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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Solving a system of equations containing complex numbers - Gaussian elimination

Problem: Determine the solutions in $\mathbb{C}^3$ of the following system over $\mathbb{C}$: \begin{align*} \begin{cases} 2x+iy-(1+i)z &=1 \\ x-2y+ iz &= 0 \\ -ix +y -(2-i)z &= 1 ...
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1answer
8 views

Generalized inverse designed to reconstruct a specific vector

Assuming we have some matrix $P\in\mathbb{R}^{m\times n}$, with $m<n$, that maps vectors $\mathbf{x}\in\mathbb{R}^n$ to $\tilde{\mathbf{x}}\in\mathbb{R}^m$ as $\tilde{\mathbf{x}} = P \mathbf{x}$, ...
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1answer
22 views

Writing a matrix in terms of a basis

I've looked for examples but found none similar to this; I have $\mathfrak{sl}(2,K)$ with the given basis $S$ as follows: $S=\{e,h,f\}$ where $e = \pmatrix{0 & 1 \\ 0 & 0}$ $h = \pmatrix{1 ...
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0answers
13 views

Two basic examples of trace diagrams?

In the wikipedia entry on Trace Diagrams (see http://en.wikipedia.org/wiki/Trace_diagram), the statement is made that "The simplest trace diagrams represent the trace and determinant of a matrix". ...
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0answers
32 views

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. [on hold]

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. I understand that the solution to $\ker A$ is the set of all solutions to $Ax=0$. I'm confused about how I ...
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2answers
32 views

A given ring of matrices has an infinite number of invertible elements

The set $\mathcal{M} = \bigg\{ \begin{pmatrix} a & 2b \\ b & a \\ \end{pmatrix} \bigg\vert a,b \in \mathbb{Z} \bigg\}$ is given. Prove that: (1) $\mathcal{M}$ is a commutative ring with ...
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2answers
22 views

Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that $T$ is bijection >iff T is injective.

Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that (a)$T$ is bijection iff (b)T is injective. Solution: show $(a)\implies(b)$ If $T$ ...
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17 views

Using semidefinite programming to solve the following problem

I am struggling with the following problem, and wonder is SDP can help: $$\mathrm{maximize\ } \alpha_{10}+\alpha_5+(\alpha_2+\alpha_8)/2 \mathrm{\ subjected\ to\ } \mathrm{T_1}\succeq0, ...
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1answer
32 views

I need help with this linear transformation.

Please let me know if my process or thinking is incorrect at any point. Let $T:P_3 \rightarrow P_3$ be the linear transformation such that $$T(-2 x^2)= 3 x^2 + 3 x,\\T(0.5 x + 4)= -2 x^2 - 2 x - 3, ...
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13 views

Write F+K as a span of some basis.

Let $F = \{(a,a+b,4b,0) | a,b\in \mathbb{R}\}$ and $K = \{ (c,2c+d,4c-d,2d) | c,d\in \mathbb{R}\}$. Write F+K as a span of some basis. Solution: The basis of $F$ is $\{(1,1,0,0),(0,1,4,0)\}$. ...
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0answers
7 views

correlation between minimum singular values of submatrices

I have one question regarding how to measure the correlation between minimum singular values of submatrices extracted from a large random matrix. Given a m*n random matrix $\mathbf{A}$, the ...
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1answer
28 views

What is the rank of the matrix in this situation?

$A$ is a $9$ by $9$ matrix in the field modulo $5$ ($\mathbb{Z}/5$). It is known that the number of solutions for the equation $Ax=0$ is between $40$ and $150$. I need to find out the rank of the ...
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1answer
27 views

Changing the basis of a transformation

Given $T: \mathbb{R}^2 \to \mathbb{R}^2 : T\begin{bmatrix} x \\ y \end{bmatrix} \to \begin{bmatrix} 2x+y \\ x-3y \end{bmatrix}$ with standard basis $\mathcal{B}$ and basis ...
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43 views

Show the equilibrium vector of a transition matrix for a Markov Chain has no zero entries

Let P be a transition matrix for a regular Markov chain and let w be its equilibrium vector. Show that w has no zero entries. I am thinking using the fundamental limit theorem for regular chains for ...
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3answers
28 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
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1answer
28 views

$K$ is a region in $\mathbb{R}^2$ where the area is $5$

Say that $K$ is a region in $\mathbb{R}^2$ where the area is $5$. Let B = \begin{pmatrix} 3 & 8 \\ 4 & 6 \end{pmatrix} Find the area of the region B$K$. Any starting hints? Is it possible ...
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3answers
24 views

