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
0answers
21 views
+50

Largest magnitude of off-diagonals of Wishart matrix

Let $X_1,\dots,X_n\sim N(0,\Sigma)$ be a multivariate normal, with sample covariance $\hat\Sigma$. Of course the diagonals of this matrix are chi-square distributed and there exist tail bounds for how ...
0
votes
0answers
24 views

Definiteness of matrix given definiteness of principal submatrix

I have a symmetric real $(J+1) \times (J+1)$ matrix $H$, with structure $$H=\begin{pmatrix} H_1 & H_2 \\ H_2^T & H_3 \end{pmatrix}$$ where $H_1$ is a symmetric $J \times J$ matrix, $H_2$ is ...
3
votes
2answers
45 views

For which values of h does a solution exist?

I am studying for a final tomorrow and didn't have this solution in my notes. Please let me know if this is correct and if not, what I did wrong. 1) For which values of h does a solution exist? 2) ...
0
votes
1answer
16 views

The Coercivity of uniformly positive definite Matrix of Sobolev function

For $u=(u^1,\ldots, u^N)\in W^{1,2}(\Omega,R^N)$ where $\Omega$ is bounded. We define $$ E[u]=\int_\Omega g_{ij}(u)\nabla u^i\nabla u^jdx$$ where $G=(g_{ij})_{1\leq i,j\leq N}$ is an given uniformly ...
0
votes
0answers
12 views

Why is this Quadratic Form Independent of its Parameter in the Limit?

For $\alpha = e^{1/N}$, I have the following upper-triangular $\left(N+1\right)\times\left(N+1\right)$-Toeplitz matrix: $$\tilde{G}^{\left(N\right)}=\begin{pmatrix} 1/2 & 0 & 0 & \cdots ...
2
votes
1answer
31 views

How many invertible matrices are in $M_{2}(\mathbb{Z}_{11})$?

I tried to solve this question but without a success. How many invertible matrices are in $M_{2}(\mathbb{Z}_{11})$? Thanks
0
votes
0answers
16 views

Express a 90 degree rotation matrix in terms of a 180 degree rotation matrix? (both anti-clockwise)

A = [-1 0 0 ] [ 0-1 0 ] [ 0 0 1 ] B = [0 -1 0 ] [ 1 0 0 ] [ 0 0 1 ] How can i represent B in terms of A?
-1
votes
2answers
47 views

How do I detect if a 4x4 transformation matrix contains reflection?

We currently check if the determinant of the upper left 3x3 values is negative to detect reflection in a 4x4 transformation matrix but we are unsure that it works in all cases (any arbitrary 3D ...
2
votes
1answer
28 views

Prove: the sum of two nilpotent and exchangable matrices is nipotent.

If $A$ and $B$ are two $n\times n$ nilpotent matrices, and they are exchangable: $AB = BA$, it is said that the sum $A+B$ is also nilpotent. Could you pls give me some hint how to prove that?
1
vote
1answer
16 views

Column space of a 3x3 matrix

Let $A$ be a $3 \times 3$ matrix $A$ with reduced row echelon form $ \left[ {\begin{array}{c} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} } \right]$. Then $\left[ ...
0
votes
1answer
13 views

$(T^{-1})^i == (T^i)^{-1}$?

I wonder if the hypothesis in the title is true. And if so, some ideas to prove it. I know $(A^T)^{-1} = (A^{-1})^T$ EDIT: Edited the title to match the generic answer. T does not have to be ...
2
votes
2answers
29 views

If $\|X(t)\|\leq M$, does this imply that $det(X(t))$ is bounded?

I am wondering if the following is true: If you are given a matrix $X(t)$ (that depends on the positive real variable $t$) which is bounded (i.e, $\|X(t)\|\leq M$ for all $t$. Can you conclude that ...
3
votes
1answer
32 views

Proof that real symmetric negative matrix is negative definite

I have a reasonably simple symmetric $p \times p$ matrix $H$, where the $(j,k)$th element is given by $$h_{j,k} = -\sum_{i=1}^n \frac{a_i}{b_i^2} x_{ij} x_{ik}$$ and we know that all $a_i \geq 0$ ...
0
votes
1answer
32 views

Determinant on 3x3 matrix and above

When finding the determinent of a matrix, what is the rationale behind multiplying the entry along the row we are deleting from times the cofactor expansion? Also how does doing cofactor expansion ...
0
votes
1answer
21 views

Are all matrices of the following form Hermitian?

If I have a matrix $A$ (and it is square and nonsingular), is $A^* A $ Hermitian? Also, does $A $ have to be nonsingular for this to hold?
-2
votes
1answer
23 views

Find a matrix representing a given linear transformation [duplicate]

$T(X) = [\{x_1-x_2+x_3\}, \{0+x_2-x_3\}, \{0+0+0\}]$ is a linear transformation from $\mathbb R^3$ to $\mathbb R^3$. Find a matrix $A$ such that $T(x) = A(x)$ Can anyone point me in the right ...
0
votes
1answer
25 views

Orthogonal columns imply orthogonal rows?

