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), ...
2
votes
1answer
124 views
Power series of matrix which is multiplied by a constant factor $c<1$?
(Important: THIS PROBLEM IS NOT DUPLICATED! Note that the case where just one row of $W$ is multiplied by constant $c$, can be handled by the Sherman-Morrison theorem, but the case where the whole ...
2
votes
1answer
80 views
Topologically equivalence of a metric on matrices
Define a function on the set of $n\times n$ matrices by $\rho(A,B)=\operatorname{rank}(A-B).$ Prove that $\rho$ is a metric that is topologically equivalent to the discrete metric.
2
votes
1answer
25 views
Matrix expression
The products $P$ and $Q$ are to be processed using two machines, $A$ and $B$. Each unit of $P$ requires $6$ hours in machine $A$ and $2$ hours in machine $B$ while each unit of $Q$ requires $5$ hours ...
2
votes
1answer
65 views
Row space and kernel in linear transformations
I am preparing a "dictionary" that translates between the "language of matrices" and the "language of linear transformations" in linear algebra. The dictionary looks more or less like this:
Language ...
2
votes
1answer
46 views
$\int_{\mathbb{R}^n} dx_1 \dots dx_n \exp(−\frac{1}{2}\sum_{i,j=1}^{n}x_iA_{ij}x_j)$?
Let $A$ be a symmetric positive-definite $n\times n$ matrix and $b_i$ be some real numbers How can one evaluate the following integrals?
$\int_{\mathbb{R}^n} dx_1 \dots dx_n ...
2
votes
1answer
48 views
Positive definite semi-ordering
I wanted to ask why is positive definite semi-ordering is well defined only for Hermitian matrices (or symmetric matrices if restricted to the reals)?
I saw an extension to the definition of positive ...
2
votes
1answer
125 views
Rank of a Vandermonde Matrix with additional weighted columns
A Vandermonde matrix:
$\left(\begin{array}{ccc}
1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\
1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\
\vdots & \vdots & \ddots & ...
2
votes
1answer
64 views
Question about how linear independence of module elements gives condtions on the determinant of a matrix
I have been trying to prove linear independence of rows implies linear independence of columns in kind of an abstract setting of modules following some exercises and notes from an old course. The ...
1
vote
1answer
51 views
Condition on a matrix sum with equal determinant and trace
Let $n$ be a positive integer, $J$ the matrix of all ones and $Q$ a symmetric positive semidefinite matrix such that
$\det(nI-Q) = \det(Q+J)$
$\rm{tr}(nI-Q) = \rm{tr}(Q+J)$
and also
$nI-Q \ne ...
1
vote
1answer
30 views
Solve $|a^H v | = \|g\|$ to $v$
I need $|a^H v | = \|g\|$ to be solved to $v$
Where $a, v, g\in\mathcal{C^{4 \times 1}}$ (standing vectors with complex elements).
I have 4 of those equation with the same $v$:
$$|a_1^H v | = ...
1
vote
1answer
37 views
Determining $u=v \times w$ using the cross product
Let $v = (3,0,0)$ and $w=(0,1,-1).$ Determine $u = v \times w$ using the geometric properties of the cross product rather than the formula.
What are the possible angles $x$ between two unit vectors ...
1
vote
1answer
34 views
A matrix decomposition
Suppose $N$ is a symmetric matrix, then show that it can be uniquely decomposed as $N=N^+-N^-$, where $N^+$ and $N^-$ are both nonnegative-definite, i.e. their eigenvalue are non-negative, and ...
1
vote
1answer
21 views
How to solve simultaneous equations using Delta?
I can solve simultaneous equations using multiple methods, but came across this new procedure while revising for my exam. I've never seen anything like this before and can't find any explanations of ...
1
vote
1answer
29 views
Rotating the gradient
Suppose I have a triangle T in 3dimensional space and i want to rotate it in arbitrary ways. The coordinates for T are given by $f: T_R \in \mathbb{R}^2 \rightarrow T \in \mathbb{R}^3 $ where $T_R$ is ...
1
vote
1answer
29 views
Calculation of determinant for differential matrix equations
Here is the differential equation $$N'_x(x)=G(x)N(x)$$ where $N, G$ are $2\times2$ matrices depending on $x$, and $G$ satisfies $\mathrm{trace} (G)=0$. My question is:
How can one then calculate the ...
1
vote
1answer
27 views
Finding a matrix with the following property
I have one $n \times n$ symmetric matrix $B$. Let $p$ be a scalar, I want to multiply the diagonal elements of $B$ by $p$. Let now $C$ denote the resultant matrix of the process described. Is there ...
1
vote
1answer
21 views
Reducing a Laplace Matrix which is also Scalar?
I had a 9x9 matrix as follows: Aim is to reduce it to Lower or Upper Matrix
...
1
vote
1answer
49 views
Linear algebra, Schur set
Can you guys give me some hints on how I can start this problem? Thanks in advance!