Eigenvectors of the matrix

I'm getting frustrated with a question. I'm trying to find the eigenvector of [[1/2,0], [0,2]] and it's been a few good years since I've had to touch eigenvectors. I get the characteristic polynomial ...
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1answer
30 views

Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
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1answer
31 views

finding column vectors - linear transformations

$L:\mathbb{R}^3\rightarrow \mathbb{R}^2$ with bases $\mathcal{S}=\left\{\left(-1,1,0\right),\left(0,1,1\right),\left(1,0,0\right)\right\} \: \text{for} \:\mathbb{R}^3 \:\text{and} \\ ...
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2answers
51 views

Why is this the eigenvector?

For the eigenvector how are they getting \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} when you have \begin{bmatrix} 0 & -1 & -1 \\ 0 & -1 & -3 \\ 0 & 0 & -2 \end{bmatrix} ...
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1answer
23 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
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2answers
50 views

How do you solve this circular system of equations in $\mathbb{Z}_2$?

I'm trying to solve a system of equations in $\mathbb{Z}_2$ that look like this: \begin{align} x_1 + x_2 = p_1 \\ x_2 + x_3 = p_2 \\ x_3 + x_4 = p_3 \\ ... \\ x_n + x_1 = p_n \\ \end{align} I know ...
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1answer
21 views

What the limit of a matrix over time shows about the future

$x_k$ is the fraction of people who prefer cake to pie at year $k$. The remaining fraction $y_k=1-x_k$ prefer pie. At year $k+1$, $\frac{1}{5}$ of those who prefer cake change their mind. Also at year ...
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2answers
34 views

Matrix with eigenvalue that should equal 1.

I have the matrix: $$A = \begin{bmatrix}4 & -2 & 3\\0 & -1 & 3\\-1 & 2 & -2 \end{bmatrix}$$ and I need to find out if $\lambda = 1$ is an eigenvalue. So I solved the equation ...
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0answers
13 views

How to check if a matrix transfer function is in Hardy-infinity space?

Just like the question says. For instance if I have a matrix transfer function $$\mathbf{G}(s) = \begin{bmatrix}s & -s \\ T & s \\ \end{bmatrix}$$ where $T$ is a positive constant, how can I ...
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0answers
14 views

Uniqueness of pseudoinverse?

Recently, I read a line of reasoning as Since $A$ (a $3\times 3$ matrix) is of rank $2$, its pseudoinverse is not unique. May I ask if there is a quickie to show this?
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4answers
51 views

A real $2 \times 2 $ matrix $M$ such that $M^2 = \tiny \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$ , then :

A real $2 \times 2 $ matrix $M$ such that $$M^2 = \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$$ (a) exists for all $\epsilon > 0$. (b) does not exist for any ...
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1answer
23 views

How can I prove that any matrix A can be expressed as the sum of two Hermitian matrices , B and C, in the form A = B + iC?

The question is in the title really. Whether or not A must also be Hermitian is not clear to me. Sorry, I am not very good with proofs of this nature.
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0answers
14 views

Using Levi-civita symbol to determine axis and angle of rotation matrix

One of the questions on the course involves finding the angle and axis of this rotation matrix; $$R = \gamma\ \begin{pmatrix} 0 & -2 & 1\\ 2 & 0 & 0\\ -1 & 0 & 0 ...
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2answers
39 views

Finding a matrix from equation

we've got the following 4x4 Matrix $$\begin{pmatrix} 4 & -2 & 3 & 2\\ 3 & 5 & 1 & -4\\ -1 & 6 & -4 & -7\\ -2 & 0 & -2 & 4 \end{pmatrix}$$ and I need ...
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1answer
21 views

Proving multilinearity of determinant [on hold]

As the title says, how we can prove multilinearity property of determinants: $$ \begin{vmatrix} p+q+r & x+y+z & u+v+w \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\\ ...
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1answer
58 views

If $A$ is positive definite then so is $A^k$

I know how to show the inverse of positive definite is positive definite but I don't know how to expand that. Suppose $A$ is positive definite then $A$ is invertible, so define $y=Ax$ for $x\neq 0$. ...
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3answers
28 views

Proving $AD_1A^{-1}=D_2$

I want to prove that if $A$ is a permutation matrix, and $D_1$ is diagonal, than $AD_1A^{-1}=D_2$ where $D_2$ is also a diagonal matrix. I have worked out that $A^{-1}=A^T$ and I can see that the ...
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0answers
26 views

centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
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1answer
29 views

