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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10 views

Looking for easygoing, well-motivated introductions to matrix norms.

I find all the various matrix norms very hard to navigate, probably because I don't know what they're used for. Question. What are some easygoing, well-motivated introductions to matrix norms? ...
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29 views

Challenge your mind with an interesting problem in probability

Let $k$-blob (or polyominos) be the number of pixels with the value of $1$ (other wise the value of any given pixel is $0$), that are attached to one another in an $n×m$ matrix. The odds of a pixel ...
0
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0answers
9 views

Eigenvalues of positive linear combination of p.d. matrices

I want to prove a property on the eigenvalues of a positive linear combination of p.d. matrices. I have the following: $$ z \in \mathbb R^m_{++} $$ $$ A(z) = \Sigma z_i A_i $$ $$A_i \in S^n_{++} ...
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14 views

Find the matrix of linear transformation $L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x)$ with respect to basis $\mathcal{B}=\{3,x-1,x^2+1\}$

Find the matrix of linear transformation $L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x)$ with respect to standard basis $\mathcal{B_1}=\{1,x,x^2\}$ and with basis $\mathcal{B}=\{3,x-1,x^2+1\}$ where ...
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0answers
29 views

Prove that $\operatorname{adj}(A) = \frac{1}{2}[(\operatorname{tr} A)^2 - \operatorname{tr}(A^2)]I_3 - [\operatorname{tr}(A)]A + A^2 .$ [duplicate]

Let $A$ be a square matrix of order $3$. Prove that $$\operatorname{adj}(A) = \frac{1}{2}[(\operatorname{tr} A)^2 - \operatorname{tr}(A^2)]I_3 - [\operatorname{tr}(A)]A + A^2 $$ where ...
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0answers
3 views

Lower bound on the sum of singular values for a sum of Hermitian matrices

Suppose $\mathbf{A}$ is a Hermitian $n\times n$ matrix with eigenvalues $\lambda_i(\mathbf{A})$, $i=1,\ldots,n$. Suppose $\mathbf{B}$ is an $n \times n$ complex-valued matrix and $b\neq 0$ is a ...
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1answer
24 views

I don't understand how Kirchhoff's Theorem can be true

Kirchhoff's Matrix-Tree theorem states that the number of spanning trees of a graph G is equal to any cofactor of its Laplacian matrix. Wouldn't this imply that all cofactors of a Laplacian matrix ...
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4answers
21 views

Show that a matrix with (I) a row of zeros and (II) a column of zeros cannot be invertible (respectively)

Show that a matrix with a row of zeros cannot be invertible. Show that a matrix with a column of zeros cannot be invertible. What I tried: I tried to show that a matrix $A \in M_n (\mathbb{R})$ such ...
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1answer
41 views

Show that $(I − Q)^{−1} $= $Q^2 + Q+ I$.

Consider $Q\in M_n (\mathbb{R})$ Assume that $Q^3 = [0] $ show that $ (I − Q)^{−1} = Q^2 + Q + I$. What I tried: I tried to use $(I-Q)(I-Q)^{-1} = I$ and use that to manipulate the left side of the ...
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0answers
10 views

What is the name of the sub-matrix?

Given a matrix $\mathbf{A}$ of size $n\times n$. Let $I=\{i_1,\ldots,i_k\}\subseteq\{1,\ldots,n\}$ for some $k\leqslant n$. How to call the sub-matrix of $\mathbf{A}$ that has its indices in $I$? (I ...
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2answers
50 views

Use matrix algebra to show $A(B^{-1}(A+B)A^{-1})B = A+B$

I've got a super simple linear algebra question for an intro college course I can't seem to figure out. Using matrix algebra and matrix identities, show that: $$ A(B^{-1}(A+B)A^{-1})B = A+B $$ ...
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1answer
12 views

2-Norm of Non-Square Matrices

So, the 2-norm of an m x n matrix for m=>n is defined by the max singular value/square of the max eigenvalue. But, if it's not square, and you're only given a matrix A (no x-vector), what do you do if ...
2
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2answers
38 views

Proving a Trick to More Quickly Calculate N-Step Transition Probabilities

So, I have been working on a homework problem all day that asks me to prove that: $P^n= \Pi +Q^n$ where P is the transition matrix of a finite-state regular Markov Chain, $\Pi$ is a matrix whose rows ...
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1answer
13 views

