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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33 views

Orthogonal Matrix question

$$A= \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} $$ is an orthogonal matrix. a) Prove that $A^{-1}=A^T$ b) show further that $a^2=d^2$ and that $b^2=c^2$. ...
1
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0answers
86 views

How to find the number of connected components of a graph by using its 16x16 adjacency matrix?

Good day, I have this exercice that provides me with the 16x16 matrix of adjacency of a graph and it asks me to find the number of connected components of the graph and I need to give a spanning tree ...
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1answer
42 views

Homogenous System

This question caught me off guard. I believe I did it the right way, but it was a bit confusing and I wanted a bit more explanation of the process and to see if what I got was accurate. This is the ...
4
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1answer
66 views

Is $\left(\sum_{n=0}^\infty\frac{1}{n!}A^n\right)v=\sum_{n=0}^\infty\frac{1}{n!}(A^nv)$?

Suppose we have a convergent power series of matrices $$A=\sum_{n=0}^\infty a_nX^n,$$ for $X\in M_n(\mathbb{C})$. Is it true that if $v\in\mathbb{R}^n$ then $$Av=\sum_{n=0}^\infty ...
4
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3answers
77 views

Are there real solutions to $\exp(X)=-I$?

As we know, the equation $$e^x=-1,\quad x\in\mathbb{C}$$ has no real solution (in fact $x=i\pi+2ki\pi$, $k\in\mathbb{Z}$). I am now considering the generalization of this question to $2\times 2$ ...
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4answers
42 views

If two invertible matrices agree on a vector, does this imply their determinant agrees as well?

As stated, if we let $A, B \in M_n(\mathbb{R})$ be invertible and there is some $v\in R^n$ such that $$Av = Bv$$ does it follow that $\det(A) = \det(B)$? Additionally, does this hold if we let $A, B ...
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3answers
51 views

Square root of determinant equals determinant of square root?

Is it true that for a real-valued positive definite matrix $X$, $\sqrt{\det(X)} = \det(X^{1/2})$? I know that this is indeed true for the $2 \times 2$ case but I haven't been able to find the answer ...
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1answer
43 views

Find the inverse of defined operation $\Delta$

We defined the operation $ \Delta $ as $ (a,b) \Delta (c,d) = (ac + \delta bd, ad + bc) $ where $ a, b, c ,d \in \mathbb{Q} $ I have already proven that this operation is both commutative and ...
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1answer
52 views

Limit of regular symmetric matrix

I've got this statement about the topic. I'm trying to figure it out as it is given without proof. I know a symmetric matrix is a square matrix $A$ such that $A = A^T$ and a regular matrix is one ...
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1answer
42 views

Symmetric matrix-Spectral theorem

Assume we have a matrix $A$ let's say $100 \times 4$. We determine the product $B=A^{T}A$ Then by the spectral theorem \begin{equation} B =U^{T} \lambda U \end{equation} $B$ is a symmetric matrix ...
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0answers
18 views

matrix range problem

[EDITED: range --> rank] Let $A, B \in \mathbb{R^{n \times n}}$. Prove that if $rank(A)=n-1$ then $rank(AB) \geq rank(B)-1$. I already know how to do this using that $rank(A)= dim(Im(A))$, and ...
0
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2answers
34 views

Prove that Det(A-E)=0 if and only if AC=C

We have some $n \times n$ matrix $A$ and $n \times 1$ vector C. Let $E$ be the identity matrix. $$Det(A-E)=0 \iff AC=C.$$ Me and a few friends have been trying to prove it, but none of us could. ...
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0answers
9 views

An algebraic relationship between inverses of the correlation and covariance matrices

Suppose that we have $p$ random variables $(x_1,\ldots,x_p)$. Stack them together as $x=(x_1,\ldots,x_p)'$ and let $V$ be the covariance matrix of $x$ and $R$ the correlation matrix. Suppose that $V$ ...
2
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1answer
52 views

If $A$ is a square matrix that is linearly independent, is $AA$?

I'm just not sure how to start this problem from Linear Algebra Done Wrong. The problem is to prove that if the columns of $A$, square matrix, are linearly independent, then the columns of $A^2$ = ...
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2answers
14 views

A question on Matrix limits

Consider $M(n, \mathbb R)$ , the set of all real square matrices of size $n$ , as an NLS with norm $||A||:=\sqrt{\sum_{i=1}^n \sum_{j=1}^n a_{ij}^2}=Trace(AA^t)$ , then is it true that a sequence of ...
2
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2answers
85 views

Why use homogeneous coordinates?