Let $ U(n) \subseteq M_n(\mathbb C) $ be the set (group) of all $ n \times n $ unitary matrices. Let $ T ...
1
vote
1answer
43 views
How would I find this eigenvalue?
I'm told to let $A$ be the matrix of the linear transformation $T$ and without writing $A$, find an eigenvalue of $A$ and describe the eigenspace. The first is to let $T$ be the transformation on ...
1
vote
1answer
75 views
problem with 4x4 matrix with big elements
I have a homework for my linear algebra class at my university the thing is that we get a 4x4 matrix A then we have to find it's Transpose which is pretty easy and then find the matrix B=(A^T)*A also ...
1
vote
1answer
65 views
irreducible, diagonally dominant matrix
I am facing a problem for irreducible,diagonally dominant matrices. How to prove that irreducible, diagonally dominant matrix is invertible? Please help me in this problem.
1
vote
1answer
85 views
How to show by induction that, for $0<\theta<\pi$, $\det A_n=\frac{\sin (n+1)\theta}{\sin \theta}.$
I need help with the underlined part.
Thanks in advance
Let $A_n$ be the $n\times n$ matrix given by
$$a_{ij}=
\begin{cases}
0 & \text{if }|i-j|>1, \\
1 & \text{if }|i-j|=1, ...
1
vote
1answer
70 views
Show that $P = Q^2$
Suppose $P$ is a positive semi-definite $n\times n$ matrix. Show that there exists a unique positive semi-definite matrix $Q$ such that $P = Q^2$.
In class we've been going over singular value ...
1
vote
1answer
40 views
Finding Pi variables from matrix. From PageRank Algorithm.
$$\pmatrix{\pi_1 & \pi_2 & \pi_3} = \pmatrix{\pi_1 & \pi_2 & \pi_3}\pmatrix{\frac{1}{6} & \frac{4}{6} & \frac{1}{6} \\ \frac{5}{12} & \frac{2}{12} & \frac{5}{12} \\ ...
1
vote
1answer
28 views
How to do QR Factorisation of a matrix
given A = $\begin{bmatrix}1 & 0 & 3\\2 & -6 & 3\\ -2 & 3 & -3\end{bmatrix}$
How would i find the QR factorisation? Well i have a guide on how to do this and have attempted ...
1
vote
1answer
39 views
Condition number of matrix order 2
I am trying to prove that
$$\mbox{cond}_2(A)\ =\ \inf_{E\in\mathscr{E}}\mbox{cond}_2(E),\;\;\; \mbox{ where }\;\; A = \left(\begin{array}{cc}
100 & 99\\99 & 98
\end{array}\right).$$
And ...
1
vote
1answer
259 views
Inverse of upper triangular matrix
I have an upper triangular matrix that I want to solve the inverse for.
I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
1
vote
1answer
64 views
Gradient with respect to a matrix variable
I want to find the gradient of the function $\mathcal{F}_1$ with respect to the matrix $\mathbf{X}$ (differentiate with respect to $\mathbf{X}$):
$$
\mathcal{F}_1 (\mathbf{X}; \mathbf{\lambda})= ...
1
vote
1answer
83 views
How to deal with such $\frac{0}{0}$ limitation?
Let $A,B \in R^{n \times n}$ and $v(k) \in R^n$. There exists a time-varying vector $v(k)$ converges to $v \in R^n$, i.e., $\lim _{k \rightarrow \infty } v(k) = v$. And, $v^TA=v^TB=0$, $Av=Bv=0$. Is ...
1
vote
1answer
89 views
Matrix inversion help
If $g(n)$ is an integer functions periodic in $a$
And $\phi(a)$ is eulers totient function
And $[r_1,r_2,r_3,...r_{\phi(a)}]$ are the postive integers less then $a$ coprime to $a$
With ...
1
vote
1answer
79 views
Least square solution based on the pseudoinverse solved efficiently with singular value decomposition
Hi apologies it's hard to type out the problem,
I have a lecture slide on neural networks. It says the fitting error gives the matrix:
N by M matrix of thi's multiplied by Mx1 weights minus Nx1 ...
1
vote
1answer
57 views
Two quick eigenvalues & complex numbers questions
A) For a vector $v\in\mathbb{C^n}$, is $Im(-v)=Im(\overline{v})$ ?
($Im(v)$denoting the imaginary part of the vector $v$)
My understanding: since every row of the vector is a complex number (say ...
1
vote
1answer
51 views
How do I show that these sums are the same?
My textbook says that I should check that
$$ \sum_{i=0}^\infty \frac{\left( \lambda\mathtt{I} + \mathtt{J}_k \right)^i}{i!} $$
is in fact the same as the product of sums
$$ \left( \sum_{i=0}^\infty ...
