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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0
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0answers
9 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 $M_2(F_p)$ ...
1
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1answer
19 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 ...
0
votes
0answers
14 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 = ...
1
vote
0answers
5 views

Inverse properties of l_1 normed matrices

Let's take the adjacence matrix $A$ of a graph $G$. We call $\bar{A}$ the row $L_1$ normalized matrix obtained from $A$. Let's take some $\alpha \epsilon [0,1)$. $(I-\alpha\bar{A})$ is strongly ...
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2answers
25 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| ...
0
votes
1answer
22 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 $P_2\in M_m(\mathbb{R})$ ...
2
votes
3answers
102 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
29 views

Fill in the missing entries of matrix $Q$ to make it orthogonal

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 ...
3
votes
0answers
16 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 ...
-2
votes
1answer
22 views

Orthogonality and inner product

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
14 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 ...
0
votes
2answers
32 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 $\begin{bmatrix}0 & X\end{bmatrix}$ ...
0
votes
1answer
22 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
votes
1answer
34 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
19 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
vote
1answer
55 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 ...
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0answers
24 views

Proving that a linear functional is matrix trace [duplicate]

Let $W=\operatorname{M}_{n\times n}(\mathbb{F})$ (square matrices $n\times n$ over $\mathbb{F}$), and $f\in W^*$. If $f(AB)=f(BA)$ for every $A,B\in W$ and $f(I)=n$ prove that ...
2
votes
0answers
19 views

Change of basis and similarity

Consider the transformation $T$ in the standard basis: $$[T]_B\begin{bmatrix} 0&3&1 \\ -1&3&1 \\ 0&1&1 \end{bmatrix}$$ Also consider the two matrices: $$A_1 = ...
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0answers
11 views

eigenvector perturbation

In the proof of Theorem 1 of (http://ai.stanford.edu/~ang/papers/ijcai01-linkanalysis.pdf), the authors cite a theorem from Steward and Sun (Theorem V.2.8) which states that if $S$ is symmetric and ...
0
votes
0answers
15 views

LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
0
votes
1answer
21 views

Which geometric operations are encoded by symmetric, positive definite matrices?

Maybe it's because I'm German and used different terminology in the past, but somehow I don't really understand what is meant by the question in the title. Didn't change the wording, just copied it. ...
0
votes
1answer
40 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
0
votes
2answers
33 views

An orthogonal projection matrix in $ \Bbb{R}^{3} $.

Consider the vector space $\mathbb{R^3}$ with usual inner product. Find the orthogonal projection matrix on the xy plane. I've found sometimes the orthogonal projection of a vector in a given ...
1
vote
3answers
29 views

Linear algebra, inner product and matrix

Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric ...
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5answers
76 views

Product of any two arbitrary positive definite matrices is positive definite or NOT? [duplicate]

Suppose that , $A$ and $B$ are $n\times n$ positive definite matrices and > $I$ be $n\times n$ identity matrix. Then which of the followings are positive definite ? (i) $A+B$ (ii) $ABA$ ...
0
votes
1answer
23 views

Inconsistent Matrices

I'm teaching myself Linear Algebra and am not sure how to approach this problem: Let A be a 4×4 matrix, and let b and c be two vectors in R4. We are told that the system Ax = b is inconsistent. ...
4
votes
1answer
41 views

Did I do something wrong solving this PDE in MATLAB?

I have the following PDE problem on a practice exam: I have completed the problem using MATLAB to the best of my ability. Here is the code I used ...
5
votes
4answers
344 views

Am I misinterpreting this matrix determinant property?

I was reading matrix determinant properties from wikipedia. The property reads $\det(cA) = c^n \det(A)$ for $n \times n$ matrix. However I am not able to realize it. What I find is $\det(cA) = ...
0
votes
1answer
17 views

Finite differences matrix and integrals

Let $f:[a,b]\to \mathbb{R}$ a smooth function. Consider a partition $a=x_1<x_2<\ldots<x_n=b$. If we put $X=(f(x_1), f(x_2), \ldots, f(x_n))$, where $x_{i+1}-x_i=\Delta x$ then: $ (f'(x_1), ...
0
votes
1answer
39 views

Overdetermined linear system solutions proof

Let $A\in M_{m\times n}(R)$ with $m>n$. Consider that the only solution of the linear homogeneous system $Ax=0_{R^m}$ is the trivial solution $x=0_{R^n}$. Show that linear system $A^ty= b$ have ...
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votes
0answers
44 views

Which of these are true? And why? [on hold]

Which of these are true? And why are they true, please answer this too, if possible?
0
votes
1answer
10 views

Is there any shortcuts in getting an H-infinity norm of a matrix expression?

