial matrices and I need to divide them, what are the steps I need to follow? I know I need to do right division and left division, and that the answer will have ...
0
votes
1answer
89 views

Quaternion isn't normal!

I use the following scheme: Quaternion: [ w, x, y, z ] Then I use Euler Angles and convert it to a quaternion, as following: ...
1
vote
0answers
24 views

Rotating two objects

I have two lines. Both created in this format: Line 1 $$line1 = \left\{ \begin{array}{c} startX, startY \\ endX, endY \end{array} \right\}$$ $$line2 = \left\{ \begin{array}{c} startX, startY \\ endX, ...
1
vote
1answer
29 views

What cases should I check when I am looking for the possible infinite solutions of a matrix?

I was reading random exercises, and found a typical Determine what values of $a$ cause the system to have no solution, an unique solution and infinite solutions. Also find the solution set for ...
2
votes
1answer
17 views

Factoring a series of Matricies

I have a graduate physics text which has an arbitrary matrix $M$. I am given the following $M^{0} + M^{1} + M^{2}...M^{n-1}$ and then it says this is equal to $(M^{n}-I)(M-I)^{-1}$, where $I$ is the ...
2
votes
2answers
224 views

How many $2\times2$ positive integer matrices are there with a constant trace and positive determinant?

The trace of a $2\times2$ positive integer matrix is a given constant positive value. How many possible choices are there such that the determinant is greater than 0? Each element of matrix is ...
6
votes
4answers
1k views

Characteristic polynomial equals minimal polynomial iff $x, Mx, \ldots M^{n-1} x$ are linearly independent

I have been trying to compile conditions for when characteristic polynomials equal minimal polynomials and I have found a result that I think is fairly standard but I have not been able to come up ...
0
votes
1answer
77 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
0
votes
2answers
34 views

Proving that an eigenvalue is a root of a polynomial

Let $A$ be an $n \times n$ matrix, and let $\lambda$ be an eigenvalue of A. Prove that if $p$ is a polynomial such that $p(A)=\mathbb{0}$ then $\lambda$ is a root of $p$.
0
votes
1answer
44 views

Eigenvalues, eigenvectors with $\sin$ and $\cos$

Consider the vectorial space defined by $E = (cos (t),sin (t))$. Consider the following derivation operators defined in $E$ by $D={\frac{d}{dt}} \wedge $$D^2=\frac{d^2}{dt^2}$ a) Show that $D$ has no ...
2
votes
0answers
69 views

$A$ positive definite iff $BAB^\intercal$ positive definite

I need to prove the following statement: $A$ is positive definite and $B$ is nonsingular if and only if $BAB^T$ is positive definite. Please let me know how this problem would be solved.
2
votes
1answer
84 views

Upper Unitriangular Matrices

Let $U$ be the group of the upper unitriangular matrices $n$-$n$ over the field of rationals $\mathbb{Q}$. I know that $U$ is nilpotent and torsion-free. It is also radicable? How it can be proved in ...
1
vote
2answers
118 views

set of symmetric positive definite matrix open?

I consider a collection of symmetric positive definite matrices of the same dimension. I've learned it's an open set but have no clue about the proof. Also, can the symmetry condition be dropped? ...
7
votes
1answer
110 views

Existence of $A^2B - BA^2 = 2A \textrm{ and } AB^2 - B^2A = 2B$. in $\mathcal{M}_n({\mathbb{C}}) $

This question arose in this classical exercise : Is there exist two matrices such that $AB-BA=I_n$ in $\mathcal{M}_n({\mathbb{C}}) $. Wich is impossible (by using trace to prove this) But if ...
1
vote
1answer
593 views

Can the identity matrix be negative?

I got the following question: Find, if possible, the inverse of the matrix: $\begin{bmatrix}3, -1\\2, -2\end{bmatrix} $ and I did the following: $\begin{bmatrix}3, -1\\2, -2\end{bmatrix}^{-1} = ...
1
vote
1answer
74 views

counting Number of matrices

We have a $2 \times 2$ matrix. We are given the trace of the matrix as $N$. Also, all elements of the matrix are greater than or equal to $1$. And, the determinant of matrix is $\geq 1$. QUESTIONS: ...