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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4
votes
3answers
176 views

Is it easier to determine that a matrix is singular than it is to determine nonsingular?

I came across this line "It is often easier to determine that a matrix is singular than it is to determine that a matrix is nonsingular. The facts below illustrate this. Fact 1.10. Let ...
0
votes
2answers
53 views

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. [on hold]

Prove that if $\|A\|<1$, then $(I+A)^{-1}=I-A+A^2-A^3+\cdots$. I'm not sure how to do this. I know the result for $(I-A)^{-1}$, but that won't help me.
0
votes
0answers
36 views

Is there a variant of Euler's theorem for matrices?

Euler's theorem says that whenever $a$ and $p$ are coprime integers, then $a^{\phi(p)} \equiv 1 \mod p$ This is particularly useful for evaluating large powers of $a$: $a^n \ \%\ p = a^{n\ \%\ ...
0
votes
0answers
10 views

Inverse of a square block matrix

I am trying to understand how to compute the inverse of a square block matrix defined as follow: $\begin{bmatrix}2I&-X\\X'&0\end{bmatrix}$, where $I$ is a TxT identity matrix, $X$ is a TxK ...
0
votes
1answer
15 views

How come the determinant of a matrix have to be 0 to find the eigenvalue and vector?

I need help understanding why if the determinant of a matrix is 0 then there exists a matrix such that multiplying it by a vector get 0 and how this relates to eigenvectors and eigenvalues. For ...
2
votes
1answer
54 views

$A$ and $B$ are positive matrices and $\|A+B\| = \|A\|+\|B\|$, then A and B have a common eigenvector

Suppose $A$ and $B$ are positive matrices such that $\|A+B\| = \|A\|+\|B\|$. Show that $A$ and $B$ have a common eigenvector, where $\|A\|$ is the operator norm of $A$. I'm wondering if there is a ...
0
votes
0answers
14 views

What will draw a shape of $L = \left\{ {\lambda \in \mathbb{C}:{s_4}(\lambda ) = {s_3}(\lambda )} \right\}$ [on hold]

Let $P(\lambda ) = \left( {\begin{array}{*{20}{c}} {{\lambda ^2} - 1} & 0 \\ 0 & {{\lambda ^2} - 2\lambda } \\ \end{array}} \right)$ and $\lambda \in \mathbb{C}$( $λ$ is a complex ...
0
votes
1answer
15 views

Lower bound for the distance between matrices of different rank.

This is a follow up question to this: Norm of diference of matrices of different rank Suppose $A$ is a norm one $n\times n$ matrix of rank $k$. What is $\inf\|A-B\|$, where the infimum is taken over ...
0
votes
0answers
9 views

Upper bound on the norm of the inverse of matrices with zero limit

Let $\{L(\sigma)\}_{\sigma}$ be a family of matrices indexed by the parameter $\sigma$ so that the operator norm $||.||$ of $\{L(\sigma)\}_{\sigma}$ satisfies $Ae^{-a/\sigma}\leq ||L(\sigma)|| \leq ...
0
votes
0answers
11 views

Identifying Symmetric Matricies

If $A$ and $B$ are arbitrary $nxn$ matrices, which of the matrices must be symmetric? $something^T$ means the transpose of "something". $A^T$$A$ $A-A^T$ $A^TB^TBA$
0
votes
0answers
12 views

Identity involving pseudoinverses (Moore-Penrose) of symmetric matrices

Let A be a symmetric $m$ x $m$ matrix of rank r, and B a symmetric $m$ x $m$ matrix of rank $m - r$, such that $AB = 0$. Show that $A^+A+B^+B=I$, where $A^+$ is the Moore-Penrose pseudoinverse of $A$. ...
0
votes
1answer
20 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
1
vote
0answers
35 views

For matrices $A$,$B$ prove that $A+cB$ is not invertible.

Let $A$,$B$ $\in M_n(\mathbb{R})$ and $B$ is invertible, then prove that there exists a $c \in \mathbb{R}$ such that $A+cB$ is not invertible. My attempt: We need to show that $\det(A+cB)=0$. So ...
2
votes
2answers
17 views

Inverse of a matrix with uniform off diagonals

Suppose that we have an all positive matrix where the off diagonal elements are all identical. Can one calculate the inverse of the matrix analytically, or more efficiently than the general case? For ...
1
vote
0answers
15 views

Linear transformations and possible dimension mismatch

The problem: Let $L: R_4 \to R_3$ be defined by $$L([u_1, u_2 ,u_3 ,u_4]) = [u_1 ,(u_2+u_3), (u_3 + u_4)]$$ Let S and T be the natural bases for $R_4$ and $R_3$, respectively. Find the ...
1
vote
1answer
33 views

How to you find out what a matrix does to an equation.

Lets say I have an equation of a plane, $$x-3y+2z=0 $$ and I get matrix to transform it with say a 3x3 matrix with just a-i as place holders for the values in the matrix. How would I find what the ...
0
votes
3answers
37 views

What is the difference between a linearly independent set and a set that spans $\Bbb{R}^m$?