Positive Semidefinite Matrices

Let $x=\left[ \begin{array}{cccc} x_{1} & x_{2} & \cdots & x_{n}% \end{array}% \right] $ be a vector with $\sum x_{i}=1$ and $x_{i}>0$. Is there an easy way to prove that ...
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0answers
17 views

A 2D smoothing convolution filter

I'm trying to find the right form of a 2D filter that will do the following to a matrix after linear convolution: Let A = [ ? ? ?] [ ? ? ?] [ ? ? ?] and B = ...
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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 ...
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2answers
30 views

Understanding matrix property

I am reading about matrix property from here. On page 2 of pdf (equation 2.2), it says if $A$ is a matrix and $U$ a row-echelon form of $A$ then $$|A| = (-1)^r \alpha |U| ...
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1answer
25 views

Sum of orthogonal matrices

Consider the subspace $\mathbb{R}^m$ with usual inner product.Let $S_1$ and $S_2$ subspaces of $R^m$, $P_1\in M_m(\mathbb{R})$ the orthogonal projection matrix on $S_1$ and $P_2\in M_m(\mathbb{R})$ ...
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3answers
106 views

How can we memorize the formula for the determinant of a $4\times4$ matrix?

This is the formula for the determinant of a $4\times4$ matrix. . 0,0 | 1,0 | 2,0 | 3,0 0,1 | 1,1 | 2,1 | 3,1 0,2 | 1,2 | 2,2 | 3,2 0,3 | 1,3 | 2,3 | 3,3 . ...
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0answers
34 views

Fill in the missing entries of matrix $Q$ to make it orthogonal [on hold]

I am given the following matrix $Q$: $Q=$ where $p1,p2,...,p8$ are unknowns. I need to make $Q$ into an orthogonal matrix. It occurs to me that $v1 =\{1,1,1,1\}$ and $v2=\{2,1,0,-3\}$, but I'm ...
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0answers
19 views

On the probability of singular matrices containing whole numbers

Today in class - my teacher was teaching determinants . He gave us problems to solve of various kinds , including various row - column operations and determinants properties. But one thing that ...
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votes
1answer
28 views

Orthogonality and inner product [on hold]

Let $A\in M_2(\mathbb{R})$ a positive definite matrix and the application $F:\mathbb{R}^2 \times \mathbb{R}^2\rightarrow \mathbb{R}$ $$F(x,y)=y^tAx$$ If ...
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0answers
15 views

Notation: column/row projection function for matrix-like objects

If we have a $n$-tuple $\mathscr x$ $$\mathscr x := (x_i)_{i\in n}=(x_0,x_1,\ldots,x_{n-1})\in \prod_{i\in n}X_i$$ where $(X_i)_{i\in n}$ is an indexed family of sets and $x_i\in X_i$. We can ...
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2answers
34 views

Nilpotent matrix similar to a matrix $[0,X]$ where $X$ is full column rank.

I am trying to prove that a nilpotent matrix $N$, which has a Jordan Form consisting only of blocks which are order 2 or greater, is always similar to a matrix $\begin{bmatrix}0 & X\end{bmatrix}$ ...
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1answer
26 views

Product of two multivariate Gaussian pdfs - normalization 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 $b$ ...
3
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1answer
36 views

Deducing that the inverse of a permutation matrix is its transpose

I would like to verify that my proof below is sound. Let $A\in P$ where $P$ is the set of all permutation matrices (only one 1 in each row and column). Also, let $(A)_{ij}$ denote the entry of $A$ in ...
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1answer
22 views

Show that a positive definite (not necessarily symmetric) matrix induces a hyperellipse

Consider $A\in M_n(\mathbb{R})$ a positive definite matrix and a matrix $B\in M_{n \times p}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. i) Show that $C=B^TAB$ is positive definite. ii) Show that ...
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votes
1answer
15 views

Null space and Matrix equations

http://studyguide.pk/Past%20Papers/CIE/International%20A%20And%20AS%20Level/9231%20-%20Further%20Mathematics/9231_s03_qp_1.pdf I would like to know the method to answer question 8. I have been having ...
1
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1answer
56 views

What is the difference between $A^{-1}$ and $A^\Theta$?

Let $A$ be a square invertible matrix. Then $$A \cdot A^{-1} = I$$. Let $A^\Theta$ be the conjugate transpose matrix of $A$. Then $$A \cdot A^\Theta = I$$. Both on multiplication with $A$ gives ...