I am having trouble understand the use of homogeneous coordinates for when describing transformations in 3D space. From what I have seen, the only difference between a transformation matrix in ...
2
votes
2answers
70 views

Minimal polynomial of an $n\times n$ matrix $A$ is $x^3+2x+2$; then $3$ divides $n$

Let $A$ be an $n × n$ matrix with rational entries such that the minimal polynomial of $A$ is $x^3 + 2x+2$. Prove that $3$ divides $n$. I think there is no rational root of this polynomial but ...
0
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1answer
31 views

Transformations between coordinate systems

I have three three-dimensional orthogonal coordinate systems, O, A and B. A and B are the result of two different transformations from O. I now want to calculate the transformation matrix R, which ...
1
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1answer
79 views

How to obtain a convergent solution iteratively for a linear system of equations?

I am working on a problem that requires an iterative procedure to solve a linear system of equations, the system of equations in matrix form is: $$\underbrace{\begin{bmatrix} r_{11} & r_{12} ...
0
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1answer
20 views

eigenvalue of a specific matrix

I am looking for a way to calculate the eigenvalues of this matrix. the last row contains complex numbers in general.
0
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1answer
34 views

Finding the value for the reproduction rate that will cause population stabilization

So essentially I have a matrix equation AB A is a 4x4 matrix containing reproduction rate, survival rate and maturity rate. B is a 4x1 matrix containing the populations for each age group. How would ...
0
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2answers
36 views

Suppose A is $n$ x $n$ and the equation A $\vec{x} = \vec{b}$ has a solution for each $\vec{b}$ in $\mathbb{R}^n$

Explain why A must be invertible. Can someone explain why? I am a little confused here.
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2answers
53 views

A real and normal matrix with all eigenvalues complex but some not purely imaginary?

I'm trying to construct a normal matrix $A\in \mathbb R^{n\times n}$ such that all it's eigenvalues are complex but at least one of then also has a positive real part (well, at least two then, since ...
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0answers
20 views

Decomposition of a matrix into a product of two-level unitaries

I am working through example 4.12 and 4.13 from 'Quantum Computing: From Linear Algebra to Physical Realizations' (Pages 83-84 and 405-406) and I can't seem to figure out some what the '*' in figure ...
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1answer
34 views

Let $x = (11, 2)^T$ . Find both reflection matrices $M$ such that $Mx$ is a multiple of $e_1$.

How would I go about solving this? I believe my professor said that it deals with householder matrices. I feel like I should calculate $v = x + ||x|| e_1$ and then calculate $u = \frac{v}{||v||}$ ...
0
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1answer
34 views

Use given identity to computer exponent of 4x4 matrix

I've been given an identity (that I don't know how to prove unfortunately), and been asked to use it to compute exp$(xM)$, where $$ M = \begin{bmatrix} 1 & 1 & 1 & 1 \\ ...
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0answers
43 views

Given a Positive Definite Matrix, find conditions of elements inside the matrix

I have a question that asks me to use the following symmetric positive definite matrix of order $n + 1$ $$B = \begin{bmatrix} \alpha & a^T \\ a & A \end{bmatrix} $$ With this matrix, I ...
2
votes
2answers
103 views

System of Equations

How would I solve a $4 \times 3$ matrix? I've tried making it into an augmented matrix but I ended up with all zeros at the bottom. Please help! $$\begin{align}\begin{cases}x_1+x_2+x_3+x_4&=1 ...
0
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2answers
63 views

Calculate $e^{xA} $

$$ A = \begin{bmatrix} 1 & 1 & -1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix} $$ I have the answer, but I don't know the ...
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0answers
40 views

Given justification for induced norm for nonsingular matrix

Help!! Determine whether the following statement is true or false and give a justifcation: If A is nonsingular, then for any induced norm, $$ ||A^{-1}|| = ||A||^{-1}$$ Thanks!
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1answer
30 views

About a determinant identity.

If $A$ is any matrix and $B$ is a rank $2$ matrix of the same dimension then it follows that for any real $t$, $det(A -B) = [1-\partial_p + \frac{1}{2}\partial_p^2 ]det(A + pB) \vert _{p=0}$ I ...
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2answers
55 views

Sherman-Morrison Formula to determine inverse of $3 \times 3$ matrix.

Given $$A = \pmatrix{2 & 0 & -1 & \\ -1 & 1 & 1 \\ -1 & 0 & 1}$$ and $$A^{-1} = \pmatrix{1 & 0 & 1 & \\ 0 & 1 & -1 \\ 1 & 0 & 2}$$ I want to use ...
2
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1answer
63 views

Approximating a matrix so that 1) all rows sum to one and 2) all values have max 6 digits.

