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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41 views

Is the following equation true about trace of a Matrix?

Is this true? $\operatorname{tr}(AB)^k = \operatorname{tr}(A^k B^k)$ If so, how can one provide proof or counter example? I tried it with the following two matrices and it turned out to be true: \...
0
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1answer
39 views

How to determine if b is a linear combination of a1,a2,a3 and the columns of matrix A?

a) Determine if $b$ is a linear combination of $a_1, a_2, a_3$, the columns of the matrix $A$ $A = \begin{bmatrix}4&-4&-16\\2&-1&-6\\-1&-1&2\end{bmatrix}$ and $b =\begin{...
0
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0answers
21 views

Inverse projection matrix 2D to 3D

I am writing a simple computer vision application in which reports the position of coloured dots on the floor. The floor is observed by a camera for which I have the correct projection matrix. I.E. If ...
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0answers
27 views

Euler Angle + Distance to XYZ coordinate

Background I'm familiar with Euler Angles and 3d space systems but I'm having trouble with Rotation Matrices. Scenario I've converted my Euler Angle to degrees for ...
1
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1answer
35 views

How do I solve for a matrix in this linear algebra expression?

In the problem I am working on I have an expression that looks like $$B^T\operatorname{diag}(Bx) y$$ where $B$ is a known $m \times n$ matrix, $x$ is an unknown $n \times 1$ vector of variables, and ...
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1answer
52 views

Mathematical expression of a rotation

I don't understand how my teacher defined an expression for the rotation adding up the two red vectors made up from the strong blue ones after rotation I especially don't understand how does the ...
1
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2answers
49 views

How to compute the product of matrix?

Compute the following product. $\begin{bmatrix}5&-1&6\\5&3&-6\\-6&2&-9\end{bmatrix}$ $\begin{bmatrix}-8\\-4\\-5\end{bmatrix} = ?$ I got: Row 1: $(5)(-8) + 8 + (-48) = -80$ ...
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1answer
45 views

Find a and b such that ? ..

Find $a$ and $b$ such that $\begin{bmatrix}-5\\-11\\14\end{bmatrix} = a$ $\begin{bmatrix}1\\-1\\2\end{bmatrix} + b$ $\begin{bmatrix}2\\2\\-2\end{bmatrix}$ a = ? b = ? I haven't done matrixes in a ...
0
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2answers
31 views

Matrices with linear mapping

It exist linear mapping $\phi_{1} : \mathbb R ^{3}\mapsto \mathbb R ^{3}$ which corresponds to a reflection about the plane $x_{1}-x_{2}=0 $. It exist linear mapping $\phi_{2} : \mathbb R ^{3}\...
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1answer
61 views

Derivative of matrix exponential at $0$

I have to show that the derivative of 'the matrix exponential' $exp: \mathbb{C}^{n\times n}\mapsto\mathbb{C}^{n\times n}$ at the zero matrix $0$ is $id_{C^{n\times n}}$, i.e. $exp(0)=id$. The above ...
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1answer
66 views

Eigevalues of block matrix of order $n$

What are the eigenvalues of the following block matrix? $$\begin{bmatrix} J_2-I_2 & J_2 & J_2-I_2 & J_2-I_2 &\cdots & J_2\\ J_2 & J_2-I_2 & J_2 & J_2-I_2 &\cdots &...
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1answer
24 views

Calculate a in dependence of b so that equation system is solveable?

