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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0
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1answer
37 views

Help with simplifying the second row of a $4 \times 4$ matrix to $(1, 0, 0 ,0)$

I wonder how you would approch the problem of simplifying the second row of the matrix: $$\left( \begin{array}{cccc} 3 & 1 & 8 & 1 \\ -1 & -3 & 0 & 2 \\ 3 & -1 & 5 ...
1
vote
2answers
399 views

Proving a matrix is positive definite using Cholesky decomposition

If you have a Hermitian matrix $C$ that you can rewrite using Cholesky decomposition, how can you use this to show that $C$ is also positive definite? $C$ is positive definite if $x^\top C x > 0$ ...
1
vote
2answers
551 views

Partial derivative with respect to a vector x for $F(x) = x^TA(x)x$

I have the next function $F(x) = x^TA(x)x$, where $x$ is a real vector with dimension $n$, and $A$ is a square real matrix $n \times n$ depending on the components of $x$. How can I compute the ...
2
votes
4answers
99 views

Invert of Matrix I-BA [duplicate]

Suppose $A$ and $B$ are two square Matrix. Let $I-AB$ be invertible. I would like to know why $I-BA$ is also invertible? Also what is invert of $I-BA$? Thanks.
5
votes
6answers
275 views

$ABCD = I$ then $B^{-1} =?$

I got this question in a practice book. A,B,C and D are $n\times n$ matrices with non-zero determinant. $ABCD = I$ , then $B^{-1}$ = ? The answer to this was $B^{-1}= CDA$. How was that answer ...
0
votes
0answers
34 views

Determine general form of control function andthus show this coul have been achieved earlier [duplicate]

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
0
votes
1answer
129 views

Calculate the determinant of the matrix ad hence prove

By computing the determinant of $\lambda I-L$ where $L$ is the Leslie matrix, derive the Euler Lotka equation. $$L= \begin{bmatrix} b_{1} & b_{2} & \ldots & b_{w-1} & b_{w}\\ s_{1} ...
4
votes
2answers
251 views

Find all $A,B$ such that $AB-BA=0$.

Can someone give me a hint how to find all $n\times n$ matrices $A,B$ over an arbitrary field, such that they commute, i.e. such that $AB=BA$ ? I found this problem in some lectures notes where the ...
1
vote
2answers
394 views

How to prove a set of positive semi definite matrices forms a convex set?

Let $C$ be the set of positive semi-definite matrices, how can I prove it is a convex set?
7
votes
4answers
2k views

How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?

Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
1
vote
1answer
989 views

Derivative of trace of matrix product including inverse

Let $A,B,X$ be n-by-n matrices, $X$ is nonsingular so $X^{-1}$ exist. What will $\frac{\partial Tr(XAX^{-1}B)} {\partial X}$ be?
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vote
2answers
219 views

reading Adjacency Matrix

How do you read product of adjacency matrix multiplying itself that has not only 1 and 0 but other numbers? For example 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 so squaring the above matrix ...
2
votes
2answers
46 views

Is there a transformation matrix A that multiplied to a B adds each two rows of B?

I have a $m \times n$ matrix $B$ and want to the $\frac{m}{2} \times n$ matrix $B'$ where each two rows of $B$ are added. Is there a $\frac{m}{2} \times m$ matrix $A$ such that $AB=B'$, and what are ...
0
votes
4answers
59 views

Sufficient condition for a polynomial to be a characteristic polynomial

Let $A\in \operatorname{Mat}_{n\times n}(F),~F$ being a field, satisfies $p(x)\in F[x]$ where $\deg p(x)=n$ and $p(x)$ is a monic polynomial. Can we say $p(x)=\chi_A(x)?$
2
votes
2answers
54 views

Question on MIT Markov Matrices video

Markov matrices are pretty new to me and I'm a little rusty with my linear algebra. My question stems from watching this video from YouTube on Markov matrices. For those who wish to skip the video, ...
2
votes
3answers
485 views

Matrix representation of a linear transformation between vector spaces

Let $v$ be an $n$-dimensional vector space over a field $F$ and $\psi: V \to V$ and isomorphism. Show that there exist bases $B_1$, $B_2$ (possibly different) such that the matrix representation of ...
2
votes
2answers
159 views

Determinants and Matrices

Suppose $A$ is a $4\times4$ matrix with $\det A=2$. Find $\det((1/2) A^T A^7 I A^T A^{-1})$ where $I$ is a $4\times4$ identity matrix. My work so far: We know that $\det A^T=\det A$. $I$ has no ...
4
votes
1answer
216 views

