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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3answers
37 views

Is it possible to solve for values in a matrix such that all rows and columns have equal sum?

Is it possible to solve for values in a grid such that all rows have the same sum and all columns have the same sum where values in the table can be any real number? meaning: ...
1
vote
2answers
57 views

If I know that a matrix $G = (X^{T}X)^{-1}$, how can I recover what $X$ is?

If I have a matrix $G$ where I know that $G=(X^{T}X)^{-1}$, is there a way to find $X$? Specifically, I would like to find $G$ where $G$ is: $$G = \begin{bmatrix} 0.125 & 0 & 0 & 0 & ...
0
votes
1answer
18 views

exponential of elementary matrix $\exp(tE_{a,b})$

$E_{a,b}$ is the elementary $n\times n$ matrix with $1$ in $(a,b)$-entry and $0$ elsewhere. Compute $\exp(tE_{a,b})$ for $a$ not equal to $b$. If $a=b$ then they would be on the diagonal, so ...
0
votes
0answers
29 views

Possible largest number of column vectors with certain structure in a rank r matrix

My question is: If $A$ is a dimension $p$ symmetric square matrix with rank $r$ ($r<p$), and $a_{ij}$ is the element in the $i$-th row and $j$-th column. How many column vectors can satisfy ...
1
vote
2answers
83 views

How to take the inverse of the matrix $X^{T}X$, when it isn't invertible?

If I have a matrix $X$ and I am trying to compute $(X^{T}X)^{-1}$, which is the inverse of $X^{T}X$. However, each time I try to do it in some computing package like R, I get that $X^{T}X$ is ...
0
votes
2answers
36 views

Computing the inverse of $I - \lambda E$ where $E^{k+1} = 0 $ for some $k \geq 1$

If $E$ is a square matrix over $\mathbb{C}$ with $E^{k+1} = 0$ for some $k \geq 1$, then show that $I - \lambda E$ is invertible for all $\lambda \in \mathbb{C}$ by explicitly computing its ...
1
vote
1answer
40 views

$Z$ coordinates disappear in the general rotation transformation matrix.

I wanted to generate the general rotation transformation matrix ($3D$). But when I did the multiplication the result didn't include the original $Z$ coordinates,I don't know why the $Z$ disappeared. ...
0
votes
1answer
22 views

How to determine which of the 6 columns of this matrix are not linearly independent when combing with the rest?

I currently have a matrix $G$ with $6$ columns from a simulation that looks like: $$\begin{bmatrix}{} 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 & 0.0 ...
1
vote
1answer
22 views

Accelerating linear solve in MATLAB for a specific type of matrices

Inside a DG solver (so far 1D) I need to solve a linear system of equations multiple times. The order of the system is rather small ($N=10..20$). I need to solve the system $Ax=b$, where $A$ is the ...
1
vote
1answer
24 views

How to prove that determinant can take any real value using only this definition of the determinant?

I was reading some facts about the determinant and refreshed my memory with the fact that the determinant of the $ n\times n $ matrix can be defined as $ \det(A)=\sum_{\sigma \in S_n} sgn(\sigma) ...
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0answers
18 views

Finding Inverse of a matrix using elementary transformations

So I have to find the Inverse of A. $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations.. The ...
2
votes
2answers
72 views

Why $x\in\ker A$ implies $x_i-x_j=\lambda \det A_{ij}$?

Suppose that $A$ is a real matrix with $n-2$ linearly independent rows and $n$ columns adding up to $0$. I can show that for any $x=(x_1,\dotsc,x_n)\in\ker A$ (that is, any $x\in\mathbb R^n$ ...
0
votes
0answers
11 views

Matrix notation: How would you apply a function to every column/row of a matrix?

Let's consider a real matrix A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ...
0
votes
0answers
12 views

What is the effect on the spectrum by addition of a matrix with that of a rank 2 matrix?

Let $A$ and $B$ be two $n\times n$ matrices with rank of $B$ equal to $2$. Then how is the spectrum of $A$ and $A+B$ related? Or whether we can say something about - which of the eigenvalues of $A$ ...
0
votes
0answers
11 views

orthogonality condition for matrices and vectors

Given two orthogonal vectors in $n$ dimensions $\vec{x}$, $\vec{y}$ ($\vec{x} \cdot \vec{y} = 0$), what are linear transformations $T$ that will maintain orthogonality of $\vec{x}$ and $\vec{y}$? What ...
1
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0answers
38 views

Numerical Algorithm for $n \times n$ Matrix Inverse

I have to write a C program in which I have to compute the matrix inverse of a $n \times n$ matrix. Is there a convenient iterative process that I can use to do that? All I see is the co factor method ...
4
votes
1answer
49 views

Finding a matrix given eigenvalues and eigenvectors.

