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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4answers
70 views

Use row reduction to show that the determinant is equal to this variable.

Show determinant of: \begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix} is equal to $(b - a)(c - a)(c - b)$ I'm not sure if you can use squares or square roots hmmm.. ...
0
votes
2answers
34 views

Show that a determinant is equal to this variable.

Show that the : determinant of: \begin{pmatrix}0&0&a_{13}\\0&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix} is equal to $-A_{13}A_{22}A_{31}$ I believe the cofactor and ...
0
votes
0answers
9 views

Is there a name for products $\Delta(v)\!\cdot\!M$ and $M\!\cdot\!\Delta(w)$?

For any vector $u$, define $\Delta(u)$ as the diagonal matrix whose diagonal elements correspond to the entries in $u$. Now, let $M$ be an $m \times n$ matrix, and $v$ and $w$ be $m$- and ...
0
votes
0answers
14 views

Prove that every reproducing kernel is a positive matrix (and vice versa)

Let $\mathcal{H}$ be a functional hilbert space (defined over a set $S$) with a reproducing kernel K. Prove that: a) $K$ is a positive matrix means the queadtric form is positive, i.e ...
0
votes
1answer
58 views

Matrix Becomes a vector space

Explain how addition and multiplication by scalar can be defined in a natural way for an $M_{m,n}$. So $M_{m,n}$ becomes a vector space. Progress I created two $m\times n$ matrices and showed then ...
1
vote
2answers
31 views

Find the standard matrix and kernel for a linear transformation.

Let $T : \mathbb{R^3} → \mathbb{R^3}$ be a linear transformation given by $T(u) = \operatorname{proj}_vU$ where $v = (2, 0,−3)$. (a) Find the standard matrix for $T$. (b) Find a basis for the kernel ...
4
votes
2answers
37 views

Find the value(s) of $k$ such that the given vectors do not span $\mathbb{R}^3$

I'm currently attempting to solve the following problem: Find the value(s) of $k$ such that the vectors $\{\vec{a}_1, \vec{a}_2, \vec{a}_3\}$ do not span $\mathbb{R}^3$, where: $$ a_1 = ...
2
votes
2answers
62 views

If $AB = I$, the identity matrix prove $\mathrm{rank}(B)$

Let $A$ be an $m \times n$ matrix and $B$ be an $n \times m$ matrix. Show that if $AB = I$, where $I$ is the identity matrix, then $\mathrm{rank}(B) = m$. I'm not exactly sure how to start this ...
0
votes
3answers
41 views

Trace of a power of a matrix product

Suppose I have two 2x2 matrices $A$ and $B$. What can I say about $Tr(A^k B^k)$ versus $Tr((AB)^k)$? I know that if there is some cyclic permutation that takes $A\cdot A\cdots A B\cdot B\cdots B$ to ...
0
votes
1answer
17 views

Finding a basis for unification of two subsets

I have this problem : $U,W \subseteq R^4$ Base of $W = \{w1 = (1,2,2,-2), w2 = (0,1,2,-1)\}$ Base of $U = \{u1 = (1,1,0,-1),u2 = (0,1,3,1)\}$ Find a basis for $U \cap W$. My solution for any $v ...
0
votes
0answers
13 views

Inverse of a 2x2 principal submatrix whose inverse is known

Let $H$ be a $n\times n$ symmetric positive definite matrix. What is the (computationally) quickest way to obtain $H_{ij}$, the $2\times 2$ matrix whose inverse is the principal submatrix of the ...
0
votes
4answers
47 views

Sin(x) + Sin(y)

When you add sound waves you are basically adding sine and cosine of certain multiples of x. Is Sin(x) + Sin(y) ... + Sin(n) = Sin(x+y...+n)? Is the same true for summation of cosines? I am making a ...
0
votes
2answers
12 views

Scalar value of similarity between Two Square Matrices

I am wondering if there is any way to compute mathematically the similarity/distance between two square matrices as a single value?
0
votes
1answer
9 views

Support of vector $w$ in graph sparsity

I'm reading about graph sparsity and I have one problem in a paper I'm reading I don't understand, maybe someone can clarify: Graph Sparsity: In graph sparsity, we have a directed acyclic graph ...
2
votes
2answers
31 views

Solve the system of linear equations by Gaussian elimination and back-substitution.

Question 1:Form the adjunct matrix and reduce it to echelon form. I dont know how to write matrices here, so i snap a picture of my operation. Did I do it right? Question 2: Use back-substitution to ...
-1
votes
0answers
45 views

Span and consistency?

