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
29 views

$a_{ij}=i $ if $i+j=n+1$ and $0$ otherwise; compute det $A$

The entries of the matrix is specified by this rule, $A=(a_{ij})\in M_n(\mathbb R)$, $a_{ij}=i$ if $i+j=n+1$ and $0$ otherwise. Compute det $A$ > I have seen ...
0
votes
2answers
34 views

What is the relationship between matrix position to array index of corresponding matrix?

I want to know the exact relationship between matrix position and array index, where array contains the matrix data in each row appended format. For example: I had a matrix of $3 \times4$ as follows: ...
1
vote
2answers
205 views

Difference between dimension and rank of matrix

Let $A = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 3 & -1 & 1 \\1 & 5 & 3\\ \end{array} } \right]$ and $V$ be the vector space of all $X\in \mathbb{R^3}$ such that $AX = 0$. Then ...
4
votes
2answers
84 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: ...
0
votes
1answer
46 views

How to prove 2x2 rotation matrix is a manifold [duplicate]

How can I prove that this matrix is a manifold? $\begin{pmatrix} \cos(a) & -\sin(a) \\ \sin(a) & \cos(a) \end{pmatrix}$ Thanks!
2
votes
1answer
30 views

Condition number of a block matrix in terms of its submatrices

Given the matrix $$ B=\begin{bmatrix} I_{n} & A\\ A^{*} & I_{n} \end{bmatrix}, $$ where $\|A\|_2<1$. I need to show that the condition number of $B$ in the spectral norm is ...
0
votes
3answers
19 views

Is $Col(A)=Col(A^k)$? for positive semidefinite $A$?

$Col(A)$ denotes the column space of $A$ I was able to show this for $k=2$, but having a hard time showing it for other integer $k$. Any ideas would be appreciated!
1
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2answers
39 views

A projection $P$ is orthogonal if and only if its spectral norm is 1

I have to show what the title says. A projection $P$ is orthogonal if and only if its spectral norm is $1$. I suppose I have to use the following identity: ...
1
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0answers
28 views

Why does $det(R) = +1$ imply right handed frame?

Let $R$ be a rotational matrix in $SO(3)$ so it satisfies $R^TR = I$ Solvng for $det(R^TR) = (det(R))^2 = 1$ yields two solutions Why does $det(R) = +1$ mean that the frame is a right handed frame? ...
-4
votes
0answers
17 views

Canonical forms, Linear algebra [closed]

Let $N$ be a $2\times 2$ complex matrix such that $N^{2}=0$. Prove that either $N=0 $ or $N$ is similar over $% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $ $\ $to $% ...
1
vote
1answer
44 views

Hessian to show convexity - check my approach please

I need to check the convexity of $f(x)$ for these two questions, using the Hessian matrix. I am aware the function can be said to be convex if over the domain of $f$ the hessian is defined and is ...
2
votes
2answers
28 views

Why is the similar of a triangular matrix unipotent

If $ A = BDB^{-1} $, $B \in Gl_n(K)$ and $ D = (d_{ij}) $ an upper triangular matrix with 1 on the diagonal line. Show that A is unipotent, using the definition that a matrix A is unipotent if there ...
1
vote
2answers
33 views

Question about dimension of nullspace of two similar matrices

There is an interesting problem I found on an old algebra prelim and it has to do with linear algebra. I want to know how I should approach it. The question is let $A,B \in M_n(F)$, where $F$ is a ...
1
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0answers
41 views

Principal matrix, Ramsey theorem

Question: Let m be given. Show that if n is large enough, then every n-by-n 0, 1-matrix has a principal submatrix of size m in which all elements above the diagonal are the same, and all elements ...
0
votes
1answer
26 views

How to quickly compute the inverse of 3x3 matrix that only has non-zero value on diagonals?

Does anyone know if there is a short cut to compute the inverse of the matrices of the form $\begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix}$ and ...
0
votes
1answer
13 views

Determining parameters for static system

So i have to determine parameters for static system: $y=o1 + o2u$ So, my idea was to pick mesurments for witch $\det[]!=0$ (2 of them as L=1 and R=2) Parameters :$n=1 u=-2 y=9$ and $n=2 u=1 y=-2$ ...
0
votes
2answers
8 views

Matrix location by indices?

