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), ...

learn more… | top users | synonyms (2)

-1
votes
1answer
8 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
3 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
13 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
10 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 ...
0
votes
0answers
36 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
28 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?($A \in {M_n}$)
-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 :)
1
vote
1answer
17 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 ...
0
votes
0answers
25 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 ...
0
votes
0answers
10 views

Find the change of basis matrix for the following basis B and D for $\mathbb{R}^2$

Find the change of basis matrix for the following basis B and D for $\mathbb{R}^2$ More or less I think I understood what is a change of basis matrix, but I am not sure how to find it. Suppose ...
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 ...
0
votes
1answer
14 views

Prove that the Reduced Row Echelon form of a bijective linear transformation is the identity matrix.

Prove that the Reduced Row Echelon form of a bijective linear transformation is the identity matrix. My professor has always said this is true, and I know it is part of the invertible matrix ...
0
votes
2answers
32 views

Matrix Calculus and Matrix Derivatives

Consider a map $f : \mathbb R^{n\times m} \to \mathbb R^{p \times l}$ between matrix spaces, what is the differential of such a mapping? I looked at a really simple example, $\operatorname{id} : ...
1
vote
1answer
31 views

What matrix transforms $(1, 0)$ into $(2, 6)$ and tranforms $(0, 1)$ into $(4, 8)$?

In the last 2 lectures of linear algebra we have talked about linear mappings and other stuff, but I missed actually the last one and I am quite in bad situation. What matrix transforms ...
2
votes
1answer
30 views

Any square matrix is equivalent to zero diagonal matrix

Let $A$ and $B$ be two square matrices of dimension $n\ge 2$. We say that $A$ and $B$ are equivalent if there exist $P$ and $Q$ invertible such that $B=Q^{-1}AP$. Is it true that every square matrix ...
4
votes
3answers
87 views

Is there any geometrical interpretation as to why matrix product is not commutative?

Is there any geometrical interpretation as to why matrix product is not commutative? Similarly, is there any geometrical interpretation of matrix product when you have matrices $A$, $B$ such that ...
1
vote
1answer
28 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
1answer
15 views

Find the change of basis matrix P from S to S'.

Consider the following bases of $\mathbb{R}^2$: $$S=\left\{\begin{pmatrix} 1\\ -2 \end{pmatrix},\begin{pmatrix} 3 \\ -4 \end{pmatrix}\right\}$$ $$S'=\left\{\begin{pmatrix} 1\\ 3 ...
0
votes
0answers
9 views

Find the singular value decomposition

Find the singular value decomposition of : $$A=\begin{pmatrix} 1 &1 \\ 2& 2 \end{pmatrix}$$ I think the singular value decomposition is $A=P\Sigma Q^T$ right? $$K=A^TA$$ $$=\begin{pmatrix} ...
0
votes
1answer
16 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
20 views

How close is Cartesian product of unit orthogonal bases of SVD to identity matrix?

If I have N unit orthogonal vectors of length N $\phi_{i,N\times 1}$ obtained from SVD of a $N\times M$ matrix $U$ : $$ U_{N\times M} = \sum_i^N \sigma_i\phi_{i,N\times 1}\times\psi_{i,1\times M}\\ ...
0
votes
0answers
21 views

Find the matrix representation of T relative to the basis

Let $T: \mathbb{R}^2\rightarrow \mathbb{R}^2$ be the linear operator defined by $$T\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} 2x+3y \\ 4x-5y \end{pmatrix}$$ Find the matrix ...
0
votes
1answer
14 views

Unique eigenvalue of maximal absolute value?

Let $A$ be an $n\times n$ matrix with $a_{ii}=0$ for all $i$, and $a_{ij}\in\{0,1\}$ for all $i\neq j$, and $a_{ij}=0\leftrightarrow a_{ji}=1$ for all $i\neq j$. Is it necessary that $A$ as a unique, ...
0
votes
1answer
12 views

Are there any general strategies to prove $K(x,y)$ is a machine learning kernel? (I.e. always defines a covariance matrix)?

So there are certain functions of two variables such as the standard Gaussian/radial function $K(x_i,x_j) = e^{-(x_i-x_j)^2}$ which are "kernels" as machine learning calls them, meaning that for any ...
0
votes
1answer
33 views

Proving that an $n\times n$ matrix is positive definite iff the eigenvalues of that matrix plus its transpose are positive

I am trying to prove that an $n\times n$ matrix $A$ is positive definite iff the eigenvalues of $(A + A^T)$ are positive. So far I have: Let $x$ be an eigenvector of $(A + A^T)$ and let $\lambda$ be ...
1
vote
1answer
37 views

Does $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$?

