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
17 views

Converting non square matrix into square matrix [closed]

Is it possible to convert non square matrix into square matrix? if so, how to convert?
2
votes
1answer
55 views

Number of ways to place exactly two kings in each column such that no king attacks another

A regular King in a chess board can attack all its adjacent 8 cells (vertical, horizontal or diagonal). Now you are given a $10 \times n$ chessboard, your task is to place exactly two kings in each ...
1
vote
1answer
43 views

Is $x^3$ in the null space of the transformation $p(x) \mapsto xp(x)$?

Let $h: P_3 \to P_4$ be given by $p(x) \mapsto xp(x)$. Is $x^3$ in the null space ? Or is it in the range space ? Also, I am having difficulty finding the null space and the range of this map, can ...
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votes
2answers
15 views

Solving systems of equations using matrices by row reduction

Solve the following system for $a$, $b$, and $c$: $$\begin{pmatrix}1 & -1 & 2\\2 & -2 & 2 \\ 3 & -3 & 2\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix} = ...
1
vote
2answers
39 views

How to solve a linear system in matrix form using Laplace transform?

How to solve this linear system using Laplace transform? $$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf ...
0
votes
1answer
39 views

Can a matrix have more than one inverse (Singular Value Decomposition)

Assume there's a matrix $A$ with SVD as below $$ A = U \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} ...
0
votes
1answer
44 views

fundamental matrix solution

$$ \frac{dy}{dt}=\left[\begin{array}{ccc} 5&1&1\\ 1&5&1\\ 1&1&5 \end{array}\right]$$ I need to solve this problem but my answers are still uncorrect.
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3answers
56 views

Symmetric Matrix Transformation

Here's the question, Let $T$ be the transformation of 2 by 2 real symmetric matrices defined by: \begin{bmatrix}a&b\\b&c\end{bmatrix}>>>>\begin{bmatrix}c&-b\\-b&a\end{bmatrix} ...
1
vote
1answer
54 views

Odd order n smaller than 27

I have a group $G$ that is a group of matrices of the form $$\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right)$$ where $a,b,c \in \Bbb Z_3$. ...
2
votes
1answer
23 views

Orthogonal decomposition of matrices

Let $M_{i,j}$ be a matrix of the standard basis of size $n>1$ (the coefficients are $1$ at position $i,j$ and $0$ elsewhere) What is the least number of orthogonal matrices required so that ...
0
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3answers
45 views

Approximate matrix by a rank 2 matrix using singular values

I only understand the singular value decomposition process. Do I apply it to this matrix? \begin{bmatrix} 0 & 0 & \pi \\ 0 & e & 0 \\ 1&0&0 \end{bmatrix} What is the idea ...
0
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1answer
18 views

Matrix Factorization of lower triangular and unit upper triangular.

So I have a matrix. $$A =\begin{bmatrix} 8 && -3 && 2 && -1\\ -3 && 8 && 0 && 2\\ 2 && 0 && 8 && -3\\ ...
0
votes
0answers
7 views

mat3x3 for orthographic projection [closed]

Is it possible to have 3x3 matrix, instead of 4x4 for orthographic projection? Assuming all my vectors 2d, and I don't use "projection offset" (left=0, bottom =0). Here is code from glm lib: ...
1
vote
1answer
39 views

Prove or disprove the statement: if all the eigenvalues of a matrix are 0, then the matrix must be the zero matrix?

Prove or disprove the statement: If all the eigenvalues of a matrix are $0$, then the matrix must be the zero matrix. What I know : If the matrix is a upper or lower triangle matrix with the ...
0
votes
0answers
5 views

Extremal singular values of $P\Phi D$

Let $A=P\Phi D$ be a matrix where $P$ is a projection matrix such that $R(P)\subset R(\Phi)$ and $D$ is a non-singular diagonal matrix. Is there any relation between $\sigma_{min}(A)$ and ...
2
votes
2answers
20 views

Let $A$ be a non-zero linear transformation on a real vector space $V$ of dimension $n$.

Let the subspace $V_o \subset V$ be the image of $V$ under $A$. Let $k = \dim (V_o) \lt n$ and suppose that for some $\lambda \in \mathbb{R}$, $A^2 = \lambda A$. Then which of the following are true? ...
2
votes
3answers
107 views

What is $\Bbb{R}^n$?

