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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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 ...
-1
votes
1answer
16 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
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 ...
1
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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
20 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 = ...
1
vote
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
1answer
22 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
80 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
34 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 ...
1
vote
5answers
80 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
24 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
43 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
348 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
19 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
44 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 ...
-8
votes
0answers
45 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
11 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
51 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
18 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
44 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
35 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 $u$.
0
votes
1answer
47 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 $z$?
1
vote
1answer
42 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
32 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
28 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
45 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
26 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
53 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
47 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 ...
-1
votes
1answer
20 views

Diagonal and anti-diagonal integral matrices: a special decomposition

Suppose we have a $3\times 3$ non-singular anti-diagonal matrix with integral entries: $A = \begin{pmatrix} 0 & 0 & a\\ 0 & b & 0\\ c & 0 & 0\end{pmatrix} , \,\,\,\,\,\,\,\, ...
0
votes
1answer
29 views

a question about linear algebra and matrix

Given a $n\times n$ matrix A,and the matrix's characteristic polynomial is $|\alpha I-A|=(\alpha-a_{1})^{r_{1}}(\alpha-a_{2})^{r_{2}}...(\alpha-a_{p})^{r_{p}}$,and $r_1+r_2+...r_p=n$. Then,as for any ...
1
vote
2answers
49 views

Determinant is the same for $A$ and $B$?

Let us have $A=(a_{ij})$ arbitrary matrix and for $B=(b_{ij})$ matrix we have $b_{ij}=(-1)^{i+j}a_{ij}$. Prove, that $det A = det B$. I tried with some examples, and the determinant is same, but how ...
0
votes
1answer
16 views

find transitive clouser of a matrix [on hold]

Find transitive closure of R if M_R is $$\begin{bmatrix}1 & 0 &0 \\0 & 1 & 1 \\1 & 0 & 1 \end{bmatrix}$$ I tried doing it like this M_R * M_R * M_R but could not get the ...
2
votes
2answers
32 views

“Simpler” geometrical description

So i was asked to find: Find the matrix that represents the linear transformation of the plane obtained by: reflecting in the line y = x, $\begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}$ then ...
-3
votes
1answer
21 views

Let $A \in {M_n}$ and $tr{A^2} = tr({A^*}A)$.why $A$ is hermitian?. [on hold]

Let $A \in {M_n}$ such as $\text{tr}(A^2) = \text{tr}({A^*}A)$. Why is $A$ hermitian?
0
votes
0answers
10 views

How to get the Riesz representative of the derivative of $L(K):=\text{tr}(\Lambda^* K A)$

$\DeclareMathOperator{\tr}{tr}K,\Lambda, A$ here are appropriate matrices. The question is not completely accurate as I can differentiate it, but I would prefer it to be in the form $⟨DL,h⟩$ for some ...
0
votes
0answers
28 views

Computations to omit from $M^TM$

I was asked to find M the the the corresponding matrix to the diagonal of S. M consists of S's eigenvectors. $$S= \begin{bmatrix} 1&2&0 \\ 2&2&1 \\ 0&1&-1 \end{bmatrix}$$ ...
1
vote
1answer
29 views

Three Simultaneously Diagonalizable Matrices

I have three symmetric square matrices $M$, $G$, and $S$ with the following properties: $S$: symmetric and positive semi-definite. $M$: Fully diagonal with positive entries. $G$: is a subset of ...
2
votes
2answers
21 views

Incremental algorithm for matrix eigenvalues

I try to solve the following problem: Given a stream of symmetric matrices $A_0, A_1, ...,A_n$ such that $A_i$ is different from $A_{i-1}$ only in one place, I want to compute the eigenvalues of ...
0
votes
1answer
42 views
+50

A problem with unitary matrices

Let $U$ be an $n \times n$ unitary matrix. And let $|\cdot |^2: \mathbb C^n \rightarrow \mathbb R_+^n$ be the function such that $|\mathbb w|^2$ is the vector which has its $i$-th entry equal to ...
-1
votes
0answers
25 views

What kind of matrices can be positive definite matrices [on hold]

Is every positive definite matrices is always expressible in form $A.A^{T}$ ?If not why?
0
votes
2answers
17 views

Similar matrices C and D: how to derive the relation $\mathbf{x} = S^{-1} \mathbf{y}$ when $C = S^{-1}DS$

D (with corresponding eigenvector $\mathbf{x}$) and C (with corresponding $\mathbf{y}$) are similar matrices, which means they have the same eigenvalues. So the relation $C = S^{-1}DS$ holds. So we ...
2
votes
2answers
59 views

Degree $2$ nilpotent matrices with non-zero product

Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and ...
2
votes
3answers
60 views

Find the inverse of a submatrix of a given matrix

I have a $A$ matrix $4 \times 4$. By delete the first row and first column of $A$, we have a matrix $B$ with sizes $3 \times 3$. Assume that I have the result of invertible A that denote $A^{-1}$ ...