The original question is: Column Vectors orthogonal implies Row Vectors also orthogonal? A counterexample with zero entries is given in one post. However, my question is whether pairwise orthogonal ...
1
vote
1answer
10 views

Definition of onto for linear transformation

I had a question ask the following: "A linear transformation is onto if and only if the columns of its standard matrix form a generating set for its range." To me that seems true but the answer was ...
4
votes
3answers
553 views

Kernel Explanation

sorry for asking so many questions lately but our lecturer is doing a terrible job explaining things. Calculate $ker(A)$ given that: $f:\{\mathbb{R}^3→\mathbb{R}^3; r→ A\vec{r}\}$ $A= ...
0
votes
0answers
18 views

Similar transformation matrix restricting determinant to be 1.

How do you prove that if restricting the determinant of a similar transformation matrix between two equivalent irreducible unitary representation of a finite group to be 1, then this transformation ...
0
votes
1answer
22 views

For an arbitrary symmetric matrix, the relation between the number of eigenvalues and the rank of the matrix?

For an arbitrary symmetric matrix $A\in \mathcal{S}^n$, $n$ symmetric space: what is the relation between the number of eigenvalues and the rank of $A$ ? If we know $rank(A) = r$, what is the ...
1
vote
1answer
37 views

If the null space contains only the zero vector, the map is one-to-one

How does finding out if the null space has only the zero vector prove one-to-one? One-to-one means that there are distinct images for each distinct vector input. $$\mathbb R^n \to \mathbb R^m$$ ...
0
votes
0answers
15 views

Searching for a definition for n-Dimensional rotation which is cosine-distance invariant

I am wondering if it is possible to define a rotation for an $n$-Dimensional space ($n=2,3,4,5,\dots$). Given any vector $\vec v$, and knowing that it should be rotated to ...
-1
votes
1answer
36 views

What is the meaning of |⋯| notation for an index subset?

I am currently working on a research project. In the attached image what does the capital $|I|$ and $|J|$ mean ?
0
votes
2answers
33 views

Why $V_{ij} = \frac {1}{2}(v_iv_j^T + v_jv_i^T),$ is rank-2 if $i\neq j$?

Can someone help me figure out the following argument ? $V_{ij} = \frac {1}{2}(v_iv_j^T + v_jv_i^T),$ is rank-2 if $i\neq j$ where $v_i,v_j \in \mathbb{R}^n$, $v_i,v_j$ are linearly indepedent. ...
1
vote
4answers
43 views

Why is this a valid definition of the dot product?

$(\vec{u},\vec{v})=u_1v_1+2u_2v_2+3u_3v_3$ I have never seen this definition before. I am used to the dot product looking something like this: $(\vec{a},\vec{b})=a_1b_1+a_2b_2+a_3b_3$ Where do the ...
0
votes
1answer
24 views

Finding vector valued function solution to differential equation

find the vector valued function x solution to the differential equation system ( X'=AX ) ( X(0) is a 3 x 1 matrix [1;2;3] ) and ( A is a 3 x 3 matrix [2 0 3; 1 2 0; 3 0 2] ) ( X is a 3 x 1 matrix ...
0
votes
2answers
45 views

Maximum eigenvalue of product of two matrices

Let $A$ and $B$ be two Hermitian matrices. I wanted to know if there is any relation between the maximum eigenvalue of $AB$ and that of $A$ and $B$. Is the following relation true? ...
0
votes
0answers
35 views

Algorithm to efficiently compute$ A^k$

If a symmetric matrix $A$ has SVD $A=U\Sigma U^{\top}$, then $A^k=U\Sigma^kU^{\top}$. What would be the most efficient algorithm to compute $A^k$ such that the worst case time complexity is as low as ...
0
votes
1answer
25 views

If a symmetric matrix $A$ has SVD $A=U\Sigma U^T$, then $A^k=U\Sigma^k U^T$ [closed]

How can I prove that for symmetric matrix for SVD, the following condition is true $$A^k=UΣ^kU^T$$
2
votes
1answer
19 views

Symmetric matrix with same diagonal elements

A paper I was reading made the claim that the eigenvector of a symmetric matrix with same diagonal elements is : $$i_n = {1, e^{jna}, e^{2jna}, ..., e^{j(N-1)na}}$$ $$ n =0, 1, 2 , ...N-1$$ $$a = ...
7
votes
2answers
70 views

Eigenvalues of $A(A+B)^{-1}$

Given a positive semidefinite matrix $A$ and a positive definite matrix $B$ of the same dimension. Can we show that each eigenvalue: $$ \lambda\{A(A + B)^{-1}\} < 1$$ (in the scalar case, this is ...
6
votes
1answer
68 views

Symmetric matrices and commutativity

Q: Let $m, n$ be positive intergers. Let $A$ and $B$ real $n\times n$ matrices. Assume that $B$ is symmetric and positive definite. If $A$ commutes with $B^{m}$, prove that $A$ commutes with $B$. So ...
2
votes
2answers
37 views

Confusion about a Linear Transformation question.