I earlier asked this question The basis of a matrix representation. I now have a another question related to the same topic. The vector space $\Bbb{R}^n$ I have seen defined as all $n$-tuples of real ...
-2
votes
3answers
38 views

what is the smallest number $n\in \mathbb N$ such that $A^n=I$? [closed]

Let $A$ be a $2\times 2$ matrix consisting of $$A = \begin{pmatrix}\sin(\pi/18)&-\sin(4\pi/9)\\ \sin(4\pi/9)& \sin (\pi/18)\end{pmatrix}$$ what is the smallest number $n\in \mathbb N$ such ...
1
vote
1answer
19 views

Replaces data in a matrix with Matlab

A is a matrix (6 ×6) defined as: A= [0.25 0.35 0.46 0.56 0.67 0.78; 0.25 0.37 0.49 0.61 0.73 0.86; 0.25 0.38 0.52 0.66 1.80 1.94; 0.25 0.40 ...
1
vote
1answer
18 views

Derivative of vector valued function

The following is given for $ ∂x^TAx/∂x $ in a book on Matrix Algebra: What I cannot understand is: Where does $A^T$ come from in the second row (in the term $ ty^TA^Tx $)?
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2answers
14 views

Variable intervals from system of inequalities

I have this system of inequalities: and I need to find possible intervals of i and j. Looking at the graph output from ...
-2
votes
0answers
45 views

$A$ be a $10\times 10$ matrix over $\mathbb R$ such that sum of each row is $1$. [closed]

Let $A$ be an invertible $10\times10$ matrix over $\mathbb R$ such that sum of each row is $1.$ Then which option is correct? A. The sum of the entries of each row of the inverse of $A$ is ...
0
votes
1answer
18 views

Finding a base field for diagonalized linear transformation and justifying

So I encountered this long question that asks you to find bases and such, I searched up Find the eigenvalues for the linear transformation and base associated to each eigenvalue. If possible find ...
2
votes
2answers
174 views

How can I rewrite this expression?

$A=\begin{pmatrix}3&-1\\-1&1\end{pmatrix}$; $U_\phi=\begin{pmatrix}\cos\phi&-\sin\phi\\\sin\phi&\cos\phi\end{pmatrix}$; ...
3
votes
2answers
63 views

The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
0
votes
2answers
49 views

Solution to a system of linear equations with an unknown matrix product

Consider the system of equations $$ Xy=Ab $$ where $X$ and $A$ are $m \times m$ invertible matrices and $y$ and $b$ are $m \times n$ matrices. The matrices $X$ and $y$ are unknown and the matrices $A$ ...
2
votes
1answer
48 views

Interlacing of eigenvalues for Hermitian matrices

This is a problem from Matrix Analysis by Horn and Johnson. Let $A \in M_n$ be Hermitian, let $a_k$$=$det$A$[{$1$, $\dots$,$k$}] be the leading principal minor of $A$ of size $k$, $k = 1, \dots, n$, ...
3
votes
2answers
68 views

Problem 11 Section 2.6 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Let $X$ be the vector space of all complex $n \times n$ matrices and define $T \colon X \to X$ by $Tx \colon= bx$, where $b \in X$ is fixed and $bx$ denotes the usual product of matrices. I know that ...
0
votes
1answer
35 views

Linear maps that are matrices

If I have the linear map $A:\Bbb{R}^3\rightarrow \Bbb{R}^3$ where $A$ is a matrix. Is the matrix $A$ (along with the vectors it operates on) in a basis or not? I think it is not, since the vectors it ...
0
votes
1answer
18 views

iterated matrix multiplication in scilab\matlab [closed]

Please help me with my code. I can't write the iteration in codes ...
1
vote
0answers
33 views

Given similar matrices $A$ and $B$, how to find $M$ such that $B=M^{-1}AM$?

I am trying to teach myself linear algebra using Strang's Introduction to Linear Algebra. I would like to know what the most (or more) efficient way to solve this problem is by hand. The question: ...
4
votes
5answers
123 views

Find the determinant of a matrix definition [duplicate]

Let $A$ be a matrix that is defined like this: $$A_{ij}=\begin{cases} \alpha, & \text{if i=j} \\ \beta , & \text{if i $\ne$ j} \end{cases} $$ So I realized this matrix looks somehow like ...
2
votes
2answers
157 views

Easy way to calculate the determinant of a big matrix?

Given this matrix: \begin{matrix} 2 & 3 & 0 & 9 & 0 & 1 & 0 & 1 & 1 & 2 & 1 \\ 1 & 1 & 0 & 3 & 0 & 0 & 0 & 9 & 2 & 3 & ...
1
vote
3answers
42 views

Find the dimension of the space of $4\times 4$ real matrices with zero trace

I'm wondering if someone can help me to understand this problem. If $S$ is the subspace of $M_{4,4}(\mathbb{R})$ consisting of all matrices with trace $0$, what is $\dim(S)$? I've created a matrix ...
-1
votes
1answer
37 views

eigenvalue problem [closed]

Prove the following statement: $\max_i{\lambda_i(A^TA)}=\max_i{\left|\lambda_i(A^T)\right|^2}$ where matrix A is a N-by-N circulant matrix and $\lambda_i(X)$ denotes the $i$-th eigenvalue of matrix ...
1
vote
1answer
42 views

What is the linear space of Eigenvectors associated with a certain Eigenvalue?