I am asked to construct a $4 \times 4$ symmetric matrix, with given eigenvalues and eigenvectors. I understand how to actually get $A$ as a product of $P^T, D$ and $P$, when $D$ is the diagonal ...
0
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0answers
22 views

What has been already done on spectrum of Hermitian matrices?

Could anyone suggest some books/articles related to the determination of eigenvalues and eigenvectors of some special complex Hermitian matrices?
0
votes
1answer
36 views

Under which condition does $Q>I_n$ result in $Q^2>I_n$?

Consider an $n\times n$ real matrix $Q>I_n$ (i.e., $Q-I_n$ is positive definite). Under which condition, $Q^2>I_n$ also holds? It is easy to show that if $Q$ is diagonalizable, $Q>I_n$ ...
0
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0answers
32 views

Solve system of two homogeneous first-order ordinary differential equa0ti0ns by eigenvectors. (7.16-1)

Please check my work and I shall have a few questions along the way. I am working out of the textbook Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons. The problem to ...
0
votes
1answer
14 views

Does pivot column include all entries within the column?

This is a quick fundamentals question. (maybe not even one) In linear algebra, a pivot column is a column where a pivot is located on. Does pivot column include all entries within the column even if ...
3
votes
0answers
55 views

What lies beyond the Möbius transform?

Consider the matrix $\pmatrix{a & b \\ c & d} ^n$ This is isomorphic to the $n$ th iteration of the Möbius transform $\frac{a z + b}{c z + d}$ when the determinant is nonzero. So I wonder ...
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0answers
20 views

Dimension and Basis of the $S_2$ set of symmetric matrices with $tr(A)=0, \forall A \in S_2$

For the following problem: Let $S_2$ be the set of symmetric matrices (with real entries) and zero trace. Prove that $S_2$ is a subspace of the space of all $M_{2\times2}$ matrices. ...
4
votes
3answers
258 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
0
votes
1answer
35 views

Exponential of matrix, taylor series

Compute $ exp(X) $for $X=$\begin{bmatrix}t&0\\0&s\end{bmatrix}, \begin{bmatrix}0&t\\-t&0\end{bmatrix} $ and $\begin{bmatrix}0&t\\t&0\end{bmatrix} The first part of the ...
-2
votes
0answers
23 views

Derivation of a matrix function

what is the derivation of the following function with respect to U: $F = {\left( {UX{U^T}} \right)^{ - 1}}UY{U^T},\,\,\,\,\,U \in {\mathbb{R}^{m \times n}},\,X,Y \in {\mathbb{R}^{n \times n}}$ ...
0
votes
0answers
33 views

Matrix Transpose SOS

I am taking my first Linear Algebra Class in college and it is one of the hardest math classes I have ever taken. It is my introduction to proofs and the semester just started. I am very lost in the ...
0
votes
1answer
22 views

Bounds on sum of entries of an idempotent symmetric matrix

Suppose that $M$ is symmetric and idempotent, dimensions $n\times n$, and trace $n-k$. Let $e$ ($n\times 1$) be a column of $1$'s. Let $$ S_1\equiv e'Me,\quad ...
0
votes
0answers
18 views

Formatting syntax for discrete coordinate reference

I hope this is the appropriate place for this, I searched elsewhere but couldn't find a better place. I have a simple formatting question. I have a an $n \ \mathrm{x}\ m$ matrix $C$ on which I'm ...
1
vote
1answer
24 views

Understanding matrix multiply analogously

When I was introduced to vectors, I was taught that we can view each element $e_{i}$ in vectors of the same size as being of the same "type". For example, if we have two vectors each of size 2, each ...
100
votes
5answers
4k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
2
votes
1answer
67 views

Prove that $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent

Consider A to be a matrix that has all eigenvalues $\lambda$ with negative real part, that is, $Re(\lambda) < 0$. a)Show that the integral $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent, ...
0
votes
0answers
37 views

Find scalar from a vector.

Given a vector $s = (s_1,s_2,s_3,s_4) \in \mathbb C^4$, find scalars $c_1, c_2, c_3$, and $c_4$ such that $s = c_1u_1 + c_2u_2 + c_3u_3 + c_4u_4$. One can obtain the column vector $c$ by multiplying a ...
0
votes
1answer
12 views

Linearly dependent columns

Assume that A, B $\in$ $M_n(\mathbb{R})$ are nonzero matrices such that $AB$ $= [0]$. Show that the columns of $A$ are linearly dependent. What I tried: I tried to arrange AB as a summation but ...
1
vote
3answers
60 views

Make $X^TAX$ identity matrix [closed]

