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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0answers
35 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
48 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
votes
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
votes
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
0answers
40 views

How to integrate over inverse of kernel Matrix [on hold]

Let $x_i\in R^m, i \in \{1,2,...,n\}$ be a vector, $x_i(t)=\left( \begin{array}{c} x_{i1}\\ .\\ .\\ x_{ip-1}\\ t\\ x_{ip+1}\\ .\\ .\\ x_{im}\\ \end{array} \right)$, $x_i\neq x_j$ for $i\neq j$ and the ...
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
52 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 ...
1
vote
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
256 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
31 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
32 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 ...
91
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
65 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, ...
-2
votes
0answers
27 views

Matrices commuting with involution like matrices [closed]

Let $A\in M_n(\mathbb{R})$ be a matrix with no real eigen values. It is possible to find a matrix $B$ commuting with $A$ and $B^2 = - I$ ?
0
votes
0answers
36 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 ...
0
votes
2answers
28 views

What does this matrix operation mean in MATLAB: X\y [closed]

I'm reading somebody's code and I'm trying to figure out what this line means X\y where X is a 100x2 matrix and y is a 100x1 vector
1
vote
3answers
59 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
36 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
43 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
29 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
80 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
20 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
45 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
30 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 ...
-3
votes
1answer
67 views

Find 2X2 matrices A and B that are not invertible but A+B is invertible. [closed]

How would I find the solution to this problem?
0
votes
0answers
33 views

How many Matrices of size 3 by 3 have a determinant of 1?

Suppose you had a matrix where each index goes from a natural value of 0 - 25 (The Hill Cipher Algorithm uses 3 by 3 matrices where each numeric value corresponds to a letter) Given a matrix of size ...
0
votes
0answers
27 views

Multiplying magic squares like matrices to hopefully arrive at another magic square

Well, the title actually describes what is the problem in question. I was just thinking a bit about magic squares and this question popped-up. It could be that it is not interesting but I do not see a ...
0
votes
1answer
39 views

Derivation wrt a vector variable: what happens to transpose of the vector?

Considering $x,y$ are vectors and $\mu,\Sigma$ are mean (vector) and covariance (matrix); how to solve: $(1): \displaystyle \frac{\partial }{\partial X}{[(y-x-\mu_N)^T\Sigma_N^{-1}(y-x-\mu_N) + ...
1
vote
2answers
71 views

Quadratic form equals zero

We have a quadratic form $x^TAx$ where $x$ is a vector in $\mathbb R^n$ and $A$ is an $n \times n$ real symmetric matrix. Define M to be the set: $$M=\{x \in \mathbb R^n| x^TAx=0\}$$ and a ...
-1
votes
0answers
32 views

Linear transformation defined by Self Adjoint matrix

Let $P$ be self adjoint matrix $P = P^\ast$ over the complex field $\mathbb{C}$. $T$ is a linear transformation $T : M_{n \times n}(\mathbb{C}) \to M_{n \times n}(\mathbb{C})$ $T(x) = P^{-1}xP$ ...
1
vote
1answer
35 views

A multiplication of vectors

From page 219, 'Machine Learning' by Murphy: ...
0
votes
1answer
58 views

Suppose a matrix valued function $A$ with $A(0) = I$, find $\frac{d^2}{dt^2} det(A(t)) $ at $t=0$

The original question is : Let $A(t)$ be a differentiable square matrix valued function with $A(0) = I$, find $\frac{d^2}{dt^2} det(A(t)) $ at $t=0$ in term of $\dot{A}$ and $\ddot{A}$. I know in the ...
10
votes
4answers
160 views

Suppose $e^A = A$, prove that $A$ is diagonalizable

Suppose $e^A = A$, prove that $A$ is diagonalizable, where A is a matrix. What I have tried to do is write $A= D + N$, where $D$ is diagonalizable, $N$ is nilpotent and $DN = ND$. Since $N$ is ...
0
votes
0answers
24 views

Set, n-Tuple, Vector and Matrix — links and differences

I know this question has been asked like 1000 times, however all supplied answers were not really satisfying to me. My question concerns the similarities and differences between these mathematical ...
0
votes
1answer
23 views

Demonstration of the uniquenes of a QL matrix factorisation

One of my teacher say to us that the QL decomposition is unique. I am convinced. How can I demonstrate the decomposition is unique?
0
votes
1answer
35 views

Show that $adj(adj(A)) = 0$

Let $n > 2$ and let $A$ be a real and singular (i.e., non-invertible) $n\times n$ matrix. Is it true that $adj(adj(A)) = 0$ ?