The following matrix $A$ has $\lambda=2$ and $\lambda=8$ as its eigenvalues $$ A = \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{bmatrix}$$ let $P$ be the ...
-1
votes
2answers
59 views

Eigenvalues of a nilpotent matrix can only be $0$ [duplicate]

Prove that the eigenvalues for a square Nilpotent matrix A can only be $0$. Definition of nilpotent A $^n$=$0$ n is a positive whole integer
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votes
2answers
65 views

Under what assumptions is it correct to say “a matrix is diagonalizable if and only if its eigenvalues are real”?

A $2\times 2$ matrix is diagonalizable if and only if its eigenvalues are real. Which statement is most correct: The proposition is true only if the eigenvalues are all greater than zero. The ...
0
votes
1answer
21 views

Algorithm for the Hill cipher (finding the inverse of the determinant of a $2 \times 2$ matrix modulo $26$)

I have a good understanding of how to do the Hill cipher on paper but putting it into program form is somewhat of a problem. Finding the the determinant is the thing I'm having problem with. On ...
0
votes
1answer
7 views

Converting nth order ODE with RHS into system of 1st order ODEs

I looked at these two questions, but they weren't directly relevant to my specific question: How to reduce higher order linear ODE to a system of first order ODE? Express differential equations as ...
0
votes
0answers
34 views

Matrix Proof relating Unitary and Hermitian Matrices

I am currently reading through my lecture notes and the following situation is given. Suppose that a ($n$ by $n$) unitary matrix $U$ can be written as $U=M+iN$ where $M$ and $N$ are Hermitian ...
0
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1answer
16 views

M22 → R Matrix Transformation Kernel

For a transformation such as this, how does one determine the form of the kernel? Is it simply making the right side equal to zero, solving for each individual variable, and then creating a matrix ...
0
votes
0answers
55 views

Finding complex eigenvectors of $n \times n$ matrix, $n\geq 3$

An example: $$ \begin{pmatrix} 1 & 2 & 0 \\ 2 & -3 & 4 \\ 4 & -8 & 7 \\ \end{pmatrix} $$ Has eigenvalues $3$, $1+2i$, $1-2i$. How ...
0
votes
1answer
28 views

If $T^k = Id$ for $k\ge 1$ then $T$ is diagonalizable [duplicate]

Let $V$ a finite dimension space over $\mathbb{C}$ and $T:V\to V$, a linear transformation such that $T^k = Id$ for $k\ge 1$. Prove that $T$ is diagonalizable. I'd be glad for an hint. How do I ...
0
votes
1answer
39 views

Is my proof correct that function is a bijection iff matrix is invertible?

For given $B\in \mathbb{C}^{n\times n}$ let's define a function $$f: \mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$$ so that $$f(A) = B^HAB$$ I have to prove that f is a bijection iff B is ...
4
votes
1answer
61 views

is this kind of symmetric matrix invertible?

Give a matrix $A=\begin{bmatrix}M&B\\ B^T&0\end{bmatrix}$, where $M\in\mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}, (m<n)$. If we know that $rank(B)=m$ and for any $v\neq 0$ and ...
0
votes
3answers
51 views

Can a matrix transformation ever make a linearly dependent matrix linearly independent?

I'm curious. Can ANY matrix transformation make some matrix with its columns linearly independent, or with an empty kernel, linearly independent? For example, if A is a linearly dependent matrix, and ...
0
votes
2answers
29 views

Inverse of Matrix?

What is the inverse of the following matrix ? Give a general formula for calculating the such matrix of dimension n-by-n. Grateful.
0
votes
1answer
20 views

Can anyone help me with “rotation matrix” and “Image of matrix”?

If A is a 3 by 3 matrix which gives a rotation about some line through the origin in R^3 , then columns of A form a basis of R^3 For any matrix A, the image of A^7 is contained in the image of A ...
4
votes
0answers
96 views

How does pointwise multiplication of two matrices affect their eigenvectors?

More specifically, suppose I have a known matrix $X\in\mathbb{R}^{d\times n}$ and an unkown vector $\alpha \in \mathbb{R}^n$. What can be said about the eigenvectors of $\alpha\alpha^T \odot X^T X$ ...