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
votes
1answer
16 views

Similar matrices NOT over the complex numbers

We say that two matrices $A,B$ with complex entries are similar if and only if there exists an invertible complex matrix $P$ so that $A = P^{-1} B P$. Does $P$ always have to be a complex matrix? ...
0
votes
1answer
12 views

Does $AS=SB\iff f_A(\lambda)=f_B(\lambda)$?

Showing the converse is straightforward: $$B=S^{-1}AS\Rightarrow f_B(\lambda)=\det(B-\lambda I_n)=\det(S^{-1}AS-\lambda I_n)=\det(S^{-1}(A-\lambda I_n)S)\\=(\det S)^{-1}\det (A-\lambda I_n)\det ...
1
vote
0answers
13 views

Is the action of the matrix UU^(t) always a projection? What can I say about I - UU^t?

The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal ...
1
vote
0answers
2 views

Convert 2D block Toeplitz matrix to convolution using the fourier transform?

How do you convert a symmetric 2D block Toeplitz problem into a convolution: something like Tx=b -> ifft2(fft2(h).*fft2(b)) ...
1
vote
0answers
9 views

A proof for a theorem related to rank and matrix product. [duplicate]

For all matrix $\mathbf{M} \in \mathbb{R}^{m,n}$ and $\mathbf{N} \in \mathbb{R}^{n,p}$, the inequality $\operatorname{rank}\mathbf{M} + \operatorname{rank}\mathbf{N} - n \leq ...
1
vote
3answers
31 views

Is any linear transformation with $\text{ker }(T)=\left\{\vec{0}\right\}$ an isomorphism?

I'm thinking no; for instance, $\exists \left\{\left.T:V\rightarrow W\right| \text{Im }(T)\neq W\right\}$. This seems counterintuitive, though. If such a $T$ with maximal rank exists, What would ...
0
votes
0answers
22 views

Compute det(A) given a function A

Suppose A is a 3×3 matrix and A = 1/3 $u_1\cdot uT_1$ + 1/4 $u_2\cdot uT_2$ + 2/5 $u_3\cdot uT_3$ with $uT_1 = (0, 1, −1)$ $uT_2 = (1, 2, 2)$ $u_3 = (−2,1/2,1/2)$ Compute det(A). I know ...
0
votes
2answers
15 views

What is the maximum value of $\text{dim ker }A$, where $A$ is $n\times m$?

True or false: "If $A$ is an $n\times m$ matrix, then $\text{dim ker }A\leq n$" My gut intuitively tells me "no"$\,\Rightarrow$ if $m>n$, $\text{dim ker }A\leq m$. I can't think of a simple, ...
3
votes
3answers
46 views

Wouldn't each addition take time $O(n)$?

I am going over the asymptotic runtime of regular matrix multiplication. Here is a lecture slide I am referencing(too much to type out, shown below), from Algorithms Everything makes sense up ...
1
vote
1answer
24 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
1
vote
1answer
17 views

Let F be a field and let $A,B ∈M_{n×n}(F)$ be a commuting pair of matrices, where B is nonsingular. Is $(A,B^{−1})$ necessarily a commuting pair?

I´m trying to solve this problem, but I can´t, I don´t know how to start. Let F be a field and let $A,B ∈M_{n×n}(F)$ be a commuting pair of matrices, where B is nonsingular. Is $(A,B^{−1})$ ...
0
votes
0answers
10 views

decomposing multiplication of two matrices to the sum of rank-1 matrices

Suppose we have two matrices: $D_{n \times k}$ and $X_{k \times p}$ I need to understand how do we decompose the multiplication DX to the sum of $k$ (am I correct about $k$?) rank_$1$ matrices. ...
2
votes
1answer
55 views

The only eigenvalue of $A \in {M_n}$ is $\lambda = 1$. Why is $A$ similar to $A^k$?

Suppose that the only eigenvalue of $A \in {M_n}$ is $\lambda = 1$. Why is $A$ similar to $A^k$ for each $k=1,2,3,\dots$?
-2
votes
1answer
47 views

Determinant Calculation Issue

Solved..found my mistakes.Thanks David for pointing out the first one to made me realize the other problem in C. I was asked to calculate the determinant for the following matrix: \begin{matrix} ...
1
vote
2answers
339 views

Strassen Multiplication?

