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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1answer
16 views

Calculate rotation/translation matrix to match measurement points to nominal points

I have two matrices, one containing 3D coordinates that are nominal positions per a CAD model and the other containing 3D coordinates of actual measured positions using a CMM. Every nominal point has ...
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0answers
13 views

row equivalency of two matrices and their homogeneous systems of linear equations

Please prove that if $A,B\in M_n(F)$ and if their homogeneous systems of linear equations have the same solutions, then $A,B$ are row equivalent. Thanks for your mention.
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1answer
36 views

If $A \in \mathbb{C}^{m\times n}$ is full-column rank matrix, then is rank($AB$) = rank ($BA$) = rank($B$)?

Let $A \in \mathbb{C}^{m\times n}$, and $B \in \mathbb{C}^{n\times k}$ complex matrices. If A is full-column rank matrix then can we say that rank($AB$) = rank ($BA$) = rank($B$)? What can we say ...
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2answers
33 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
22 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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1answer
26 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
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1answer
13 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
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1answer
47 views

Inverse of a matrix is expressable as a polynomial?

Let $A$ be an $n \times n$ matrix. Prove that if A is invertible, then there exists a polynomial $p$, such that $A^{-1}=p(A)$ Thus far: Let $W$ denote the $k$ dimensional A-cyclic subspace spanned ...
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0answers
50 views

derivation of ML-estimator (statistics)

I have the following likelihood function: I'm given this information about the $\Omega$ matrix ($\boldsymbol{1}$ is a $T \times 1$ vector of ones): I would like to be able to show that the ...
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2answers
31 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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0answers
43 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.
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1answer
24 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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1answer
31 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 ...
7
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1answer
80 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 ...
2
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1answer
28 views

Jordan similar matrix

I have matrix B = ((1 1 -2 0), (2 1 0 2), (1 0 1 1), (0 -1 2 1)) I found the characteristic polynomial (1-x)^4 and was able to get my Jordan Matrix J=((1 1 0 0), (0 1 0 0), (0 0 1 1), (0 0 0 1)). I ...
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0answers
19 views

Coefficient of Determination (Alternative Solution)

Consider the following problem - where I have purposefully omitted the numbers in question since it is of no interest for my question. From $40$ observations on \begin{align} ...
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1answer
31 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 ...
1
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1answer
60 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: ...
7
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1answer
98 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 ...
2
votes
3answers
101 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 ...
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2answers
31 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? ...
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0answers
4 views

Production Capacity

single product "X", 2 processes for completion "A" & "B", Product must complete both processes in any order, Process "A" can do 250 products per hour per person, Process "B" can do 300 products ...
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1answer
23 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 ...
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1answer
31 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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16 views

Matrix Representations of Chordal Graphs and Uses in Linear Algebra

Chordal graphs have the property of perfect elimination ordering. In Knuth's 2012 Christmas lecture ~1:12:10 he mentions that when the coefficients of a linear algebra problem can be written as a ...
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1answer
43 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 ...
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0answers
33 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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1answer
28 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 ...
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0answers
19 views

About the rank of (sub) matrices whose entries are roots of unity

Let $\Omega$ be a matrix with entries $a_{jk}=\omega^{jk}$, where $0\leq j,k\leq n-1$, and $\omega=e^{-2\pi i/N}$, with $N\in \mathbb{N}$, so $\Omega$ looks like $$ \Omega=\begin{pmatrix} 1 & 1 ...
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1answer
40 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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2answers
31 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 ...
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1answer
18 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 ...
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2answers
33 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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0answers
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+50

Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...
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2answers
38 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 ...
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2answers
48 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
166 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
votes
2answers
18 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 ...
0
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1answer
35 views

What does it mean when a system of linear equations have no solution?

$$ A = \left( \begin{array}{ccc} 10 & 29 & 41 \\ 23 & 27 & 42 \\ 24 & 28 & 48 \\ \end{array} \right) $$ $\det (A) = -1748$. Now $B$ is formed when the second column is ...
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2answers
62 views

If $ \det A$ is nonzero then $A$ is invertible

The problem is prove if $A$ is an $n\times n$ matrix with $\det A\neq 0$, $$A^{-1} = \frac{1}{\det A} ...
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0answers
13 views

how to use Householder matrices to solve this problem?

Any guidance and/or advice to solve this problem will be appreciated: Suppose $$H\_{n} ... H\_{1}$$ can be used to determine a lower triangular L1 from $$R^{n by n}$$ so that: $$H\_{n} ... H\_{1}L = ...
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1answer
29 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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2answers
40 views

Why is the determinant of any triangular matrix always the multiple of the main diagonal?

Is there a mathematical proof or a conclusion explaining as to why it is that?
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1answer
28 views

Linear Algebra Eigenvalues question

This question doesn't look too hard but I just can't seem to figure it out. Let $A$ and $B$ be n x n matrices. Show that if none of the eigenvalues of A are equal to 1, then the matrix equation $XA ...
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2answers
195 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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1answer
23 views

Image of $\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$

This question is related to the question I previously asked: Kernel. The following function is given: $$\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$$ ...
1
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1answer
19 views

Find the parameters for which the matrix…

$\left(\begin{matrix} 1 & 2 & -1 \\ 3a & 2 & b \\ 1 & 2& 1 \end{matrix}\right)$ For example I have such a matrix. Can you, please explain how can I find for what values of "a" ...
2
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1answer
14 views

PSD matrix characteristics

A very stupid question that would solve a lot of troubles in my life: Is the product of vector of positive values with a positive semi definite matrix always positive vector? I have to analyze when ...
1
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1answer
37 views

Calculating the kernel of a function $\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}$

I am kind of confused at the moment. The following function is given: $$\phi: \mathbb{Q}^{2\times 2} \rightarrow \mathbb{Q}^{2\times 2}, \ A \rightarrow A + A^t$$ The task is to prove: $\ker{\phi} = ...
2
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1answer
62 views

Conditions for “$AA^T=A^TA$ implies $A$ symmetric” to hold.

This claim arose in this question Show that $A$ is symmetric, with $A \in M_n(\mathbb R)$ where it is assumed additionally that $AA^TA$ is symmetric. I'm considering weakening hypotheses. Let $A$ be ...