$0 * \Lambda ^{1/2} \neq 0$ (?) Problems with matrix multiplication

Suppose $\Lambda$ is a diagonal matrix of size $n > 1$ and rank $1$, let's denote the sole element on the diagonal as $\lambda$. Consider the following equation: $\Lambda ^ {-1/2} = 1/\lambda * ...
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0answers
10 views

Stationary points with matrix

I have an exercise but I do not even know where I should start: Consider the normalised quadratic form $(x^T Ax)/(x^T x)$ where $x∈R^2$, $A$ is a general 2x2 matrix. Find the vectors that make this ...
4
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1answer
34 views

Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
2
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0answers
33 views

Under What Intervals Is A Matrix Positive Definite, Positive Semi-Definite, Indefinite, Negative Definite and Negative Semi-Definite?

Suppose we have a matrix which represents a quadratic form. $$ \begin{matrix} a & -a & -3a \\ -a & 2a & 2a \\ -3a & 2a & (9a+2) \\ ...
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0answers
8 views

Find a basis for the column space - why not reduce to RREF first?

Related to Understanding how to find a basis for the row space/column space of some matrix A. . When asked to find the basis for the column space of a matrix, can I first reduce to RREF, and then use ...
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2answers
48 views

Find $\det(A)$ of Matrix and condition on a and b

Let $$ A=\begin{bmatrix} a & b & 1 \\ b & 1 & b \\ 1 & a & a \\ \end{bmatrix} $$ Find $\det(A)$ in terms of $a$ and $b$, and write down ...
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1answer
16 views

L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of ...
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3answers
32 views

$A$ has more columns than rows and has full row rank, show there exist infinitely many $B$ s.t. $AB=I$

If A $\in M_{m\times n}(R)$ such that $n>m$. Prove that if $\text{rank} (A) = m$ then there are infinitely many matrices $B \in \ M_{n\times m} (R)$ such that $ AB = I_m$ So the question is ...
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0answers
55 views

Prove that $A+2I$ is invertible [duplicate]

Given $A$ is a square matrix such that $A^{3} = 2I$ Prove that $A+2I$ is invertible and find its inverse. How do I prove that $A+2I$ is invertible? For proving $A-I$ is invertible, I use ...
0
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1answer
26 views

Differentation of vector with respect the another vector [on hold]

$y$ is $m \times 1$ vector $y=Ax$. $A$ is $m \times n$ matrix in function of $z$. $x$ is $n \times 1$ vector in function of $z$. And $z$ is vector $r \times 1$. How can i find $\frac{dy}{dz}$? I ...
3
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3answers
35 views

Intuitive understanding of vector / matrix calculcation

I am currently dealing with calculations done on vectors and matrices. For some parts I have gained an intuitive understanding, for others I didn't. E.g., when we are adding two vectors, you can ...
2
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1answer
33 views

Does $A-\lambda I$ have rank smaller than $A$?

Consider $\lambda$ as one eigenvalue of $A$, can we say that $A-\lambda I$ must have rank smaller than $A$? Or equivalently, $A-\lambda I$ spans a space which is a subset of $A$?
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1answer
23 views

Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal

Let $A$ be an $n \times n$ matrix over a field $F$. Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal. I don't understand how to apply this when it comes to ...
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0answers
5 views

Difference between the SCM converging to the Marcenko-Pastur distribution and Johnstone's result about the top eigenvalue

I have a confusion which I suppose must be rather basic. As I understand, in the 60s/70s it was known that the empirical eigenvalue distribution of the sample covariance (of $n$ i.i.d. standard ...
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0answers
79 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix?
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1answer
23 views

L2 Matrix Norm Upper Bound in terms of Bounds of its Column

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1} $ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a ...
2
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0answers
29 views

Find (linear) transformation matrix using the fact that the diagonals of a parallelogram bisect each other.

This is the first time I post something on this website. I'm on this question already for hours. I'm clearly not asking the community to do my homework, I'm hoping someone can explain me how I should ...
2
votes
1answer
49 views

The root system of $sl(3,\mathbb C)$

I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem? I know that the root-system is $A_2$ but I want to see a ...
0
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2answers
20 views

Finding a matrix inverse when an equation involving it is a multiple of the identity matrix

Say you had a matrix $A$, and you did an equation like $A^2 - A$, and proved that it was a multiple of $I$. How could you find $A^{-1}$ in the form $rA + sI$ after proving that? I want to do it ...
4
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1answer
63 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ ...
0
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1answer
30 views

Operation count, LU-decomposition

I'm having trouble with an assignment question. The question is as follows: Determine the total number of multiplications and divisions (as a function of $n$) required to compute the LU-decomposition ...
2
votes
1answer
19 views

Is it true that for all matrices $A$ and all traceless matrices $T$, there exists a traceless matrix $T'$ such that $AT = T'A$?