Given is following equation system: $\begin{pmatrix} 2 & 1 & 1 \\ 3 & 2 & 3 \\ 4 & 3 & 5 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} =\begin{pmatrix} a \\ b \\ 1 ...
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2answers
48 views

Condition check for matrices

If a matrix $A= \begin{bmatrix}2a & 2b \\ 2c & 0 \\ \end{bmatrix} $ and matrix $B=2 \begin{bmatrix}a & b \\ c & 0 \\ \end{bmatrix} $, then how is $A=2B$ also explain how is this ...
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1answer
52 views

How to show bound ${\rm Tr} ( {\bf A} ({\bf I}+ b {\bf A})^{-1}) \le \frac{n {\rm Tr}(A)}{n +b {\rm Tr}(A) }$

Let ${\bf A} \in \mathbb{R}^{n \times n}$ be symmetric positive definite. Can one prove the following inequality for some positive constant $b$? \begin{align} {\rm Tr} ( {\bf A} ({\bf I}+ b {\bf A})^{...
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2answers
32 views

Determine kernel of matrix

The following matrix is given: $A=\begin{pmatrix} 2 & 1 & -2 \\ 0 & 1 & 0 \\ 0 & 3 & 4 \end{pmatrix} $ a) Determine kernel of matrix A I did this, but I always end up with ...
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3answers
64 views

Solving a third degree matrix equation

Let $X, A, B$ be square, matrices, and let $X$ be an invertible covariance matrix (symmetric, square, positive definite). Is it possible to solve for $X$ the following equation? $$ A=X(I+B-XX) $$ ...
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1answer
6 views

Eigenvalues of block matrix of oreder $2n$ related

What are the eigenvalues of following block matrix? $\begin{bmatrix} A_0 & A_1 \\ A_1 & A_0 \\ \end{bmatrix}_{2n \times 2n}$ Where $A_0$ and $A_1$ are $n \times n$ matrices.
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2answers
26 views

System of linear equations with parameter - strange result, does this make sense

I have a system of linear equations $Ax = b$ where $A \in \mathbb R^{3\times 3}$, and $x,b = \in \mathbb R^{3 \times 1}$. $A$ has some parameter $\alpha$ in its entries. I was asked to find for ...
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2answers
86 views

Very very simple matrix multiplication formula, don't go harsh on me please :)

So I'm studying from "Linear algebra and it's applications 3rd edition" - Gilbert Strang. He gave out this formula to find $$ \sum_{j=1}^n a_{i,j}x_j $$ matrix multiplication of the matrices below ...
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2answers
44 views

$A \in M(n,\mathbb R)$ has all its eigenvalues real , then is it true that all the eigenvalues of $A^2-A$ are also real?

Let $A \in M(n,\mathbb R)$ has all its eigenvalues real , then is it true that all the eigenvalues of $A^2-A$ are real ? [ Note that if $k$ is an eigenvalue of $A$ then $k^2-k$ is an eigenvalue of $A^...
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2answers
75 views

All eigenvalues of $A \in M_n(\mathbb R)$ are real , then are all eigenvalues of $A^2$ real ( and also non-negative)?

Let $A \in M_n(\mathbb R)$ has all its eigenvalues real $1.$ Is it true that all the eigenvalues of $A^2$ are real and non-negative? $2.$ If $k$ is an eigenvalue of $A^2$ then is one of $\pm\sqrt k$ ...
4
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3answers
89 views

Without calculating $A^4$ prove that $A^4\in Span\{A,I\}.$

Let $$A= \begin{bmatrix} -1 & 6 & -9 \\ -11 & 24 & -33 \\ -6 & 12 & -16 \\ \end{bmatrix} $$ a) Without calculating $A^4$ prove that $A^...
2
votes
2answers
45 views

A proof involving invertible $n\times n$ matrices

I'm new to studying linear algebra... doing ok with the computation type questions but struggling through some of the proof questions. I'm just not sure how to get started and what format this type of ...
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1answer
38 views

Prove the following: $[1-\lambda \operatorname{sum}(A^{-1})][1-\lambda \operatorname{sum}(B^{-1})]=1$

$\newcommand{\s}{\operatorname{sum}}$Problem: Let $E=[1]_{n\times n}$ and let $\s(X)$ be the sum of all elements of matrix $X$. If $A$,$B$ $\in$ $M_{n \times n}$ $(\mathbb C)$ are regular matrices so ...
2
votes
1answer
42 views