This is more of a conceptual question. Here's what I know about a linearly independent set of vectors: A set of vectors $\{v_1, ..., v_p\}$ is linearly independent if the equation $$x_1v_1 + x_2v_2 + ...
2
votes
1answer
33 views

Idempotent and nilpotent matrices are defined differently. Why?

We call $A$ idempotent if $A^2$ is $A$. But we call A nilpotent if $A^k$ is $0$ for some integer $k$. Why are not they defined uniformly like both with power 2 or both with power some integer $k$.
0
votes
1answer
21 views

Gradient Chain Rule: Applying Gradient in the case of a Series of Matrix operations (Neural Net Gradient Calculation)

I have the following situation: I need to calculate the gradient of the Error of a CNN a few layers deep by hand. Starting with the Error function, The $\operatorname{Error}[readoutX]= -\sum_i ...
-1
votes
0answers
18 views

Which of the following are subspaces of $Mat_2(\mathbb{Q})$? [on hold]

Which of the following are subspaces of $Mat_2(\mathbb{Q})$? (i) The set of matrices which have zero trace. (ii) The set of matrices which have zero determinant. Do matrices refer to all matrices ...
3
votes
2answers
39 views

Property of group of $k \times k$ orthogonal matrices

Does the group of $k \times k$ orthogonal matrices lie on an $(k^2 - 1)$-sphere? If so, of what radius? If not, does it lie on some sort of other object?
0
votes
1answer
22 views

Linear Regression without X? :

(Have been working in matrix algebra) Given model: $ y_i = a + e_i$ ( $y_i= α+ϵ_i$ ) That is $y$ subset $i$ and error term subset $i$ Where the expected value of each error term for each entry ...
-3
votes
2answers
21 views

An example of unitary matrix which is $3\times 3$ and complex

Please give me an example of unitary matrix which is $3\times 3$ and complex. If I get this example, i will finish my thesis.
0
votes
1answer
12 views

Get vertex points of transformed rectangle knowing bounding box and transform matrices

(I'm not a mathematician so talk down to me). I have a rectangle that has been transformed by a series of matrix transforms. I can recover the transform matrices and get the x,y coordinates of each ...
0
votes
0answers
8 views

Solve a generalized eigenvalue problem in LDA

http://www.facweb.iitkgp.ernet.in/~sudeshna/courses/ML06/lda.pdf Page 6. I don't quite understand how that can be solved... I have tried following general one $$det(S^{-1}_{w}S_B-JI)=0$$ But I am ...
2
votes
0answers
22 views

Prove that $J_n(0)$ and $(J_n(0))^t$ are similiar

Prove that $J_{n}(0)$ and $(J_{n}(0))^t$ are similar ($J_n(0)$ is a $n \times n$ Jordanian block which belongs to the eigenvalue $0$). Use your answer and Jordanian form to prove that every matrix $A ...
3
votes
1answer
50 views
+100

How to prove this result about the interlacing of eigenvalues.

Let $A$ be a real symmetric matrix of order $n$ with eigenvalues $\mu_1\ge \mu_2,\ldots, \mu_{n}$. $B=\begin{bmatrix} x&x&x&x&x \end{bmatrix}$, where $x$ is non zero column vector in ...
-4
votes
1answer
31 views

Is it unitary matrix or not? [on hold]

$A = \begin{bmatrix} \frac{i}{3^{1/2}} & \frac{1+i}{3^{1/2}} & 0\\ \frac{-1}{2^{1/2}} & 0 & \frac{i}{2^{1/2}}\\ \frac{1-i}{3^{1/2}} & \frac{1}{3^{1/2}} & 0 \end{bmatrix}$ Is ...
0
votes
1answer
13 views

Hexagonal - number of cells

For $n = 2$; We have something like this: https://zapodaj.net/0cc6e3c190f32.png.html and number of calls is equal 7. But how designate for $n$ ? For $n = 3$; we have 19
0
votes
1answer
18 views

Columns of a matrix linearly independent and spans

Let $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n \in \Bbb R^n$ and let $P$ be the $n\times n$ matrix whose columns are $\mathbf v_1, \mathbf v_2, ...,\mathbf v_n$ I'm wondering why the followings are ...
0
votes
0answers
3 views

partial order and equivalence relation question [on hold]

Let A = ℤ+ x ℤ+ and R be a relation on A (that is, R ⊆ A xA) defined as follows. (a,b) ~ (x,y) if and only if a + y = b + x. Is R a partial order? Is R an equivalence relation?
0
votes
2answers
18 views

Proof the inequalitiy for the two Matrices $A, B$

Let $ A,B \in C^{nxn}$ then $$ 1 \le || (\lambda I-B)^{-1})(A-B)||\le ||(\lambda I-B)^{-1}||*||(A-B)||$$ for any eigenvalue $\lambda $ of $ A $ which is not an eigenvalue of $ B$ and any operator ...
2
votes
1answer
47 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
-1
votes
0answers
19 views