Fix a real number $n$. By a "matrix", I mean an $n \times n$ real matrix. Now let $A$ denote a matrix. Is it true that for all traceless matrices $T$, there exists a traceless matrix $T'$ such that ...
1
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1answer
48 views

Proving matrix is invertible using the Banach Lemma

I have an assignment question that goes like this: Consider the $n \times n$ matrix $$ \begin{pmatrix} 2 & 1 & 2^{-1} & 2^{-2} & 2^{-3} & 2^{-4} & \cdots & ...
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0answers
34 views

Minimizing the error by finding optimum step-size

I need to recheck a proof for minimizing the error by finding optimum step-size. I re-checked the proof many times but still can't find a mistake although the number I am getting in Matlab is not ...
0
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0answers
11 views

Perturbation theory for a symetric rank-one update

I know perturbation theory of the eigenspectrum/singular value decompostion of a symetric matrix $A$ under a symetric perturbation $E$, that besides being symetric has no other structure. Is there ...
0
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1answer
15 views

Prove of identity: $(Av) × (Aw) = CofA (v × w)$ [on hold]

How can I prove that for each $A \in M^{3×3}$ and $v, w ∈ \mathbb R^3$ $(Av) × (Aw) = CofA (v × w)$
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4answers
217 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
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0answers
41 views

XOR binary matrix multiplication $AX=B$? [on hold]

Let $A$, $B$, and $X$ be binary matrices (in F2 ), where $A$ and $B$ are of size $n \times m$ with $n > m $. $X$ is an $m \times m$ matrix. Compute $X$ such that $AX=B$. ps: $A$ is not a ...
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1answer
43 views

Prove that a real matrix is a matroid

Problem $A$ real matrix, size $m\times n$ $M$ some structure, possible matroid $E(M)$ set of all columns of $A$ (we're considering them vectors) $I(M)$ set of all linearly independent columns of $A$ ...
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2answers
14 views

Prove a covariance matrix is positive semidefinite

Given a random vector c with zero mean, the covariance matrix $\Sigma = E[cc^T]$. The following steps were given to prove that it is positive semidefinite. $u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = ...
0
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1answer
21 views

Prove that a small shift in the diagonal term leads to smaller spectral radius (for Perron-Frobenius theorem)

On Wikipedia, the proof for Perron Frobenius theorem in the strictly positive case has a confusing step: Suppose $T=A^m-\epsilon I$, where $\epsilon$ is smaller than the smallest diagonal term of ...
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1answer
25 views

How polynomials are represented in matrix form for Univariate Polynomial. [on hold]

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
0
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1answer
56 views

Wolfram|Alpha refuses to find the inverse of a large 6x6 matrix.

Just to be clear, this isn't a question on how to find the inverse of a matrix, I just don't want to find the inverse by hand (I hope you see why). $$ \begin{pmatrix} 1 & 2006 ...
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votes
0answers
17 views

show i-th projection is a linear transformation

For $i ∈ {1,2,...,m}$, define $\pi : F_m → F$ by $\pi(x_1,x_2,...,x_m) = x_i$ (the $i$-th projection). (a) Show that it is a linear transformation. (b) If $T : F_m → F$ is a linear transformation ...
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0answers
6 views

Why does “up to scale” make homograph matrix lose one freedom?

Can anyone explain "if H is up to scale, then dof(H)=8" in the following discussion? degree of freedom of Homography matrix Thank you!!!
1
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0answers
26 views

Derivative of an Euclidean-Vector norm.

Consider: x a $N \times 1$ vector , with elements $x_i$ b a $N \times 1$ vector , with elements $b_i$ A a $M \times N$ matrix , with elements $a_{ij}$ ( Symmetric matrix - Block Circulant ) As we ...
1
vote
1answer
38 views

Sylow's theorem for group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$

Let $G=SL(2,\mathbb{F_3})$ - group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$. (a) Find the order of $G$. I think it is $24$ but not sure how to verify it. (b) ...