From problems I've completed, it seems as if a consistent matrix does not imply spanning $R_m$(rows). Is this correct? So if we have col $1 = (1, 0, 0, 0)$ and col $2 = (3, 2, 0, 0)$ and col $3 = ...
1
vote
2answers
97 views

How to find transformation matrix which converts matrix to simple standard form

I have a matrix A$$ \left( \begin{array}{ccc} 0 & 1 \\ a^2 & 0\\ \end{array} \right) $$ Using eigen values, I convert it into simple standard form B: $$\left( \begin{array}{ccc} a & 0 \\ ...
1
vote
2answers
50 views

If $AB+BA=0$, then $A^2B^3=B^2A^3$?

If I have a matrix $A$ and $B$ such that $AB+BA=0$ is it true that $A^2B^3=B^2A^3$? I think that it is false.
2
votes
0answers
50 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
4
votes
3answers
49 views

Does a symmetric matrix $A^2$ imply a symmetric $A$?

Does a symmetric matrix $A^2$ imply a symmetric $A$? Any help would be much appreciated.
0
votes
1answer
21 views

Eigenvalue and eigenvector of $A'A$

Suppose that $\mathbf{A}\in\mathrm{R}^{m\times m}$ is a square but not necessarily symmetric matrix whose eigenvalues and eigenvectors are $\lambda_i$ and $\mathbf{x}_i,$ $i = 1,2,\cdots,m$. Is ...
0
votes
1answer
37 views

Convert coordinates to a different coordinate axis

Sorry for any forum rules I have broken, I needed a quick answer. I want to create a plane including 3 nonlinear points on a 3d coordinate system, one being the origin. I also need to create a ...
2
votes
2answers
46 views

Invertible: A non-square matrix?

So I am doing a question were I have the set column matrix 1 = (3, -8, 1) and column matrix 2 = (6, 2, 5) and the question is asking if this is either a bases for R2 or R3. Can I just say that since ...
1
vote
0answers
17 views

Variance-covariance matrix of a linear regression model

In finding the covariance matrix of a linear regression model I don't understand this step: $$ E[(b-\beta)(b-\beta)']=E[(X'X)^{-1}X'\epsilon\epsilon'X(X'X)^{-1}] $$ where we've been given that $$ ...
2
votes
2answers
41 views

For what values of k and h does this system of equations have a unique solution?

Here's my system of equations: $x−3y+2z=5$ $2x−5y−3z=9$ $−x−y+kz=h$ So I have $ \begin{bmatrix} 1 & -3 & 2 & 5 \\\\ 2 & -5 & -3 & 9 \\\\ -1 & -1 & k& h ...
1
vote
1answer
23 views

Do all symplectic transformations give rise to skew symmetric matrices?

Suppose that $ \Delta(x,y) = x^T\Delta y $ where $ \Delta$ is a symplectic matrix of form given in https://en.wikipedia.org/wiki/Symplectic_matrix If I define an inner product $ \alpha(x,y) = ...
0
votes
1answer
16 views

Write Generator Matrix (2,4) of Reed Muller code of (2,4)

I was wondering how to go about this sum, since I wasnt able to figure out how to solve this. Could any one help me out on this?
4
votes
1answer
142 views

Proof of the conjecture that the kernel is of dimension 2

I already asked this question which has been answered. This question may seem very similar but the required matrix manipulations are probably very different here due to the addition of the matrix ...
-1
votes
2answers
36 views

Linear Alegbra - inverse matrix multiplication

I have a general question. If there is a matrix which is inverse and I multiply it by other matrixs which are inverse. Will the result already be reverse matrix? My intonation says is correct, but ...
0
votes
2answers
47 views

Solutions of $Ax=b$ of square matrix $A$

If A is a $5 \times 5$ matrix and the equation $Ax = b$ is consistent for every b in $R^5$; is it possible that for some $b$, the equation $Ax = b$ has more than one solution? Why or why not?
0
votes
0answers
6 views

Monotone operator without symmetry?

A function $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ is monotone with respect to $P = P^\top\succcurlyeq 0$ if $$ \left( f(x) - f(y) \right)^\top P (x-y) \geq 0 $$ for all $x,y$. Now suppose that ...
0
votes
0answers
12 views

Represent row normalized product of matrices, by doing something to matrices first

I have two matrices $A$ and $B$, I need to get the same result as $C = normR(A * B)$, where $normR$ is a normalization along the row axis, dot product of very row of $C$ is equal to 1 (unless all ...
-4
votes
2answers
34 views

Why the columns of an $n \times n$ matrix $A$ span $R^n$ when $A$ is invertible

Why the columns of an $n \times n$ matrix $A$ span $R^n$ when $A$ is invertible? It's really sad that you see people down voting your question because of their inferiority complex. 4 people have ...
2
votes
1answer
41 views

“Degrees of freedom” of some low-rank skew-symmetric matrices

Let $n$ be an even integers. Let $r\in \mathbb R^n$ and $e=[1,1,\dots,1]^T$. If $$A = re^T - er^T,$$ then $A\in \mathbb{R}^{n\times n}$ is of rank 2 and skew-symmetric, i.e., $$A = -A^T.$$ This ...
2
votes
1answer
25 views

Parallelism in Golub & van Loan's Jacobi algorithm for symmetric eigenvalue problems

In Matrix Computations by Golub and Van Loan (3rd edition, page 433) an algorithm is given for a parallel version of the classical Jacobi algorithm for solving a real symmetric eigenvalue problem. The ...
0
votes
2answers
16 views

Iteration of a function related to the minimal polynomial of a matrix

Let $M$ be a singular $n \times n$ matrix over some field. In order to find a matrix $N$ s.t. $MN=0$, I do the following : $p(x)=$ minimal polynomial of $M$. Then the constant term of p is zero ...
1
vote
1answer
29 views

What does this matrix operation mean?

