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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2answers
12 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
8 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
45 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} ...
0
votes
1answer
22 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
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1answer
32 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 ...
1
vote
1answer
52 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 $k=1,2,3,\dots$?
0
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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 ...
0
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1answer
20 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
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0answers
15 views

Compute det(A) given a function A

Suppose A is a 3×3 matrix and A = 1/3 u1*uT1 + 1/4 u2*uT2 + 2/5 u3*uT3 with uT1 = (0, 1, −1), uT2 = (1, 2, 2), u3 = (−2, 1/2 , 1/2 ). Compute det(A). I know how to find determinants of 3x3 ...
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0answers
33 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
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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?
0
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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 ...
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
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 ...
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
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
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 $V$ ...
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
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1answer
32 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
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: ...
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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
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0answers
7 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
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
16 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
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
32 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?($A \in {M_n}$)
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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
19 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
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 ...
0
votes
0answers
11 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
17 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
41 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
33 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
90 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
32 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
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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} ...
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
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
13 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
38 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
15 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
7 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
44 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
25 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 ...