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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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
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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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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
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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 ...
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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 ...
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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 ...
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2answers
750 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 ...
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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 ...
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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: ...
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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, ...
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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 ...
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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
225 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 ...
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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
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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 ...
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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 $p$.
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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 ...
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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 ...
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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
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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 ...
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1answer
595 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} = ...
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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: ...
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1answer
154 views

Prove that determinant of matrix equal $\pm1$ or $0$

We are given square binary matrix $A_n$. Data contained by A comply the following rule: if row has any 1's then they would appear there only successively (row $(1\space 1\space0\space1 )$ is ...
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2answers
54 views

Equation with multiplication of a matrix by a column vector

How do I solve this matrix equation? $$\begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & 4\end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}5\\7\end{bmatrix}.$$ I know ...
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votes
1answer
15 views

Building a square matrix having in the j-th column specific data

Why if we want to build the square matrix $A$ having in the j-th column the ordinates $y_i^j$ in the y5 vector, we have to elevate every column as it follow: ...
2
votes
3answers
119 views

Why is a matrix $A$ that fulfils $AA^t = I$ invertible?

Given a square matrix $A$ that fulfils $$AA^t = I$$ Justify why must $A$ be invertible. The answer, according to my book, is simply $$AA^t = I$$ $$A^t = A^{-1}$$ I don't ...
2
votes
1answer
89 views

How does Laplace expansion work?

\begin{bmatrix} 1 & 2 & 0 & 0 & a\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0\\ 1 & 0 & 1 & 1 & 1 ...
3
votes
1answer
53 views

Show that any invertible matrix has a logarithm.

I was trying to remember how to show that any invertible matrix has a (possibly complex) logarithm. I thought what I came up with was kind of cool, so I thought I'd post my answer here.
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2answers
60 views

Determining the necessary values for a matrix' coefficients to achieve a certain rank.

I'm having a headache with this... Given the augmented linear system matrix: $$A = \begin {cases} 1 & 0 & 0 & 2 \\ 0 & a-2 & 0 & 0 \\ 0 & 0 & b + 1 & c \\ 0 ...
1
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1answer
395 views

2x2 Fibonacci matrix singular value decomposition

$A = \left[\begin{array}[c]{rr}1 & 1\\1 & 0\end{array}\right]$ I am supposed to find all the eigenvalues and vectors for this matrix so that $Av=σu$ and then form a singular value ...
0
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1answer
54 views

Find 1's submatrices

In the following 0/1 matrix I'm trying to identify every largest submatrices formed by 1's as shown in the picture. Submatrices can have just one row only if they have more than 3 columns. Submatrices ...
2
votes
0answers
39 views

Is a matrix a subgroup of a group when its the inverse matrix “looks different”?

I have the to prove whether a subset of a group is a subgroup. The following subset is given: $$U = \left\{ \begin{pmatrix} a & b & 0 \\ 0 & 1 & c \\ 0 & 0 & d \end{pmatrix} ...
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vote
1answer
48 views

Transformation Matrix $M_B^B$ of $P_3$ for $B = (1,x,x^2,x^3)$. Is that correct?

I have the following task and just wanted to check weather this is (written) correct(ly). Let $V$ be the vector space of all polynomials of grade $\le 3$ and $f: V \rightarrow V, p \rightarrow p'$ an ...
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1answer
20 views

Operator norm of real matrix

I've been looking through my workbook in preparation for the next set of classes and I'm stuck on this problem and don't know how to possibly proceed with it. The hint isn't helping and there isn't ...
0
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2answers
48 views

About matrix $R$, what is this called: $R^TR$? What is it for?

I am doing singular value decomposition on a matrix $R$. The first step is to compute such a matrix $R^TR$. What is this matrix? A reference told me this is cross product of matrix R. I use a ...
13
votes
4answers
313 views

Is it true that all matrices in $M_2(\mathbb R)$ is the sum of two squares?

I recently show that every polynomial with real coefficient and $P$ is always positive. is a sum of two squares of polynomials. These questions also appear often in arithmetic. What if we change ...
0
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2answers
133 views

Calculating the determinant of $-2A^{-1}$ given the determinant of $A$.

If $A$ is a square matrix or size $3$, where $\left | \ A \ \right| = -3$ How do you calculate something like $$ \left | -2A^{-1} \ \right |$$ ? Well, for starters, I believe that the determinant ...
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2answers
252 views

If a matrix is not invertible, is it still possible to find a left and/or right inverse?

I was recently asked to find the right inverse of some matrixes. I found that all three of them were invertible, so it was just a matter of finding their inverses, which would be exactly the same as ...
2
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2answers
36 views

About inverse matrixes

I've been reading about invertible matrixes. I have a few questions: One theorem says The rank of an invertible matrix of size $n$ is $n$. So, is it safe to say that all invertible matrixes ...
8
votes
4answers
2k views

Proof of elementary row operations for matrices?

I'm taking a Linear Algebra course, and we just started talking about matrices. So we were introduced to the elementary row operations for matrices which say that we can do the following: ...
2
votes
3answers
777 views

Reverse rotation matrix

I have to write an algorithm that can given a rotation matrix, find $k$ and $f_i$. $R = \text{rotationMatrix}(k, f_i)$ I am given $R$ and need to find $k$ and $f_i$, but i don't know how to do this, ...
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0answers
733 views

Linear Algebra - Finding the matrix for the transformation

Find the matrix for the transformation which first reflects across the main diagonal, then projects onto the line $2y+\sqrt{3}x=0$, and then reflects about the line $\sqrt{3}y=2x$. Reflection about ...
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1answer
72 views

Finding an $D$-dimensional orthonormal change-of-basis matrix given a $D$-$2$ transformed points.

This is frustrating, I should be able to solve this but I'm having a mental fog. I want to find an orthonormal change of basis: given a single point $(x_1,y_1)^T$ and its image $(x_2,y_2)^T$, find ...
2
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2answers
1k views

Find the matrix of the given linear transformation $T$.

Here's the specific scenario: $T(M) = \begin{bmatrix}1&2\\0&3\end{bmatrix}M$ from $U^{2 \times 2}$ to $U^{2 \times 2}$ with respect to $\mathfrak{B} ...
2
votes
1answer
41 views

Find the basis for $\text{Im} \, ψ$ of a matrix transformation

Let $\psi\colon\mathrm{Mat}_{ 2\times 2 }(\mathbb R) \to \mathrm{Mat}_{ 2\times 2 }(\mathbb R)$ be defined by $$\psi\colon \pmatrix{a&b\\c&d}\to\pmatrix{a+b&a-c\\a+c&b-c}.$$ Find ...
1
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1answer
3k views

Find the Matrix A of the Linear Transformation

Can anyone walk me through the steps to complete this problem? I am unsure of where to start to solve the problem. I get that the resulting matrix $A$ should be a $2 \times2$ matrix, should I be ...
0
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1answer
294 views

Finding the matrix of a rotation transformation

The rotation transofrmation is defined as some composition of rotatation along the $x,y,z$ axes. Assuming $T$ is a rotation transformation in $\mathbb{R}^{3}, ...