The formula for location number by successive rows for an $n$ by $m$ matrix is $f(x,y)=m(x-1)+y$ That is, for example a matrix with $n=3, m=4$ \begin{bmatrix} 1 & 2 & 3 & 4\\ ...
0
votes
0answers
14 views

Symbolic expression of eigenvalues for this symmetry 3x3 matrix

Can anyone suggest if the analytical expressions of the eigenvalues for this symmetry real matrix $L$ exist or not? All variables are real. $$\begin{align} g_{11}&= ...
0
votes
0answers
67 views

Can the transpose of a matrix times itself reveal the original matrix

If I only have $AA^T$ and I know that $A$ has the dimensions $m \times n$ where $m \gg n$, is there a way to get the matrix $A$? Also, what if it was $A^TA$. Would I be able to get $A$ out of $A^TA$? ...
2
votes
2answers
22 views

Compact subset of space of matrices and compactness verification of a set of eigenvalues

Let $M_n(\mathbb R)$ be the vector space of real matrices of size $n$ , identified with $\mathbb R^{n^2}$ ; let $X \subseteq M_n( \mathbb R)$ be a compact set ; let $S \subseteq \mathbb C$ be the set ...
1
vote
1answer
48 views

Which of the following cannot be the value of $g(x)$

Let A = $\begin{bmatrix}1 & \tan x\\-\tan x & 1\end{bmatrix}$ then let us define a function $f(x)=\begin{vmatrix}A^{T}A^{-1}\end{vmatrix}$ then which of the following cannot be the value of ...
0
votes
1answer
13 views

Finding the intersection between 2 lines using matrices

My professor uploaded some notes, and there's a step in his explanation of a Linear Programming Problem which I do not understand. He takes 2 lines and converts them into matrices in order to find the ...
0
votes
1answer
16 views

Covariance matrix of linear transformation of eigenvector matrix

Let $K_t=E'R_t$ where $E$ is a matrix with the eigenvectors of the covariance matrix of $R_t$. According to my book, then the following holds: ...
0
votes
0answers
10 views

The number of scalar additions required to compute P(QR)

Let P, Q, R be matrices of order $3\times5, 5\times7$ and $7\times3$ respectively. What is the number of scalar additions requiered to compute P(QR)? Do we have any formula to compute this? If we had ...
2
votes
1answer
56 views

Can someone direct me to prove the following three claims about a rotation group $SO(3)$ and its Lie Algebra?

In class my prof made three claims about a group and its Lie algebra. I cannot find direct reference to these claims because they are delivered in verbatim (im not even sure if I have them jogged down ...
2
votes
2answers
55 views

Singular matrix geometric sum

What is a fast way to calculate the sum $M + M^2+M^3+M^4+\cdots+M^n$, where $M$ is an $n \times n$ matrix whose cells are either $0$ or $1$? I have researched an alternative way which makes use of ...
0
votes
0answers
22 views

Proving subgroups of the unitary group U(n) are Lie groups

The problem: Let $K$ be a complex $n \times n$ matrix such that $K$ and $K^\dagger$ is non-singular and $K + K^\dagger$ is positive definite. Show that the set of complex $n \times n$ matrices $G$ ...
0
votes
0answers
27 views

Prove Faculty Determinant

so I got this following task from my professor: n element N Matrix An = (aij) element M^nxn(K) with aij = 1 if i = j -j if i = j + 1 i if i = j - 1 0 for the rest I must determine the ...
1
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4answers
55 views

Given A is a nil-potent matrix (given $ A^k=0 $), prove that A-I is invertible. Is my proof correct?

Given $A$ is a nil-potent matrix (given $A^k=0$), prove that A-I is invertible. I have proved the statement using contradiction, and I want to know if it is correct: Let $ A-I \neq I.$ ...
1
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2answers
53 views

Inverse of a unipotent matrix

Show that all unipotent matrices are invertible. Also, specify a formula for the inverse of a unipotent matrix. Now, I've tried to approach the problem using the determinant: a matrix is unipotent, ...
1
vote
1answer
45 views

Finding the eigenvector of a matrix $2\times 2$ in two ways?

I need some help. I dont understand why the eigenvector $v_1$ of the matrix \begin{pmatrix} 0 & 1 \\ -1 & -2 \end{pmatrix} is the vector $(1,-1)$ and no the vector $(-1,1)$. $\lambda_1 = ...
4
votes
3answers
161 views

Nipotent matrix over a ring

This question is linked to this one: nilpotent elements of $M_2(\mathbb{R})$, $M_2(\mathbb{Z}/4\mathbb{Z})$ Let $R$ be a commutative ring with unity and let $A\in M_2(R)$. Show that $A$ is a ...
0
votes
0answers
31 views

Proving Determinants with induction

so I got this following task from my professor: n element N Matrix An = (aij) element M^nxn(K) with aij = 1 if i = j -j if i = j + 1 i if i = j - 1 0 I ...
2
votes
0answers
15 views