Determine basis of matrix

I have this example and by now I am able to do only a) and maybe b). I would be thankful is someone could help me with the rest. a) Determine rotation matrix with angle $\pi$ What I did: $$D=\begin{...
0
votes
1answer
36 views

Linear transformation dimensions of domain and range

$A$ is an $m\times n$ matrix. If $v$ is an $n\times 1$ vector and it is an element of a subspace of dimension $$d \le n $$, then must vectors of the form $Av$ of dimension $m\times 1$ be elements of ...
0
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1answer
26 views

On the equivalence of two traces

If we are given $$\rm{Trace}\{ G \: a \: a^T\} = \rm{Trace}\{H \: w \: w^T\}$$ where $a$ is $N \times 1$ vector, $G$ is $N \times N$ symmetrical matrix, and $w^T = [a^T \: t^T \: 1]$ and $t$ is ...
0
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1answer
21 views

Significance of Alpha and Beta with regards to matrix multiplication.

I'm learning a library for doing GEMM on the GPU. There are some things in the library which I don't quite understand. This is the function call, for reference. The function asks for alpha and beta, ...
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1answer
35 views

Prove “If $\text{rank } A = k$ with $A\in R^{n\times m},\ \ m>n$, there is a $k\times k$ submatrix with rank $k$”

How to prove the title? My work: The title implies there exists $k$ independent columns in $A$. Delete other $m-k$ columns. By elementary row operation, we can obtain a reduced-row-...
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0answers
21 views

Finding a basis and the dimenson of $Im(f)$ and $ker(f)$.

$B_1=(p_1(x),p_2(x),p_3(x),p_4(x))$ , basis of $R_3[x]$. $B_2=(v_1,v_2,v_3,v_4)$, basis of $R^4.$ $p_1(x)=1+x$, $p_2(x)=1-x+x^2,$ $p_3(x)=x-2x^3,$ $p_4(x)=3+x^3,$ and $v_1=(1,2,0,3)$, $...
1
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1answer
29 views

'Bounds' on the Covariance Matrix

We define covariance of random vector ${\bf X}$ as \begin{align} Cov({\bf X})=E \left[ \left( {\bf X}-E[{\bf X}] \right) \left( {\bf X}-E[{\bf X}] \right)^T \right]. \end{align} In the scalar case ...
6
votes
2answers
61 views

Find the value of special tridiagonal determinant

Let $A_{n}$ be the following tridiagonal determinant of order $n:$ \begin{vmatrix} a_{0}+a_{1}& a_{1}& 0& 0& \cdots& 0& \quad0\\ a_{1}& a_{1}+a_{2}& a_{2}&...
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0answers
31 views

Get content of transformation matrix from transformed vectors

In the following example: $$ \begin{pmatrix} X\\ Y\\ \end{pmatrix} = \begin{pmatrix} \cos\alpha & 1\\ 0 & \sin\beta\\ \end{pmatrix} \begin{pmatrix} A\\ B\\ \end{pmatrix} $$ $X$, $Y$, $A$ and $...
1
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1answer
51 views

Calculate the determinant of $\det(5(AB^{-2})^T)$

I have a matrix $$A = \begin{pmatrix} 0 & 0 & −2 & −7\\ 2 & 2 & 0 & 0\\ 0 & 0 & 1 & 3\\ 5 & 6 & 0 & 0\\ \end{pmatrix}$$ ...
0
votes
1answer
17 views

Addition of Two positive definite matrices [duplicate]

Let A and B be two positive definite matrices.Then is it true that A+B is also positive definite? My view:I have taken several examples and every time it indicates that A+B is p.d.But haven't able to ...
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0answers
101 views

Converging matrix and its form

I'm studying converging matrices now and following case is important for me. How to prove statement (if it is always true) that for any matrix $A_{n\times{n}}$ of rank $n$ where entries in any its ...
3
votes
1answer
49 views