Construct a matrix transform

consider $\frac{dx}{dt} = Ax$ where $A$ is the matrix $$ \begin{bmatrix} 1 & 0 & 1 \\ 0 & 0 & -2 \\ 0 & 1 & 0 \\ \end{bmatrix} $$ ...
4
votes
2answers
206 views

Prove that function is bijective

Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$. ...
4
votes
2answers
88 views

How to recover a shuffled matrix

Suppose that I have a matrix $A$. $A$ can be a rating matrix. That is, $A(i,j)$ is the rating user $i$ has given to item $j$. Suppose that I shuffle the rows and columns of matrix $A$ and get ...
1
vote
2answers
54 views

Why if unity is not an eigeinvalue of A then (I - A) is nonsingular?

Seems so obvious but I can't get it: If unity is not an eigenvalue of $A$, then $(I - A)$ is nonsingular. How can I prove this?
1
vote
1answer
74 views

Looking for solution of a linear equation (which is a very important lemma in my research).

Given $M\geq 2$ and $1<\beta_i<2$, $1\leq i \leq M$ and the equation: $h_1\beta_1^{L-1}+h_2\beta_1^{L-2}+\cdots+ h_{L-M-1}\beta_1^{M+1}=h_L+\beta_{1}^{L}$ ...
3
votes
3answers
379 views

Factorize Positive Definite Symmetric Matrix

Let's start from the assumption of disposal of a positive definite symmetric matrix of size $\ (N,N) $. For some reason I have to factorize this matrix: I am already aware of the ...
3
votes
1answer
81 views

For this matrix $A$, what is $A^n$?

$$A = \begin{pmatrix}0 & a & b \\ 0& 0 & c \\ 0& 0 &0\end{pmatrix}$$ What is $A^n$ (for $n\geq 1)$?
5
votes
3answers
213 views

Show that $\operatorname{rank}(A^2+A+I_3)=1$

If $A \in M_3(\mathbb{R}), A \ne I_3 $ and $A^3=I_3$ Show that $\operatorname{rank}(A^2+A+I_3)=1$. What I have reached so far is that $\operatorname{rank}(A-I_3)+\operatorname{rank}(A^2+A+I_3)\le ...
2
votes
1answer
68 views

Are there any maps which preserve addition and multiplication over Matrics?

I don't know whether the title is correct, cause English is not my native language. What I mean is: Suppose there is a function, say $f$, which maps Matrix $A$ into Matrix $A'$, and satisfies ...
2
votes
1answer
80 views

For what A, If $A+A^T>0$ then $A^2+A^{2T}>0$?

let me know if I am wrong with the next with a real square matrix A. $A+A^T = \sqrt{A^2+A^{2T}+AA^T+A^TA} > 0$ This square root exists right? And because of this, the sum of all its elements are ...
0
votes
2answers
103 views

Canonical form of a Matrix question involving a conjugacy

How do i find the canonical form of this matrix, my attempt is to use it in a conjugacy for flow. $$A=\begin{pmatrix} 0&1&0 \\ -1&0&0\\ 1&1&1\end{pmatrix}$$ do i need to ...
0
votes
3answers
119 views

Find a nonzero $3\times 3$ matrix with all 0 eigenvalues. Is there a systematic way?

After playing around for a bit I found one: $$ \begin{bmatrix} 0&1&0\\ 0&0&1\\ 0&0&0 \end{bmatrix} $$ but I couldn't find a good systematic way.
0
votes
2answers
632 views

Finding eigenvectors and eigenvalues of a matrix with complex numbers

I need to find eigenvectors and eigenvalues of $\begin{pmatrix} 1 & i \\ -i & 1 \end{pmatrix}$. Attempt: When I find the equation which I have to solve for the eigenvalues I get $(\lambda ...
0
votes
5answers
800 views

How do I show that this statement is true?

If I have the matrices $A$ and $D$ where if $D$ satisfies $AD=I$ ($I$ is the identity matrix) then $D=A^{-1}$, how do I show this is true using matrix algebra? I wanted to just say that if $D=A^{-1}$, ...
1
vote
2answers
279 views

How to prove that det($A^{T}A$) is nonnegative?