If we have a $n \times m$ matrix $X$ where $m<n$, and a $A$ $n \times n $ matrix. Given $X$ , In which case that $A$ can make $ X^T A X$ identity matrix? Note: what about if we consider $A$ as ...
0
votes
3answers
38 views

Show that set of all $2 \times 2$ matrices forms a vector space of dimension $4$

I have this question: Show that the set of all $2 \times 2$ matrices with real coefficients forms a linear space over $\Bbb R$ of dimension $4$. I know that the set of the matrices will ...
0
votes
0answers
14 views

Nilpotent Matrix Sign Patterns given by Existence of Nonlinear Multivariable Polynomial Solution

I am currently doing a little exploring in sign patterns in nilpotent matrices, and am trying to determine whether or not an ambiguous sign pattern has a solution (i.e permits a nilpotent matrix). ...
2
votes
2answers
44 views

Maximum number of idempotent independent matrices

What is the maximum number of idempotent and linearly independent matrices in $M_n(F)$ (considered as a vector space over the field $F$). My attemp: computer check in low dimensions shows that the ...
0
votes
1answer
30 views

Show endomorphism $\phi$ is determined by $\phi(e_1)$

Say $\phi \in End_{M_n(D)}(D^n) $ I'm trying to show $\phi$ is determined by $\phi(e_1)$ and that $\phi(e_1)=de_1$ where $d \in D$ To show it's determined by $\phi(e_1)$ I have used the property that ...
0
votes
1answer
20 views

What does $End_{M_n(D)}(D^n)$ mean? Where D is a division ring

What does $End_{M_n(D)}(D^n)$ mean? (D is division ring) I know it's the homomorphisms from $D^n$ to itself, but what role does ${M_n(D)}$ play? Does that mean over the nxn matrices over D? What does ...
0
votes
2answers
19 views

Determinant property $|c \cdot A| =c^n \cdot |A|$

$$\begin{array}{|ccc|} x & 2 & 4 \\ x & 1 & 2 \\ x & 4 & 0 \\ \end{array} = x \cdot\begin{array}{|ccc|} 1 & 2 & 4 \\ 1 & 1 & 2 \\ 1 & 4 & 0 \\ ...
2
votes
1answer
87 views

Matrix induction proof

Given the following $\lambda_{1}=\frac{1-\sqrt{5}}{2}$ and $\lambda_{2}=\frac{1+\sqrt{5}}{2}$ How do I prove this using induction: $\begin{align*} A^k=\frac{1}{\sqrt{5}}\left(\begin{array}{cc} ...
0
votes
2answers
26 views

Why is the Jacobian matrix equal to the matrix associated to a linear transformation?

Given the linear transformation $f$, we can construct the matrix $A$ as follows: on the $i$-th column we put the vector $f(\mathbf e_i)$ where $E = (\mathbf e_1, \ldots, \mathbf e_n)$ is a basis of ...
2
votes
1answer
44 views

Given a column vector, can we add columns, continuously dependent on the given one, to get an invertible matrix?

Given a vector $x$ in the $n=6$-dimensional Euclidian space $\mathbb{R}^n$, do there exist $n-1$ continuous functions $f_1$ to $f_{n-1}$ such that the matrix $$(x,f_1(x), … ,f_{n-1}(x))$$ is ...
0
votes
1answer
22 views

Supremum Infimum of Norm

Let $A\in\mathbb{R}^{n\times n}$ be an invertible matrix and $\mathbf{x}\in\mathbb{R}^n$. I am trying to prove that ...
0
votes
0answers
24 views

Cholesky Decomposition of the Hilbert Matrix

I would like to have an analytical expression for the Cholesky decomposition of the following matrix: \begin{equation} \mathbf A = \left [ \begin{array}{cccc} 1/1 & 1/2 & 1/3 & 1/4 ...
2
votes
1answer
46 views

In common tongue, what is the differences between sparse and dense matrices?

What are the differences with sparse and dense matrices in practice, so as to offer some insight to new learners on a more intuitive level. Obviously everyone knows about the dictionary definition of ...
2
votes
1answer
38 views

Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?

Suppose $A,B\in\mathbb{R}^{n\times n}$ are matrices such that $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$. I couldn't come up with a ...
1
vote
1answer
31 views

How to neatly summarize indexes of a matrix where there are a lot of i's x j's [closed]

As you can see from the subject line, I can't even think of the word of what I need to do. I am trying to write in text that I multiplied columns of a matrix (n columns, i = 1:n). There are many ...
1
vote
2answers
16 views

Inverse of sum of matrices (SVD, ridge regression)

Looking at these slides, I've found the following: $X=UDV^T$, where $U$ and $V$ are orthogonal matrices, $V$ is a square matrix, and $D$ contains the singular values of $X$. The author then writes ...