1
vote
1answer
51 views
Matrix equation for $SL(4,\mathbb{C})$
Suppose $E=\{X\in M_4(\mathbb{C}): X^T=-X\}$ and that there exists $a\in SL(4,\mathbb{C})$ such that for all $X\in E$
$$
aXa^T=X
$$
I want to show that it follows that $a=\pm I$. This can be done by ...
1
vote
1answer
73 views
How to get Euler angles with respect to initial Euler angle
I have a sensor which gives me Euler angles (roll,pitch,yaw). There is a baseline value of Euler angle (assume it is $5,10,15$) at the beginning.I want to calibrate from this baseline values all ...
1
vote
1answer
39 views
Finding an D-dimensional orthonormal change-of-basis matrix given a D-2 transformed points.
This is frustrating, I should be able to solve this but I'm having a mental fog.
I want to find an orthonormal change of basis: given a single point $(x_1,y_1)^T$ and its image $(x_2,y_2)^T$, find ...
1
vote
1answer
37 views
Prove that $(e^{At}−I)/t→A$ as $t→0$, meaning $\|(e^{At}−I)/t−A\|\to 0$ as $t\to0$ for all $A\in C^{n×n}$.
a) Prove that $(e^{At}−I)/t\to A$ as $t\to 0$, meaning $\|(e^{At}−I)/t−A\|\to 0$ as $t\to 0$ for all $A\in C^{n×n}$. Hint: You may use the inequality $\|A^k\|\leq n^{k−1}\|A\|^k$
I've been ...
1
vote
1answer
100 views
Finding all possible solutions to an $n \times n$ matrix, given row and column sums.
What I am ultimately attempting to do is find the solution that maximizes a given equation so I need to find all possible solutions so I can check them. I need all possible solutions to an $n \times ...
1
vote
1answer
81 views
Why do we assume that a matrix in quadratic form is Symmetric?
I am looking to the review document for linear algebra and the part of the quadratic form (pg17) mentions about an assumption of being symmetric for a matrix in quadratic form. It also includes some ...
1
vote
1answer
66 views
Change in singular values of matrix after left-multiply with a diagonal matrix
Say that we have an SVD for a matrix $X = U \Sigma V^T$, giving trace norm $||X||_{tr} = ||\Sigma||_{tr} = \sum \Sigma_{ii}$. I am wondering what happens to the SVD and/or trace norm if we left ...
1
vote
1answer
51 views
Good source for self study of matrix decompositions
What is a good source for study of various types of matrix decomposition, which is both comprehensive and also includes applications? It should at least cover LU, RQ, SVD, spectral, Schur, and ...
1
vote
1answer
36 views
On the uniqueness of the real logarithm of a real matrix
I was wondering about the uniqueness claim in the paper, on the exitence and uniqueness of the real logarithm of a matrix, to answer the questions but I have not been able to understand the ...
1
vote
1answer
87 views
Stabilizer of a 4 by 4 skew symmetric matrix by orthogonal matrix
Matrices are over the field of complex numbers, and $X^t$ means transpose of a matrix $X$.
Consider the group action of $O(4)=\{P\mid PP^t=I\}$ on $SK(4)=\{M\mid M^t=-M\}$ by $(P,M) \rightarrow ...
1
vote
1answer
42 views
There are $n!$ different unitary similarities that give $L_i \sim L_j$. Or are there more?
Given a matrix $L$ of dimension $n \times n$ which is lower triangular, and with distinct elements along the diagonal, there is a way to unitarily transform it into a different lower triangular matrix ...
1
vote
1answer
38 views
Confusion in matrix manipulation
I have a vector $y = \frac{-x^TB}{C}$
Substituting y in $x^TAx + 2x^TBy+y^TCy$ I am supposed to get
$x^T(A-BC^{-1}B^T)x$
I am just beginner with matrix stuff. Obviously, substituting y in the ...
1
vote
1answer
42 views
Linear transform of parameterization, but implicit?
Suppose I have a $2\times 2$ matrix $M$ in implicit equation form, where $(x,y)$ is transformed to $( xOut,yOut )$:
$$
ax + by = xOut, \quad cx + dy = yOut
$$
Now additionally I have a line $L$, also ...
1
vote
1answer
88 views
Criteria for the difference of two matrices to be positive semidefinite when the eigenvectors are known
Let $A$ be a rank 1 positive semidefinite matrix and $B$ a Hermitian matrix. Suppose I know the eigenvectors of both $A$ and $B$ and that $A-B$ is also positive semidefinite.
Apart from Weyl's ...
1
vote
1answer
30 views
Stabilization property of an operator in a finite-dimensional vector space?
Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$.
For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$.
(a)
How do we prove the stabilization ...
1
vote
1answer
49 views
Help regarding a weird Matrix
Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on ...
1
vote
1answer
158 views
Subspaces, transformation matrices exercise
I have trouble understanding the following exercise so I would really appreciate any help you could give me:
Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be ...