Solving a structured partitioned linear system

I am trying to solve the following partitioned linear system, where each letter represents a block $\begin{pmatrix}-H & A^T & I_n \\ A & 0_1 & 0_2 \\ z_D & 0_2^T & ...
3
votes
2answers
84 views

For which values of $a$ the matrix is diagonalizable

Given the following matrix: $$B=\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & a^2 \\ 1 & 1 & 0 \end{bmatrix}$$ I tried to find for which values of $a$, the matrix $B$ is diagonalizable. ...
0
votes
1answer
27 views

solve a system of equations by matrices

Is there any way to solve the system of linear equations by matrices ?? for example: $$\left\{\begin{aligned} 2x+3y-z&=p \\ px+(p+1)y-pz&=1 \\ x+y+pz&=2 \end{aligned} \right. $$ Thank ...
1
vote
2answers
44 views

Does it always hold, that the product of the eigenvalues of a matrix is it's determinant?

I know that this relation holds in several cases but I'm not sure about the full scope of it. So are there any cases when it doesn't hold?
0
votes
1answer
16 views

Projection and trace of the matrix

For a given vector: $$ x= \begin{pmatrix} 3e^{j^{45^\circ}}\\ 2-3j \end{pmatrix} $$ I need to find projection matrix $P$ of $\hat x$ then trace of the matrix $tr P$ Could anyone advice?
4
votes
1answer
41 views

Does PSD imply on average diagonal dominant?

Suppose $A$ is a $N \times N$ positive semidefinite matrix. This does not necessarily imply that $A$ is diagonally dominant. But does it imply the following "average diagonal dominance" i.e. ...
0
votes
0answers
22 views

Important question about calculating determinants

so I was browsing through StackExchange in the hope to find some inspirations for my tasks from university and I came across this one which should help me: Calculate the determinants of the following ...
8
votes
1answer
118 views

nilpotent elements of $M_2(\mathbb{R})$, $M_2(\mathbb{Z}/4\mathbb{Z})$

Let $R$ be a commutative ring. We define $\mathfrak{N}(R)$ to be the set of nilpotent elements in $R$. Find $\mathfrak{N}(R)$ for: $R = M_2(\mathbb{R})$ $R = M_2(\mathbb{Z}/4\mathbb{Z})$
2
votes
4answers
95 views

Why is positive (semi-)definite only defined for symmetric matrices?

When we are defining positive (semi-)definite matrices, we do so for symmetric matrices only. Why do we need symmetry in the definition?
1
vote
1answer
45 views

Expressing $C(x) = \tilde{x} = (\langle x,a_i \rangle )$ as a product of matrices in the form $Cx = \tilde{x}$

Le that $(a_i)^{n}_{i=1}$ be an orthonormal basis and $C(x)$ be a transformation defined as follows: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k ...
1
vote
2answers
63 views

All matrices which commute with all $2\times 2$ matrices

I would like to find all matrices which commute with all $2\times2$ matrices. I started solving problem in this way: 1) I have this matrix $A$ with real numbers: $$A=\left[\begin{array}{cc}a ...
1
vote
0answers
41 views

Is there an interesting interpretation of the ROWS of an affine transform matrix?

Context: I have a question about affine transform matrices in 3-space. Matrices are 4x4, with the right-most column being the translation, and the bottom row being [0,0,0,1]. In discussions I read ...
0
votes
1answer
17 views

Geometric interpretation of 2D-Translation's Matrix representation

I just learned the trick of writing a translation of a 2-dimensional real vector as a matrix multiplication in a 3-dimensional space - wikipedia explains it here. Basically it shows: $$ ...
1
vote
1answer
27 views

Matrix Inversion acceptable Condition Numbers

When considering matrix inversion it is worth while worrying about the condition number of the matrix you wish to invert. Matrices that are poorly conditioned can often create inaccurate results. This ...
0
votes
1answer
18 views

Can the Kernel of the commutator of two matrices with empty Kernel sets be non-empty?

The motivation for this question arises from the following: Is it possible, given two Quantum Mechanical observables $A$ and $B$ with associated operators $\hat{\mathbf{A}}$ and $\hat{\mathbf{B}}$ ...
3
votes
3answers
43 views

If $f_A(x) \ne m_A(x)$ and $A^3=I$ then $A=I$?

Suppose $A \in M_{3\times3}(\mathbb R)$ and $f_A(x) \ne m_A(x)$ where $f_A(x)$ is the characteristic polynomial of $A$ and $m_A(x)$ is the minimal polynomial of $A$. If we were to assume that ...
1
vote
0answers
20 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...