If T is matrix what is this operation? What's name of operation?
2
votes
0answers
76 views

Jacobian Matrix Requirement for Linear Approximation

It is my understanding that when searching for a linear approximation of a nonlinear function, using the Jacobian (matrix) could help. I did some reading, and read that there is a condition: the ...
2
votes
2answers
19 views

Preservation of rank implies Invertibility

Show that if the rank of $XY$ (where $Y$ is an $n\times n$ matrix) is the same as the rank of $X$ for every $m\times n$ matrix $X$, then $Y$ is invertible. I thought I had found a counterexample: $$ ...
4
votes
1answer
77 views

Prove that the kernel is of dimension 2

"Experimentally", I found that the kernel (null space) of the following matrix is of dimension 2. I'd like to prove it, but haven't managed yet: \begin{equation} \text{for almost all } t>0,\quad ...
0
votes
0answers
12 views

Matrix of Adjoint operator (Hermitian conjugate)

Can someone tell me what I have to do? Operator $D$ of the matrix $D_f=\begin{bmatrix} 2&1\\ 2&0\end{bmatrix}$ with basis $f_1=(1,1), f_2=(0,1)$ of vector space $\mathbb R^{2\times 2}$ with a ...
1
vote
1answer
25 views

Negative determinant

Let $$ A = \begin{bmatrix} -a_{12}-a_{13}-a_{14} & a_{12} & a_{13} & 1\\ a_{21} & -a_{21}-a_{23}-a_{24} & a_{23} & 1\\ a_{31} & a_{32} & -a_{31} - a_{32} - a_{34} & ...
0
votes
1answer
16 views

Help understanding formula $score(K) = \sum_{i,j} | d_{ij} - e_{ij} |$

I am trying to write some code to perform an equation based on the formula below, however I am having a hard time understanding mathematic syntax. The formula is as follows: $$ score(K) = ...
1
vote
0answers
27 views

Suppose that A and B are nxn matrices with A row equivalent to B. Prove that if A is nonsingular, then B is row equivalent to In.

Suppose that A and B are nxn matrices with A row equivalent to B. Prove that if A is nonsingular, then B is row equivalent to In. I answered this on a test and it seemed right to me, but got zero ...
0
votes
1answer
27 views

Why does the discrete cosine transform as matrix multiplication work this way?

I have read that the DCT can be computed as a matrix multiplication. The 8x8 DCT matrix is: $D=\frac{1}{2}\left[\matrix{ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & ...
1
vote
3answers
41 views

Determinant of linear transformation

Given a linear transformation $T:V\rightarrow V$ on a finite-dimensional vector space $V$, we define its determinant as $\det([T]_{\mathcal{B}})$, where $[T]_{\mathcal{B}}$ is the (square) matrix ...
3
votes
3answers
79 views

If $A=\pmatrix{1 &0\\-1&1}$, show that $A^2-2A+I_2=0$. Hence find $A^{50}$

If $$A=\pmatrix{1 &0\\-1&1},$$ show that $$A^2-2A+I_2=0,$$ where $I_{2}$ is the $2x2$ Identity matrix. Hence find $A^{50}$. We have $$A^2-2A+I_2=A(A-2I_3)+I_=\pmatrix{1 ...
0
votes
1answer
26 views

how to differentiate this equation (contains absolute and norm)

how can I differentiate the following wrt $\mathbf{d}_i$? $\frac{|\mathbf{d}_i^T\mathbf{d}_j|}{\|\mathbf{d}_i\|_2\|\mathbf{d}_j\|_2}$ Thanks in advance.
0
votes
0answers
41 views

Characteristic polynomial and Jordan normal form of permutation matrices

What can be said about the characteristic polynomial and the Jordan normal form of permutation matrices?
0
votes
1answer
18 views

Can a general time-dependent finite-dimensional Schrödinger equation with complex Hamiltonian be transformed to one with real Hamiltonian?

Consider a general-form time-dependent Schrödinger equation: $$i\partial_tv=\hat Hv,$$ where Hamiltonian $\hat H$ is an Hermitian matrix (finite-dimensional for simplicity), and $v(t)$ is a complex ...