Let $K$ be a field, $K^n$ a vector space over $K$. Is the following true? $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$ Does this change if $K$ is a ring, and $K^n$ a module over $K$?
0
votes
0answers
14 views

Diagonalization of Markov Matrices

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b ...
0
votes
0answers
6 views

Markov chain ergodicity

$Xn$ is a discrete-time, time-homogenous Markov chain. I have have the following transition matrix and want to show whether the chain is ergodic. P = \begin{pmatrix} \frac{1}{2} & 0 & 0 ...
1
vote
1answer
43 views

How to prove that $I+A^{T}A$ is invertible [duplicate]

Let $A$ be any $m\times n$ matrix and $I$ be the $n\times n$ identity. Prove that $I+A^{T}A$ is invertible.
3
votes
1answer
23 views

Prove matrices are of equal rank

Suppose $P$ and $Q$ are $n \times n$ matrices of real numbers such that $P^2 = P$, $Q^2=Q$ and $I-P-Q$ is invertible, where $I$ is the $n × n$ identity matrix. Show that $P$ and $Q$ have the same ...
1
vote
1answer
35 views

Bound for eigenvalues of some special matrix

Let $Tridiagonal(a, c, b)= \begin{vmatrix} c & b & 0 & \ldots & 0 \\ a & c & b & \ldots & 0 \\ 0 & a & c & \ldots & 0 \\ \vdots & \vdots & ...
1
vote
2answers
29 views

Determinant of a square matrix in a field [duplicate]

\begin{array}{rrrrr|r} b & a & a & \cdot \cdot \cdot & a \\ a & b & a & \cdot \cdot \cdot & a \\ a & a & b & \cdot \cdot \cdot & a \\ ...
0
votes
0answers
9 views

Coppersmith-Winograd algorithm

I'm interested in algorithms to compute matrix multiplications. Is the Coppersmith-Winograd algorithm similar to the Strassen algorithm ? I have two other questions: 1) Are the multiplications done ...
0
votes
1answer
23 views

Finding the matrix representation of a linear transformation $ T: P_{3} \to \text{M}_{2 \times 2} $.

Define a function $ T: P_{3} \to \text{M}_{2 \times 2} $ by $$ T \! \left( a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} \right) = \begin{pmatrix} a_{3} & a_{0} \\ a_{2} & a_{1} \end{pmatrix}. ...
0
votes
1answer
11 views

Eigenvector / eigenvalue pairs for a Markov Matrix

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b ...
2
votes
0answers
19 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
0
votes
0answers
16 views

Find a basis and the dimension of the eigenspaces of the matrix

Find a basis and the dimension of the eigenspaces of the matrix $$ \left( \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 3 & 0 \\ 2 & 0 & 1 \end{array} \right) $$ given that the ...
0
votes
1answer
10 views

Transitions of matrix

$T: \Bbb R^3 \to \Bbb R^3$ and $S: \Bbb R^3 \to \Bbb R^4$ are matrix transformations whose standard matrices are $$T=\begin{bmatrix} 1 & 0 & 2 \\ 2 & 3 & 4 \\ 1 & 5 & ...
0
votes
1answer
21 views

Solving a variable in a matrix equation?

I am having trouble solving for a in the problem below. I've simplified it down to: $e^{14} = ln(e^e \cdot a)$. I'm not really sure where to go from here.
1
vote
2answers
32 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} ...
13
votes
11answers
250 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 & ...
1
vote
1answer
29 views

What Method is used for Projecting the Rauzy Fractal?

I am trying to construct the Rauzy Fractal (http://en.wikipedia.org/wiki/Rauzy_fractal), I have a Tribonacci word generator and have the stairs constructed but I can't seem to get the projection onto ...
0
votes
1answer
17 views

“distance” metric between two bases modulo determinant, rotation and chirality

I'd like some kind of metric that tells me how similar two complete, not necessarily orthonormal bases (represented by non-singular matrices $B_1, B_2 \in \mathbb{R}^{n \times n}$) are to each other, ...
0
votes
1answer
14 views

Diagonal factorization of upper triangluar matrix to unit uppper triangular matrix

How to compute Diagonal factorization of upper triangluar matrix to unit uppper triangular matrix. i.e U = D*M where U is upper triangular; D is diagonal; M is unit upper triangular.
0
votes
0answers
21 views

Bounding cosine of angle between vectors

Let $M$ be a symmetric, positive definite matrix such that $0\lt c_1 \le \lambda_{min}(M)\le\lambda_{max}(M)\le c_2$. I am trying to show that $\dfrac{v^TMv}{||Mv||||v||}\gt 0$ for $v\ne 0$ I ...
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
1answer
37 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 $b$ ...
5
votes
1answer
53 views

Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?

Let $A \in {M_n}$ and $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
9
votes
7answers
3k views

Why is the inverse of a sum of matrices not the sum of their inverses?

Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$? I know the answer is no, but don't get why.
24
votes
9answers
1k views

What does it mean to represent a number in term of a $2\times2$ matrix?

Today my friend showed me that the imaginary number can be represented in term of a matrix $$i = \pmatrix{0&-1\\1&0}$$ This was very very confusing for me because I have never thought of it ...