Let $\beta := [M_1, M_2, M_3, M_4]$ be the ordered basis of $R^{2×2}$ defined by: $$ M_1 := \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}, M_2 := \begin{pmatrix} 0 & 1\\ 0 & 0 ...
0
votes
0answers
6 views

Motivation for Low Rank Matrix Approximations

I was wondering if anyone could shed some light onto the motivations for using a low rank approximation of a matrix? I don't think it is computationally more efficient to use a low rank matrix so why ...
0
votes
1answer
30 views

Yet another n-Dimensional Rotation question: is there a definition for n-Dimensional rotation which is cosine-distance invariant? (TO CLOSE)

(Moved to Searching for a definition for n-Dimensional rotation which is cosine-distance invariant, flagged it to delete it) I'm wondering if there exists a rotation definition by which the vectors ...
0
votes
4answers
49 views

Triangular matrices proof

Let $A$, $B$ be $n\times n$ lower triangular matrices. Prove that $AB$ is also a lower triangular matrix. How do I prove this for every $n$?
7
votes
1answer
81 views

$GL_n(\mathbb{F})$ contains a copy of $\mathbb{F}^{n-1}$

It is a fact of matrix multiplication that $$\left( \begin{matrix} 1 & a & b \\&1&\\&&1 \end{matrix} \right) \left( \begin{matrix} 1 & a' & b'\\&1&\\&&1 ...
0
votes
2answers
29 views

How many orthogonal matrices map one vector to another?

Say you have two real vectors $u$ and $v$ and $\|u\| = \|v\|$ How many real square and orthogonal matrices $A$ are there such that $Au = v$? Assuming $u$ and $v$ are not parallel/antiparallel, there ...
0
votes
3answers
36 views

A good source for linear algebra on matrices

I am studying for an Algebra qualifying examination to be taken in one month. I need a good source (a book) where I can find the most important theory and examples on linear algebra on matrices. Does ...
0
votes
0answers
5 views

Optimal Rectangular Matrix Multiplication

I'm trying to find optimal algorithms for non-square boolean matrix multiplication, and I was wondering if people could point me towards some of the literature? Right now, most of the papers and ...
0
votes
0answers
33 views

Does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$?

For $p$ prime, does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$? (It does seem to be the case for $p=3,5,7,11$) If yes, what are the generators? I guess this is simple, but my ...
4
votes
1answer
64 views

Det(AB)=0: what is the determinant of A and B

True or false. If the determinant of AB is zero, then the determinant of A is zero or the determinant of B is zero. I put true in my exam. After all det(A)det(B)=det(AB). Why was I wrong? The answer ...
1
vote
0answers
49 views

Inequality in inverse Laplacian

I have the following problem, which is motivated by geometric diffusion on a directed graph. Conjecture. Let $A \in [0,1]^{n\times n}$ be strictly substochastic - i.e. $\forall i ~ \sum_j A_{i,j} ...
0
votes
1answer
32 views

Matrix addition

How do I solve the following? [2x1 -3x2 + x3; 4x1 - 2x3] + [x1 +2x2; 0x1 - 2x2; 4x1 + x2]^T When I do the transpose of the second matrix and try to add them together I get lost. Should I consider x1 ...
0
votes
1answer
14 views

if L is the cholesky decomposition, then what is L' L?

Let $L$ be a lower-triangular matrix such that $LL^T=A$. Then $B=L^TL$ has the same eigenvalues as $A$, but different eigenvectors. 1) Why are the eigenvalues the same? 2) Is there an analytical ...
0
votes
0answers
20 views

Gram matrix of Gaussian kernel is not positive definite

I am developing a machine learning software, where I am trying to apply kernel methods. I have N uniformly sampled scalar values, $\{x_1,\dots,x_N\}$ from a given interval $[a,b]$. My aim is to ...
2
votes
1answer
28 views

Determining diagonalizability of a linear transformation defined by a matrix.

Suppose $A\in M_n(\Bbb C)$ satisfies $A^6-A^3+I=O$. Prove that if a linear transformation $T:M_n(\Bbb C)\rightarrow M_n(\Bbb C)$ is given by $T(B)=AB$, then $T$ is diagonalizable. How to prove it? ...
1
vote
2answers
349 views

is this always identity matrix?

do you think the following matrix multiplication results in I? $R(R^TR)^{-1}R^T$= I or diag(I, O) R is not necessarily square and may not have an inverse.