Dynamical Systems problem

I have a problem that have been trying to solve but it's not going so good. I would like some guidelines on how to work myself around this problem: Two neighboring countries spy on each other and ...
0
votes
0answers
11 views

A Projection Matrix which projects to a space $V$ with $dimV=2$ has $3$ eigenvalues which span a space of dimension=3

I have found an exercise involving a $3x3$ projection matrix which projects to a space $V$ with $dim(V)=2$. The matrix(or operator) is defined as $P=v*(v, v)+u*(u, u)$. So, in my understanding it ...
0
votes
0answers
19 views

Matrix Norm Confusion

I am looking at my textbook which considers an example but I am not sure how it derived the matrix norm with $||A|| = \sqrt{9/2 + (1/2)\sqrt{65}}$ and was hoping someone could provide the calculations ...
1
vote
2answers
24 views

Norm of diference of matrices of different rank

Suppose $A$ is a $n\times n$ matrix of rank $k$ that has Euclidean norm equal to $1$. Given $p<k$, and $\epsilon>0$, can we always find a norm one matrix $B$ of rank $p$ such that ...
0
votes
1answer
21 views

representation of a map with respect to 2 bases

From Heffron, p.231 Consider the two linear functions $h:$ ${R}^3$ $\longrightarrow$ $\mathcal{P}_2$ and ${g}: \mathcal{P}_2 → M_{2x2}$ $ \left( \begin{array}{ccc} a \\ b \\ c \end{array} ...
0
votes
1answer
16 views

Frobenius norm of matrix $A^{T}A$ is $trace(A^{T}A)$? Where all of the values in the matrix $A$ are real

We know that frobenius norm of a matrix $A$ is given by $\|A\|_{F}=\sqrt{trace(A^{T}A)}$. Can we write frobenius norm of matrix $A^{T}A$ to be $\|A^{T}A\|_{F}=trace(A^{T}A)$, that is I am effectively ...
0
votes
0answers
12 views

Given the 3x3 matrix D2 is a second order differentiation matrix, for a function u defined and twice differentiable on the interval [a,b].. [on hold]

Given that the 3x3 matrix D2 is a second order differentiation matrix, for a function u defined and twice differentiable on the interval [a,b].. u"(a) ≈ D2 * u(a) u"((a+b)/2) ≈ D2 * ...
1
vote
0answers
52 views

Cramer's rule doesn't work here?

I tried to solve the following system: $$A_2\cdot 2\mathrm{i}\sin( \beta a) = B_3\exp(- \alpha a)$$ $$\mathrm{i} \beta A_2 2\cos( \beta a) = - \alpha B_3\exp(- \alpha a)$$ Then I got $A_2=0 ...
1
vote
1answer
23 views

Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space ...
0
votes
2answers
24 views

The determinant of the transposing endomorphism

Let $K$ be a field and $f$ the endomorphism of $\mathcal M_n(K)$ that sends a matrix to its transpose. I want to determine the determinant of $f$. I know that since $f^2=id$ then $det(f)=1\ or \ -1 $ ...
5
votes
2answers
71 views

If $BA = I$, prove that $AB = I$ (using determinants)

I've seen this problem around here, but I wanted to check if this particular solution is right. So, if $BA = I$, then $det(B)det(A) = 1$, meaning neither $det(B)$ or $det(A)$ are equal to $0$. ...
0
votes
0answers
14 views

Find 2 point getting far away each other from their intersection point

I want to know how to find 2 aircraft getting far away from their intersection point, from Dataset such as aircraft 6-7,11-2, 10-6,37-36,etc. Dataset: my algorithm is: calculate direction ...
2
votes
1answer
138 views

log norm inequality for lower triangular part of matrix

Suppose $L$ is the lower triangular part of a matrix $A \in \mathbb{C}^{n\times n}$. Prove that $||L||_2 \leq ||A||_2 \log_2(2n)$. Here $||\cdot||_2$ is the matrix norm induced by the $p=2$ vector ...
3
votes
1answer
64 views

Result about Matrices of form $B(AB)^{-1}A$

I am trying to prove the following result. So far my only idea was to try using the formula for inversion of block matrices, but that did not get me very far. Any help will be much appreciated. ...
0
votes
0answers
29 views

Change of Basis between linear Transformations

I am trying to get a better understanding in change of basis with matrices and linear transformations, therefore I am using several linear Transformations $^{i-1}A_i=\begin{bmatrix} \cos\theta_n ...
0
votes
1answer
31 views

Rationnal canonical form of the matrix $A$

Let the matrix \begin{equation} A=\begin{bmatrix} 2 & 1 & 2 \\ -2 & -1 & -4 \\\ 1 & 1 & 3 \end{bmatrix}. \end{equation} So far I found the characteristic polynomial ...
1
vote
2answers
177 views