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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1
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2answers
22 views

Determining similar matrices

I have this matrix $$A= \begin{bmatrix}1 &0& 2\\0&-1&-2\\2&-2&0\end{bmatrix}$$ I found the eigenvalues to be $0, 3, -3$ I am tasked with finding if $A$ is similar to a ...
0
votes
1answer
17 views

When diagonalizing a matrix, in what order should you arrange the the eigenvectors to form the invertible matrix $P$?

I was following this example online to diagonalize a matrix. It lists the eigenvectors as $\lambda =3,2,4$ (note the order). It then arranges each eigenvalue's corresponding eigenvector (3 column ...
0
votes
3answers
43 views

What do I conclude if I found the eigenvalues of a matrix, then noticed that one of those eigenvalues resulted in a zero eigenvector?

By definition: $$Ax = \lambda x$$ for $x \neq 0$ I was using this to calculate the eigenvectors for $A$: $$A = \begin{bmatrix}2 & 0 & 0\\1 & 3 & 0\\ 2 & 3& 4\end{bmatrix}$$ ...
7
votes
1answer
88 views

Does $AB+BA=0$ imply $AB=BA=0$ when $A$ is real semidefinite matrix?

This is a general question that came to my mind while listening to a lecture(although its framing may make it look like a textbook question). Suppose that $A$ and $B$ be real matrices. $A$ is ...
0
votes
0answers
12 views

sum converge, matrix, norm

Let $A_j$ be a sequence in $\mathbb{C}^{n\times n}$. Show that $ \sum_{j=0}^\infty A_j$ converges if $ \sum_{j=0}^\infty ||A_j||$ does.($||A||= sup_{|x|=1} |Ax|$ with euclid norm) Hello, Be ...
0
votes
0answers
14 views

Generate Lie-Algebra $su(N)$

Let $A \in \mathbb{C}^{n \times n}$ be a diagonal matrix with $Tr(A)=0$ and all eigenvalues (purely imaginary eigenvalues) are mutually different from each other. Hence, in particular $A \in ...
0
votes
3answers
62 views

If $A$ and $B$ are arbitrary $n \times n$ matrices, prove that $(A^TB^TBA)$ is symmetric

My attempt: $(A^TB^TBA)^T$=$(A^T)^T(B^T)^TB^TA^T$=$(AB)B^TA^T$ $\ne$ $(A^TB^TBA)$ therefore $(A^TB^TBA)$ is not symmetric.
0
votes
1answer
24 views

If $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many solutions

I was reading this pdf: https://www.math.ohiou.edu/courses/math3600/lecture10.pdf and it tells you that if $A$ is a singular square matrix, then $Ax = b \neq \vec{0}$ has $0$ or many ...
0
votes
5answers
72 views

Why, if a matrix $Q$ is orthogonal, then $Q^T Q = I$?

I was looking at the definition of an orthogonal matrix, which is as follows: Square matrix $Q$ is orthogonal if its columns are pairwise orthonormal, i.e., $$Q^TQ = I$$ Hence also ...
1
vote
0answers
22 views

Linear Map of an ellipsoid in $\mathbb{R}^N$ into another ellipsoid in $\mathbb{R}^n$, with $n<N$

Starting from the closet set describing an ellipsoid in $\mathbb{R}^N$: $$\Omega_x = \{ x \in \mathbb{R}^N : (x-x_0)^T\Sigma_x^{-1}(x-x_0) \leq \varepsilon^2 \}$$ where $\Sigma_x \in ...
0
votes
0answers
24 views

Operator norm of a diagonal matrix

I want to prove that the operator norm of a diagonal matrix $D$ is less than or equal to its largest value. I've tried the following but I don't know if it is correct. ...
0
votes
2answers
35 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: $\begin{bmatrix}2{\bf I}&-{\bf X}\\{\bf X}'&{\bf 0}\end{bmatrix}$, where ${\bf I}$ is a ...
0
votes
1answer
19 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 ...
0
votes
0answers
32 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
17 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
1answer
35 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 $A$. ...
1
vote
2answers
68 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.
1
vote
0answers
41 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 ...
1
vote
0answers
19 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 ...
2
votes
2answers
20 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 ...
3
votes
4answers
311 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
3answers
41 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 + ...
1
vote
1answer
42 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 ...
2
votes
1answer
34 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 $k$.
3
votes
1answer
62 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
1answer
23 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 ...
0
votes
1answer
19 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
10 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 ...
-4
votes
1answer
35 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
19 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 ...
-3
votes
2answers
30 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
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
0answers
4 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?
2
votes
0answers
25 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 ...
-1
votes
0answers
21 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
2answers
19 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
49 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
vote
1answer
32 views

A projection matrix which projects to a space $V$ with $\mathrm{dim}V=2$ has $3$ eigenvalues which span a space of dimension $=3$

I have found an exercise involving a $3\times 3$ projection matrix which projects to a space $V$ with $\mathrm{dim}(V)=2$. The matrix(or operator) is defined as $P(x)=v*(x, v)+u*(x, u)$. So, in my ...
0
votes
0answers
40 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
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
26 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
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
23 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
53 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 ...
0
votes
2answers
25 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 $ ...
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 ...
0
votes
1answer
42 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 ...
1
vote
1answer
29 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
1answer
13 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
31 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 ...