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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1
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1answer
96 views

Column space of complex matrix

Let $A=\begin{pmatrix} 1 & \alpha & \alpha^2\\ 1 & \beta & \beta^2 \end{pmatrix}, \alpha,\beta\in\mathbb{C},\alpha\ne\beta$. I know that the column space of A is supposed to ...
2
votes
1answer
218 views

Jacobi's Method for a matrix

Say we have a $2\times 2$ matrix $A$: $$A=\begin{pmatrix} 1&2 \\ 3&1\end{pmatrix}$$ What is the spectral radius of $A$? So I get the eigenvalues of $A$, and the maximum eigenvalue ...
3
votes
0answers
148 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
1
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2answers
43 views

Inverting all values in matrix

Lets say I have a matrix: $$\left[\begin{array}{cc} 2 & 4 \\ 3 & 7 \\ \end{array}\right] $$ And my maximum range value is $10$, how would I go about creating another matrix that ...
3
votes
0answers
113 views

Counting symmetric unitary matrices with elements of equal magnitude

Let $X$ be an $n\times n$ symmetric unitary matrix with elements of equal magnitude and the elements of the first row (and the first column, of course) are $1/\sqrt{n}$, i.e. $X_{j,k} = e^{i ...
0
votes
2answers
327 views

Reverse rows in a matrix

To rotate a matrix 180 degrees around the center point, what I am planning to do is first transverse the matrix, then reverse the rows and then do it again to produce the final result. This works and ...
1
vote
1answer
90 views

the matrix in example is following which non-negative matrices ?

The matrix in below is following which non-negative matrices irreducible, stochastic, and primitive (or ergodic) matrices? i know it is a stochastic row matrix as row sum =1 but i have no idea about ...
1
vote
1answer
146 views

Matrix representation of linear transformation

I'm having trouble with part of homework exercise. Let $\mathcal{P}_3(\mathbb{C})$ be the vector space of complex polynomials of degree 2 or lower and $\alpha,\beta\in\mathbb{C},\alpha\ne \beta$. Let ...
2
votes
1answer
102 views

Trace matrix inequality

Let $A,B$ be positive definite matrices, and assume that $$ a_{i,j}<{b_{i,j}} $$ for all $1\leq i,j\leq n$, where $a_{i,j}$ is the $(i,j)$ element of the matrix $A$ and $b_{i,j}$ is the $(i,j)$ ...
0
votes
0answers
52 views

Solve system of linear equation

Can anyone help we solve the following: Find $X$ so that $F'(X)=0$ s.t. $F(X)=hX^T-\frac{1}{2}X^T \Lambda_p X$ Where $X$ is a vector with elements $x_i$ and $Λ_p$ is a matrix with diagonal ...
3
votes
1answer
123 views

eigenvalues of block matrix with the eigenvalues of one block already known

Give a matrix which can be decomposed into 4 parts $B = \left[\begin{matrix}A &I \\ -I &0\end{matrix}\right]$ where $I$ denotes the identity matrix and $0$ is a zero matrix. It's easy to ...
5
votes
3answers
2k views

Eigenvalues and Eigenvectors of Large Matrix

Computing eigenvalues and eigenvectors of a $2\times2$ matrix is easy by solving the characteristic equation. However, things get complicated if the matrix is larger. Let's assume I have this matrix ...
2
votes
1answer
112 views

Row space and kernel in linear transformations

I am preparing a "dictionary" that translates between the "language of matrices" and the "language of linear transformations" in linear algebra. The dictionary looks more or less like this: Language ...
0
votes
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
490 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 $x^\top C x > 0$ ...
2
votes
2answers
594 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
112 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
291 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
137 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
267 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
544 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
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1answer
1k 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?
1
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2answers
255 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
47 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
63 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
56 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
553 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
177 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
220 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
220 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
91 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
56 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
426 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
82 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
218 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
85 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
104 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
147 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.
1
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2answers
1k 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
981 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
404 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
103 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
300 views

norm equivalence

Exercise 7. of Terence Tao's blog on random matrix, specifically on eigenvalues of Hermitian matrices ...
2
votes
1answer
392 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
420 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
94 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 ...