Nilpotence and conjugacy in $M(p,\mathbb F_p)$

I have to solve the following problem: Characterize matrices $X\in M(p,\mathbb F_p)$ (note that $p$ is the dimension and the characteristic of the field) such that there exists $Y$ with the ...
0
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3answers
39 views

Linear transformation defined by an $m$ by $n$ matrix

I came across these terms on an MIT open course on youtube: An $m\times n$ matrix defines a linear transformation from the $\mathbb{R}^n$ vector space to $\mathbb{R}^m$ vector space. Can ...
3
votes
3answers
803 views

Why is this matrix neither positive nor negative semi-definite?

After some search here and on Google, I couldn't find a way to determine the definiteness of this matrix: \begin{bmatrix}0&1\\1&0\end{bmatrix} My understanding is that it should be negative ...
1
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1answer
76 views

minimum number of dependent rows in a matrix

Does the minimum number of dependent rows in a matrix have a specific name? (the way "rank" refers to the maximum number of independent rows). This comes up in calculating distances of codes. There ...
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2answers
49 views

Inverse of a Matrix(shortcut and tricks)

Can someone tell me if there is any shortcut or trick of finding the inverse of a matrix and not by elementary operations? Also is it possible to judge an inverse of a matrix by judging the options ...
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1answer
28 views

A question on Involuntary matrices [closed]

If A is a square matrix such that $A^2= I, then A^{-1} $is equal to what? (where I is the identity matrix)
6
votes
5answers
93 views

Let $A$ be a $2 \times 2$ real matrix such that $A^2 - A + (1/2)I = 0$. Prove that $A^n \to 0$ as $n \to \infty$.

Question: Let $A$ be a $2 \times 2$ matrix with real entries such that $A^2 - A + (1/2)I = 0$, where $I$ is the $2 \times 2$ identity matrix and $0$ is the $2 \times 2$ zero matrix. Prove that $A^n \...
4
votes
2answers
95 views

Matrices problem: $AB=B$ and $BA=A$, what is $A^2+B^2$?

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$, then $A^2+B^2$ would be equal to?
2
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3answers
47 views

Finding the matrix of this particular quadratic form

I have been working on problems related to bilinear and quadratic forms, and I came across an introductory problem that I have been having issues with. Take $$Q(x) = x_1^2 + 2x_1x_2 - 3x_1x_3 - 9x_2^...
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0answers
35 views

Similarity of matrices with Jordan Form

My problem consists in determining whether the following matrices are similar or not: $$ A = \left(\begin{array}{cccc} -1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 4\\ -1 & 0 & 1 &...
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0answers
19 views

Projected to the null space of the matrix?

I am the moment trying to understand how this is projected to the null space of the matrix? This snippet is from this paper : http://www.golems.org/papers/StilmanIROS07-task-constrained.pdf Could ...
2
votes
1answer
54 views

Solve for third rank linear tensor equation $C_{[ij]k}U^jU^k=A_i$

Is there a way to solve a general tensor equation of the form, written in an arbitrary frame \begin{equation} C_{[ij]k}U^jU^k=A_i, \end{equation} for a tensor field $C$ of type $(0,3)$ (the square ...
1
vote
1answer
18 views

Compute covariance matrix random walk

Consider a random walk on the square lattice $\mathbb{Z}^2$ with diagonal jumps of size $2$, i.e. the jump probabilities are $$P(X_1 = x) = \begin{cases} \frac{1}{4} & \quad \text{if } ...
1
vote
1answer
33 views

Linear application

Let $f:\mathbb{R}^3\to\mathbb{R}^3$ be a linear application and let $\{e_1,e_2,e_3\}$ the canonical basis of $\mathbb{R}^3$. We know that $\operatorname{Im} f=\langle(1,1,3), (0,1,1)\rangle$ and that ...