How are the values of the 7 new matrices derived? I'm referring to the values that reduce matrix multiplication to 7 multiplications per level: $M_1 = \left(A_{1,1} + A_{2,2}\right)\left(B_{1,1} + ...
0
votes
1answer
24 views

Find $a$ and $b$ in a 4 equation system

$a, b \in\mathbb{R}$. I have four equations: $$x+3y-2z+t=-3$$ $$3x+11y+az+5t=2$$ $$3x+12y-6z+6t=b$$ $$4x+15y-8z+8t=-5$$ I have to find out the values of $a$ and $b$ where the system is solvable (has ...
1
vote
1answer
34 views

Show that a nonzero $2 \times 2$ matrix $A$ such that $A^2 = 0$ is similar to $\begin{pmatrix}0&1\\0&0\end{pmatrix}$

Let $A$ be a $2 \times 2$ non-zero matrix such that $${A}^{2}=0.$$ How do I find an invertible matrix P such that $${P}^{-1}AP=\begin{bmatrix}0&1\\0&0\end{bmatrix} ?$$ Anyone? Please provide ...
0
votes
1answer
13 views

prove the following property related to singular value decomposition

Suppose $A$ is a $n\times n$ matrix. Show that the following are equivalent:(i), $A^2=BA$ for some non-singular $B$. (ii) $rank(A)=rank(A^2)$. (iii), $$Range(A)\bigcap Ker(A)=\{0\}$$, (iv) there ...
0
votes
1answer
21 views

Calculating the adjoint

I am having some trouble understanding the idea of cofactors and adjoints of matrices. From my understanding the adjoint of a matrix is the transpose of the matrix of cofactors? $A=\begin{bmatrix} 1 ...
0
votes
0answers
10 views

Differntiating matrix functions $f : \mathbb R^{n\times m} \to \mathbb R^{p\times q}$

How would you differentiate matrix functions $f : \mathbb R^{n\times m} \to \mathbb R^{p\times q}$ like for example $f(X) = X^T \cdot X$? There are no directional derivatives in the usual sense, and ...
1
vote
1answer
33 views

Positive semi-definite Matrix, Schur complement

Let $\mathbb{R}^{n \times n} \ni C = C^\top \succ 0$. Let $A \in \mathbb{R}^{m \times n}$ with $\text{rank}(A) = m$, where $m \leq n$. How do I show that \begin{equation} C - CA^\top(ACA^\top)^{-1}AC ...
0
votes
0answers
34 views

Let $\text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}$. Why are $A$ and $B$ similar?

Let $A$ and $B \in M_n$ be two matrices such that $$\forall k=1,2,\dots,n,\ \text{Rank}{(A - \lambda I)^k} = \text{Rank}{(B - \lambda I)^k}.$$ Why are $A$ and $B$ similar?
-1
votes
2answers
11 views

$A$ is similar to $cA$ for some complex scaler with $\left| c \right| \ne 1$.why dose all eigenvalue of matrix $A$ are zero?

Let$A \in {M_n}$ and $A$ is similar to $cA$ for some complex scaler with $\left| c \right| \ne 1$.why dose all eigenvalue of matrix $A$ are zero?Is this true that matrix$A$ is nilpotent?
1
vote
1answer
17 views

Find the standard matrix representation of the linear transformation T in M2,2

let $T: M_{2,2} \rightarrow M_{2,2}$ be a linear transformation defined by: $$T \left(\begin{bmatrix} a & b\\ c & d\\ \end{bmatrix}\right) = \begin{bmatrix}a + b& ...
0
votes
5answers
6k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
votes
1answer
26 views

If $A$ is negative-definite, then for a sufficiently big $k>0$ the eigenvalues of $M = kA + B$ are all with negative real part?

I want to prove the next statement: "If $A$ is a symmetric negative-definite matrix, then for a sufficiently big $k\in\mathbb{R}^+$, the eigenvalues of $M = kA + B$ are all with negative real part, ...
0
votes
1answer
12 views

Diagonally dominant matrix for Cholesky?

I have a $10^6 \times 10^6$ dense SPD matrix, which I am called to invert, by using Cholesky factorization. However, I came across this statement: We start with the Cholesky and LU ...
2
votes
0answers
82 views

Bound on entries of L in A=LDL Cholesky factorization for diagonally dominant spd matrix A

I've been wondering about the following: Conjecture: If $A$ is a (strictly) diagonally dominant symmetric positive definite matrix, and $A=LDL^T$ is its square-root free Cholesky factorization, the ...
1
vote
1answer
24 views

relation between conformal and orthogonal matrices in 2D

I want to show that if a matrix $T \in \text{GL}(2, \mathbb{R})$ is conformal, i.e. $$ \text{arccos} \left( \frac{\langle Tv,Tw \rangle}{|Tv||Tw|} \right) = \text{arccos} \left( \frac{\langle v,w ...
0
votes
0answers
33 views

Ideals in the ring of $n\times n$ complex matrices [duplicate]

I want to find the left and right ideals in the ring of $n\times n$ complex matrices. Let's start with the left ideals: A subset $I$ of $R$ is called a left ideal of $R$ if it is an additive ...
1
vote
1answer
130 views

Matrix structure; maybe you can see something I can't…

I am attempting to find a 'smarter' way to solve a matrix, in the form $Ax=B$, where $B_{i}=F_i*N$ $A_{i,j}=-F_i/K_{j,i}$ where $N$ is constant, $K$ is a constant matrix, and $F$ is a vector of; ...
2
votes
0answers
41 views

How to calculate the eigenvalue of the following general matrix [duplicate]