Let consider a big matrix with values ranging from 0 to 1 (included). Each row sums to values that are lower than 1, extremely ...
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2answers
56 views

Norm of orthogonal projection

Consider $\Bbb R^n$ with the standard inner product and let $P$ be an orthogonal projection defined on $\Bbb R^n$. It is known that the operator norm of $P$ induced by the inner product is less than ...
1
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3answers
135 views

Prove that $trace(A^TA) = 0$ if and only if $A = 0$.

Given that $A_{m \times n}$ has real entries, I want to prove that $trace(A^TA) = 0$ if and only if $A = 0$. In other words, I want to show that the only way for the trace of $(A^TA)$ to be zero is if ...
1
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3answers
47 views

Matrix Algebra simplify $(A^{T} A)^{-1}A^{T}(B^{-1}A^{T})^{T}B^{T}B^{2}B^{-1}$

I'm sorry, this is probably very basic... I'm trying to review stuff to make sure I dont forget things. The question is simplyfy the below as much as possible: $(A^{T} ...
1
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1answer
37 views

Finding eigenvalues of a matrix with two unknowns

I've been asked to find the eigenvalues of the following matrix: $$ \begin{bmatrix} 0&1&1\\ 0&0&1\\ 216k^3&-108k^2&18k \end{bmatrix} $$ I'm just not sure how to work it out as ...
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0answers
13 views

Testing if a matrix is MDS?

In the Wikipedia page that explains the concept of an MDS matrix, the following condition for a given $m \times n$ matrix being MDS is mentioned: it is MDS if and only if all the square sub-matrices ...
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0answers
25 views

Can i from the Smith Normal Form conclude which Rows are Linear Dependend?

If i have calculated for an integer matrix A: $$A = V*S*T$$ So that S is Smith Normal form, and V,T, are unimodular matrixes. The rank of S is equal to the rank of A. Can i somehow decide which ...
1
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1answer
79 views

Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $

With the definition of $ \lVert A \rVert_2$ and $\lVert A \rVert_1$ and $\lVert A \rVert_ \infty$ that is: \begin{gather} \lVert A\rVert_1 = \max_{j} \sum_{i=1}^m \lvert a_{ij}\rvert\\ \lVert ...
0
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2answers
31 views

Matrices proof by induction

For any 3x3 matrix $A$, prove by induction that $$(A^T)^n=(A^n)^T$$ for all $n∈ℕ$ I'm not sure how I do this.
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2answers
70 views

Hints on how to calculate $A^{99}$.

\begin{equation} A=\frac{1}{3} \begin{bmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \\ \end{bmatrix} \\ \end{equation} $A$ is orthogonal. ...
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1answer
16 views

$C(A)\subseteq C(B)$ Meaning To Linear Transformations

Let there be $A,B\in M_{m\times n}(F)$ and $C(A)\subseteq C(B) $ where $C(A)$ means the column space Prove: there is $C\in M_{n\times n}(F)$ so that $A=BC$ That can be shown by matrix ...
0
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0answers
60 views

Which method of calculating team rankings using matrices is best?

I am trying to work out which method of calculating team rankings when using matrices is best (specifically dominance matrices but If you have a better way please share) I have tried it two ways so ...
0
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1answer
32 views

How do I do about solving this matrix equation?

Solve the matrix equations $CYA = D$ in which $$A=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 2 & 1 \\ 0 & 1 & 1 \end{pmatrix} \qquad D = \begin{pmatrix} 1 & 1 & 2 \\ 2 & 0 ...
2
votes
1answer
78 views

Simulate correlated $\chi^2$ distribution

I understand that when one have multiple independent variable that follows $N\sim(0,1)$, denoted as $A$ if we have a correlation matrix $R$, we can generate correlated variables $B$ that are normally ...
0
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0answers
52 views

Matrix and Vector Norm with diagonal matrices

I don't even know where to begin with a problem like this. Where should I start? A thorough (no shortcuts )answer and explanation is greatly appreciated. Let A be symmetric positive definite. Show ...
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1answer
45 views

Matrix gauss-jordan / gaussian

I am a bit confused in terms of doing gaussian and gauss-jordan elimination for a system of equations. For example let's say we have the following system of equations: We get the following: ...
15
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1answer
144 views

When do two matrices have the same exponential?

Let $A$ and $B$ be $n\times n$ hermitean matrices. When do we have $e^{iA}=e^{iB}$? Can we somehow classify those pairs of matrices that have the same exponential? Here are some observations that I ...
9
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7answers
3k views

Why is the inverse of a sum of matrices not the sum of their inverses?

Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$? I know the answer is no, but don't get why.