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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24 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 ...
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48 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 $u = \frac{v}{||v||}$ ...
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2answers
46 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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11 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 $V$ ...
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252 views

Why is the rank of a matrix invariant under row operations?

Prove that the rank of a matrix ($m\times n$) doesn't change if we apply row operations. For example if we multiply a row with a nonzero number $k$.
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131 views

Rank of vectors

Prove that the rank of a system of vectors from $E^n$ does is not bigger than the dimension of the vectors. For example the vectors $a,b,c$ are from $E^n$ so each of them has $n$ components (the ...
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113 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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15 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 ...
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1answer
75 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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1answer
30 views

Transition functions of sheaf tensor product

Suppose $\mathcal{L},\mathcal{M}$ are invertible sheafs on a scheme $X$. I've seen an abstract construction of $\mathcal{L}\otimes_X \mathcal{M}$, but I'm having trouble connecting this with a more ...
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2answers
136 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 ...
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1answer
55 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 ...
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2answers
87 views

Expression for arbitrary powers of a particular $2\times2$ matrix

Given$$\mathbf{M}= \begin{pmatrix} 7 & 5 \\ -5 & 7 \\ \end{pmatrix} $$, what's the formula matrix for $\mathbf{M}^n$? The eigenvalues and eigenvectors are ...
2
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1answer
115 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 ...
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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.
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1answer
37 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 ...
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2answers
37 views
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43 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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2answers
104 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 ...
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1answer
46 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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26 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 ...
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1answer
40 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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2answers
66 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
51 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 ...
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1answer
31 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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52 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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2answers
879 views

How to prove that 2-norm of matrix A is <= infinite norm of matrix A

Now a bit of a disclaimer, its been two years since I last took a math class, so I have little to no memory of how to construct or go about formulating proofs. My tutor unfortunately is very and ...
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2answers
111 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 ...
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2answers
129 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 ...
2
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1answer
83 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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3answers
322 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 ...
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3answers
49 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} ...
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0answers
15 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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1answer
91 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 ...
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2answers
32 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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1answer
42 views

Determinant is product of different primes

Let $M_{1}$, $M_{2}$ be two $n \times n$ matrices with entries in $\mathbb{Z}$ such that $\det(M_{1})=\det(M_{2}) = p_{1}p_{2}\cdots p_{m}$, where $p_{j}$ are distinct prime numbers. I need to show ...
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74 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
52 views

Find the approximation of an inverse of sum of two special matrices?

Given $K$ is a psd symmetric covariance matrix. Is there any way that we can approximate $(I+K)^{-1}$ by $K$. Where $I$ is the identity matrix. Best,
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1answer
19 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 ...
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0answers
91 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 ...
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1answer
133 views

Find Representative Matrix of a Linear Transformation

Given $L:R^{3}\to R^{3}$ for which $L(\begin{bmatrix}x ,y ,z\end{bmatrix})=\begin{bmatrix}2x+y+3z\\x+2y+z\\x-y+2z \end{bmatrix}$ and the basis $U={[2,0,1],[1,1,0],[0,-1,1]}$ I have to find the ...
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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 ...
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2answers
453 views

Weirdly written matrix equation

The matrix $A$ is a $3 \times 3$ matrix. $$A = \left [\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 2 & 1 \\ 0 & 1 & 1 \\ \end{array} \right ]$$ How can I solve the following ...
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0answers
91 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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2answers
62 views

Scaling of a matrix?

Lets say you have two column vectors: $$\begin{array}{|r r|} 3 & -4 \\ 4 & 3 \\ \end{array}$$ How to find the scaling of this matrix? I thought that it can be any number but apparently it ...
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1answer
66 views

How to do the derivative of log matrix with respect to scalar?

I am trying to find the good cost function for my optimization problem and I come across the logarithm of the matrix. $$\log{(t\mathbf{Z})}$$ where $\log$ is a matrix logarithm and the matrix $t$ ...
3
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2answers
61 views

How to compute the trace of an exponential and diagonal matrix?

I would like to understand how to compute the trace of an exponential and diagonal matrix. For instance, what is: $$ \mathrm{Tr}\left[ \exp \begin{pmatrix} 5 & 0 \\ 0 & 8 \end{pmatrix} ...
2
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3answers
60 views

Find Formulas for $M^{n}$ of Matrix $M$

Find formulas for the entries of $M^n$, where $n$ is a positive integer. $M$ is the following matrix: $$ \begin{bmatrix} 4 & -2 \\ 4 & 10 \end{bmatrix} $$ So if $M=PDP^{-1}$ then it follows ...
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1answer
89 views

Is it possible to compute row and column sums of $A^{-1}$ given row and column sums of $A$?

The question is simple: We have a symmetric matrix "A" with all diagonal entries 1. Unfortunately Off-diagonal entries are unknown, but we know the row and column sum of A. Now we just need the row ...
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1answer
19 views

QR Decomposition when u3 is zero

I need to find decomposition for a matrix. I am using Gram Schmidt. $ A = \begin{bmatrix}1&1&7\\1&2&8\\1&3&9\end{bmatrix}$ I am able to find $ u_1 ,u_2, e_1, e_2 $ $ u_3 ...