One of the past exam problems I was solving, has this in its official solution: Usually, to calculate the $H_{\infty}$ norm of any matrix expression $M$ I'd first calculate the eigenvalues of ...
0
votes
2answers
47 views

Propositions of elementary matrix

i'm trying to solve a question about elementary matrix. When given $A_{m,n}$ and $B_{n,p}$ which differ from the Zero matrix. Also, multiplying of $A$ and $B$ is the zero matrix, that is: $AB=0$; ...
3
votes
1answer
50 views

Verifying whether a number is the determinant of a matrix

What is the (computationally) fastest way to determine whether a number is the determinant of a given real matrix? I am wondering if I have an upper bound on the absolute value of the determinant of ...
1
vote
1answer
17 views

Degree $2$ nilpotent matrices with non-zero product and subset products not depending on the order

Follow up to this answered question. Let $n$ be sufficiently large positive integer. Let $S=\{M_i\}$ be a set of $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/m \mathbb{Z}$. ...
1
vote
0answers
17 views

How to find the determinant of a parallelogram using the vertices. (Using a matrix)

What it says on the tin. I already know how to find the determinant of a parallelogram using the vector components in a matrix, however, I am curious if there is a way to do it simply through the ...
3
votes
2answers
39 views

Prove that matrix is symmetric and positive definite given the fact that $A+iB$ is.

I have some questions regarding the following problem Let $ A + iB $ - hermitian and positive definite, where $A, B \in \mathbb R^{n\ \times\ n} $ show that the real matrix $$C =\begin{pmatrix} A ...
1
vote
1answer
32 views

Relation between Hermitian Matrices

For a given Hermitian matrix $A$, what is the semidefinite positive matrix $B$ such that $$\mathbf{y}^{H}B\mathbf{y} = \left | \mathbf{y}^{H}A\mathbf{y} \right |, $$ for all $\mathbf{y} \in ...
0
votes
1answer
43 views

Abstract Linear Algebra Inner Product [on hold]

Let $u\in\mathbb{R}^n$ be a vector such that $\|u\|=1$ (for the usual inner product). Prove that there exists an $n\times n$ orthogonal matrix whose first row is $u$.
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1answer
39 views

Eigenvalue of $B=uv^\text{T}+wz^\text{T}$

We have $u,v,w,z \in R^\text{n}$, how can we express the eigenvectors and eigenvalues of $B=uv^\text{T}+wz^\text{T}$ by analyzing over $u,v,w$ and $z$?
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vote
1answer
40 views

$A$ is the set of all $n \times n$ matrices where $\operatorname{tr}(A)=0$, is $A$ a subspace of $M_{nn}$ (where $n\ge2$)?

$\newcommand{\tr}{\operatorname{tr}}$For $A =$ zero matrix, $$W=\{ A \in M_{nn} : \tr(A) = 0 \}$$ I can proof that the set of all n x n matrices A with $\tr(A)=0$ is a subset of $M_{nn} $for$ \ n \geq ...
2
votes
1answer
42 views

Do all singular $n\times n$ matrices form a vector subspace when $n\ge2$?

I want to prove or disprove that the set of all $n\times n$ singular matrices form a vector subspace of $M_{nn}$ when $n\geq 2$. So, let: $$ A_{n,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & ...
1
vote
0answers
31 views

Is there a method to measure the similarity between undirected graph vertices?

I'm doing some research on User Identity Resolution. Assume i can get two undirected graphs of a person, one is the friendship in Twitter of that person, the other ...
1
vote
0answers
24 views

a question about prove an exponential matrix function can be infinitely differentiable

If I have an exponential matrix exp(t(U+sH)), can someone tell me what is the dirivative with respect to s? I am really confused. (where U and H are matrices,and s,t are real numbers). Thus,if I let ...
2
votes
2answers
44 views

Prove that the eigenvectors of this matrix are a basis in $\mathbb{R}^n$

Let $A \in \mathbb{R}^{n \times n}$ and $w \in \mathbb{R}^n$. Suppose that, $w_i>0$ and $a_{i,j} = w_i / w_j$ for all $i,j=1,\dots,n$. Note that from the construction comes that ...
0
votes
1answer
17 views

Prove that the columns of the similarity matrix of a diagonalization are the eigenvectors

I'm interested in eigendecomposition of a matrix. It is clear for me, that you can eigendecompose a matrix if and only if it is diagonalizable. But I don't know how to prove, that in the similarity ...
0
votes
1answer
23 views

Vector notation for shifting the elements of a vector

I'm looking for a suitable notation to express a "shift operator" which shifts all elements of a vector forward and sets the first element to zero, e.g., $$\begin{eqnarray*} (1,1,0,1) & ...
2
votes
1answer
52 views

Why do $n$ linearly independent vectors span $\mathbb{R}^{n}$?

Suppose we have $n$ linearly independent vectors $\mathbf{v}_{1}\ldots\mathbf{v}_{n}$ in $\mathbb{R}^{n}$. I know that they do span $\mathbb{R}^{n}$, because we can easily specify a non-singular map ...
1
vote
3answers
45 views

Is a matrix with complex entries invertable?

This is merely a question of interest and not for something I am doing in school. I have never seen a matrix with complex entries in class before, but mind you it was a limited linear algebra class, I ...