Why is the determinant of the product of a matrix and its transpose nonnegative?
1
vote
1answer
81 views

Linear dependence of functions of t

Let $\ \vec x_1=(e^t,te^t)$ and $\vec x_2=(1,t)$. Show that they are linearly dependent at each point $t\in [0,1]$. Nevertheless, show that they are linearly independent on $t\in [0,1]$. Attempt: ...
1
vote
1answer
232 views

norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
2
votes
1answer
310 views

How to get Euler angles with respect to initial Euler angle

I have a sensor which gives me Euler angles (roll,pitch,yaw). There is a baseline value of Euler angle (assume it is $5,10,15$) at the beginning.I want to calibrate from this baseline values all ...
4
votes
2answers
398 views

Diagonalise a matrix and show the formula

I have diagonlised P to get $$P=\left(\begin{matrix} -1 &0 &0\\ 0 &0 &0\\ 0 &0 &1 \end{matrix}\right)$$ however am unsure on how to proceed, would appreciate any help! By ...
2
votes
2answers
88 views

How do I know if a discrete time-invariant homogeneous dynamic system will reach, at some point, an equilibrium point?

Is this even possible? Given a time-invariant homogeneous dynamic system: $$x(k+1) = Ax(k)$$ My textbook defines an equilibrium point of the system as: A vector $\bar x$ is an equilibrium point ...
0
votes
1answer
183 views

Proving the following fact about the matrix exponential.

Assume the formula $\det(e^A)=e^{\operatorname{tr}(A)}$ for all matrices $A \in \mathbb{C}_{n\times n}$. Show why this implies that the exponential always yields a regular matrix.
4
votes
2answers
50 views

Why is this transformation's standard matrix the way it is?

So I know for a simple rotation of $2$ vectors, I know that the vector $\left[\begin{matrix}1\\0\end{matrix}\right]$ rotates to $\left[\begin{matrix}\cos\phi\\\sin\phi\end{matrix}\right]$ and the ...
1
vote
1answer
488 views

Projecting a point on a plane through a matrix

I need to render some shadows in opengl, one way to do this is to render your object twice, a first time multiplying it by a special "shadow matrix" that flat your object on a plane generating the ...
1
vote
2answers
46 views

Solving symbolic equation from determinant numerically

In this problem: Intersection of conics using matrix representation , a situation arises where there are two matrices (for example:) $$Q_1 = \begin{bmatrix}1 & 0 & 0\\0 & -1 & 0\\0 ...
1
vote
1answer
55 views

Rank of matrix products

Suppose $A_1$, $A_2$ and $A_3$ are $3$ by $3$ matrices of rank $2$ such that their kernels are linearly independent. Is the following true? Define: $V_1=A_2A_1$, $V_2=A_3A_2$ and $V_3=A_1A_3$. Then ...
0
votes
1answer
549 views

computing the inverse of a special sparse matrix

Given a high-dimensional symmetric postive-definite matrix with only the main diagonal and several other diagonal (say, 1st, 5th and 100th) above and below the main diagonal to be non-zero and all ...
2
votes
1answer
51 views

Is there any geometric interpretation of a non-invertable matrix?

My question is basicly what the title says. Also is there any non-invertable n by n matrices except the all 0-element matrices? Thanks in advance. =)
0
votes
1answer
159 views

Is the determinant of a matrix preserved under permutations of the rows/columns of a matrix?

Is the determinant of a matrix preserved under permutations of the rows/columns of the matrix? If not, is its absolute value preserved?
1
vote
1answer
82 views

Flow of D.E what is the idea behind conjugacy?

I got some kinda flow issue, ya know? well enough with the bad jokes let A be a 2x2 matrix, T a change of Coordinate matrix, and $B=T^{-1}AT$ the canonical matrix ascoiated with A. Show that the ...
3
votes
2answers
154 views

Real symmmetric positive definite matices have all diagonal entries positive?

If a real symmmetric matrix $ A $ is positive definite. Is it true that all diagonal entries are positive? I have only prove it for a matrix of order 2. How can I prove or disprove it, please ...
0
votes
1answer
111 views

Using Maple to work out error terms

I have the exponential matrix using a Jordan form matrix and transition matrix: $E(t) = e^{At}= \left[ \begin{array}{ccc} e^{3t}+3te^{2t} & 3e^{3t}-3e^{2t}+6te^{2t} & ...
1
vote
2answers
410 views

Changing parameters in a 3x3 Matrix trying to find the general solution.

Consider the system $$X^{'}= \begin{pmatrix} 0&0&a \\ 0&b&0\\ a&0&0 \end{pmatrix}X$$ depending on the two parameters a and b. 1) find the general solution of this system. ...
1
vote
2answers
109 views

Basic questions regarding matrix algebra.

I had two true/false questions on my exam of which I missed. $1)$ The map $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $T(x)=x+e_1$ is a linear transformation. I know this to be false, because ...