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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2 views

Find equations such that the solution space is the Image of T

Suppose T is a linear transformation such that (x1, x2, x3) -> (3x1 + 4x2 + 2x3, x1 + 2x2, 2x1 + x2 + 3x3, -x1 + 5x2 - 7x3) Find a homogeneous system of equation such that Image(T) = Solution space ...
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1answer
7 views

Finding a transformation matrix given a basis of matrices

Edit: bear with me, for some reason formatting got messed up. I am given the following linear transformation: $$ T\left( \begin{bmatrix} a & b \\ 0 & d \\ ...
3
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1answer
17 views

On the product of involution matrices

Let $F$ be a field and let $A\in M_n(F)$ be a matrix with $det(A) = \pm 1 $. How can I show that $A$ is a product of involutions ? Of course the converse is true and clear. By involution I mean a ...
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2answers
32 views

Find 2x2 matrix such that its inverse equals its transpose

Find some matrix $B\in GL_2 (\mathbb{R})$ such that $B^{-1} = B^T$ and $B \neq I$ What I tried: I tried to create a simultaneous equation i.e. if B = $\begin{bmatrix} a&b\\c & ...
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2answers
16 views

Prove constant times invertible matrix is also invertible

Let $B\in GL_n(\mathbb{R})$ and $\beta \in \mathbb{R}$ with $\beta \neq 0$. Show $\beta B \in GL_n(\mathbb{R})$ What I tried: I know it intuitively makes sense that this would be the case, but I ...
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1answer
10 views

How to find scalar multiples that would make sum of matrices the zero matrix

What are all the possible values of $c_1$,$c_2$,$c_3$ $\in$ R such that $c_1$$\begin{bmatrix} 1&0\\ -1&0 \end{bmatrix} $ + $c_2$$\begin{bmatrix} 2&1\\ -2&2\end{bmatrix} $ ...
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1answer
27 views

System of sequences

How can I solve this system of two sequences, $$ \begin{cases} a_{n+1}=0.9a_n+0.3b_n \\ b_{n+1}=0.1a_n+0.7b_n \end{cases} $$ with $(a_1,b_1)=(1000,1000)$ using matrix ? I know the answer is ...
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0answers
10 views

Computation of general continued fractions by $2 \times 2$ matrix multiplication - is it the best way?

There are two main ways to compute a continued fraction (or its $n$th convergent). Let's say we have a general fraction: $$ x= a_0 + ...
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2answers
17 views

Cancellation law for invertible matricies

Show that the cancellation law holds for invertible matrices. i.e. if $A \in GL_n(R), B, C \in M_{n×m}(\mathbb{R})$ and $AB = AC$, then $B = C$. What I tried: I know that I can prove this by ...
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0answers
22 views

Let $U$ and $V$ be vector spaces of dimensions $n$ and $m$ over $K$. Find the dimension and describe a basis of $\operatorname{Hom}_K(U,V)$ [duplicate]

I am given vectors spaces $U$ and $V$ of dimensions $n$ and $m$ over $K$. How can I find the dimension and basis of $\operatorname{Hom}_K(U,V)$ ?
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5 views

Unique least squares solution for bounded variables of overdetermined rank-deficient linear system?

I am trying to solve an overdetermined linear system $A x = b$ where $A \in \mathbb{R}^{m \times n}$ $m > n $ $rank(A)<n$ $0 \leq x \leq u $ (all entries are bounded) $A, b \geq 0 $ (all ...
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1answer
22 views

Let $T$ be a defined linear map. Write down the matrix of $T$ using the standard basis of $\mathbb{R}^2$ and secondly using the basis $(1,-1),(0,-2)$. [on hold]

So I am given a linear map $T$ which is specifically defined. I have to find a matrix of $T$ using the standard basis and then using the given basis. I am not sure how to approach this problem?
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0answers
15 views

Gramian Matrix Eigenvalues--Stronger Statement than Non-Negative

I'm struggling to find conditions under which this holds: $AA^T - B \succeq 0\,.$ If it helps, A is not necessarily square and $A_{ij} \in \{-1, 0,1\}$. B is diagonal and I would like to find ...
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0answers
26 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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0answers
33 views

What are the odds of havinh R number of polyominos in the matrix [on hold]

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 ...
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1answer
17 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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0answers
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
30 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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1answer
28 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
26 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
46 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
14 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
55 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: $$ A(B^{-1}(A+B)A^{-1})B = A+B $$ ...
0
votes
1answer
15 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
votes
2answers
49 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
11 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
votes
1answer
37 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 ...
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0answers
36 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
49 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 ...
0
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1answer
17 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 ...
3
votes
3answers
39 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
votes
1answer
28 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 ...
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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
25 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
87 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix? I replaced each row starting from the thrid with the difference of the one before and it. In this way i transformed it into an almost diagonal matrix ...
1
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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 ...
4
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1answer
65 views
+50

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
54 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 ...
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2answers
21 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
votes
1answer
64 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
35 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
vote
1answer
51 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 & ...
1
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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
votes
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 ...