Let the $n\times n$ matrix $Z$ with $(i,j)$-element defined by $Z_{i,j}=i+j$. How to calculate the eigenvalue of $Z.$? I have used Matlab to calculate it. I find no matter how bigger n is, there are ...
1
vote
1answer
33 views

Matrices $P$ such that $A$ is symmetric $\Longrightarrow $ $PAP^{-1}$ is symmetric

Let $M_n(\mathbb{R})$ be the (vector) space of all $n\times n$ matrices over $\mathbb{R}$. Let $Sym_n(\mathbb{R})$ denote the subspace of symmetric $n\times n$ matrices. $GL(n,\mathbb{R})$ acts on ...
0
votes
0answers
3 views

The function that connects SVD(X) to SVD(XX')

I am very curious to know if there is any relationship between SVD(X) and SVD(X'X). Assume X is a $m \times n$ matrix. So my question is about the function that connects $V_{XX'}$ to $V_X$ where $V$ ...
14
votes
11answers
267 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$?

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
0
votes
0answers
9 views

How to formally describe the lowest values of a vector / sorted vector?

I have a distance matrix D and would like to describe that I am just taking the mean (or median) of the 5 lowest values for each column. The programming implementation e.g. in R is fairly easy: ...
1
vote
1answer
30 views

Decompose an invertible $4 \times 4$ real matrix into product of $4 \times 3$ and $3 \times 4$

If we have an invertible matrix $M$ that is $4 \times 4$ and $\left| M \right| \neq 0$ (i.e. it is invertible), is it possible to decompose it into two matrices $4 \times 3$ and $3 \times 4$ ...
-1
votes
0answers
11 views

Question related to matrix and it's transpose.

Prove: For any matrices A and B and any scalars a and b, $(aA+bB)^t$ = a$A^t$ + b$B^t$.
1
vote
0answers
14 views

Inverse properties of $L_1$ normed matrices

Let's take the adjacence matrix $A$ of a directed graph $G$. We call $\bar{A}$ the row $L_1$ normalized matrix obtained from $A$. (i.e. we divide each elements of the row by the sum of the elements of ...
0
votes
0answers
14 views

real similar matrices [duplicate]

If real matrices $A$ and $B$ are similar to each other, prove that there is a real matrix $S$ such that $A=SBS^{-1}$. As we know, when $A$ and $B$ are similar to each other, then there exits complex ...
2
votes
0answers
8 views

Eigenvalues and positivity of Hermitian Toeplitz matrices

I want to check the eigenvalues (and also the positivity) of the $n \times n$ complex Toeplitz matrix \begin{equation} T = \begin{bmatrix} r & z_1 & z_2 & z_3 &\cdots & z_{n-1}\\ ...
0
votes
0answers
44 views

Finding $Q$ for any $A$ s.t. $QAQ^\top = I$

Given an invertible and PSD matrix $A$, I am looking to find $Q$ such that: $$ QAQ^\top = I $$ What is a/the right/efficient way to do this? Here is what I did: SVD gives $$ A \approx U S V^\top ...
2
votes
0answers
17 views

Efficient computation of matrix determinant in finite field

I am trying to implement generalization of Hill cipher. My idea is very simple: the size of key matrix should be arbitrary number not only three. All steps of this cipher is trivial except computation ...
2
votes
0answers
33 views

$trc(A)=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices? [duplicate]

Let $trc(A)=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?($A \in {M_n}$)
1
vote
2answers
33 views

Problem implementing a QR factorization

I'm trying to write a Fortran subroutine to compute a QR factorization using the Householder method. To test my routine, I compute the factorization of the following matrix: $$ A = \begin{pmatrix} ...
-5
votes
2answers
21 views

Let $A$ be any $m\times n$ matrix, and let $B = AA^T$ Prove B is symmetric [on hold]

Let $A$ be any $m\times n$ matrix, and let $B = AA^T$ What is the size of the matrix $B$? Justify your answer. Prove that $B$ is symmetric. Any help with this is appreciated :)
0
votes
0answers
26 views

Using axis coordination to represent rotation matrix instead of angles

Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently: $$\vec ...
4
votes
4answers
23k views

Find the standard matrix for a linear transformation

If T: $\Bbb R$3→ $\Bbb R$3 is a linear transformation such that: $$ T \Bigg (\begin{bmatrix}-2 \\ 3 \\ -4 \\ \end{bmatrix} \Bigg) = \begin{bmatrix} 5\\ 3 \\ 14 \\ \end{bmatrix}$$ $$T \Bigg ...
0
votes
1answer
38 views

Product of two multivariate Gaussian pdfs - normalizing constant

https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf contains expressions (p.2, 6e, 6f) for the normalization constant for the product of two multivariate Gaussian pdfs, with mean vectors $a$ and $b$ ...
1
vote
1answer
10 views

changing bases/rotating axes to find reflection across y=2x

Find the (exact) reflection of the vector v = (5, 1) across the line: y = 2x. Hint: A sketch of v and the line may suggest an approach. I found the matrix -3/5 6/5 4/5 2/5 which seems like it gives ...