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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2answers
19 views

Norm Used in Perturbation Matrix Thoery?

My question is that what is the type of 2-Norm used in Weyl's theorem for relative perturbation? Is that a induced norm, or a entry-wise norm? $\epsilon=\|X^T X-I\|_2$, where relative difference in ...
2
votes
0answers
30 views

Why can matrices be reversed when implementing the hypothesis function?

I'm learning about the hypothesis function used in linear regression. $$h(\theta) = \theta_0X_0 + \theta_1X_1$$ Where $\theta$ is a $1\times 2$ matrix and $X$ is a $n\times 2$ matrix (with the first ...
1
vote
1answer
14 views

Orthogonal complex matrices: polar decomposition

Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)? thanks ...
0
votes
1answer
42 views

Rotation Matrix and programming [on hold]

I am actually programming in Android. An android tablet as a lot of sensors including one that gives the rotation vector of the tablet. (See ...
0
votes
3answers
26 views

How to determine if the set of vectors are linearly dependent or independent

Determine if the following sets of vectors are linearly dependent or linearly independent $$V1=\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0\end{bmatrix}$$ $$V2=\begin{bmatrix}0 & 0 & ...
1
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0answers
28 views

Solving linear equation for low rank matrices

Consider $Ax=b$ where $A$ is invertible, so we have $x=A^{-1}b$. Now, let's consider a low-rank approximation of $A$, say $\bar{A}$ such that $rank(\bar{A})\leq r$ and $||A-\bar{A}||_F\leq \epsilon$ ...
0
votes
2answers
36 views

eigenvalues of A - aI in terms of eigenvalues of A

I am stuck with this question of my assignment where given that A is nxn square matrix and a be a scalar it is asked to - Find the eigenvalues of A - aI in terms of eigenvalues of A. A and A - aI ...
2
votes
0answers
44 views

Is there a name or symbol for the matrix division resulting in a scalar?

I am not talking about the inverse matrix, $A^{-1}$ which gives $A\times A^{-1}=I$, but rather the operation $\frac{1}{n}tr(\space\cdot \times A^{-1})$, which gives 1 when applied to a $n\times n$ ...
0
votes
2answers
47 views

Matrix notation in handwriting

I understand that typically matrices are printed in bold to distinguish them from other mathematical entities with the same symbols. However I find it difficult to actually handwrite in bold. With ...
0
votes
2answers
67 views

Show that $\det(A) > 0$

Let $(a_{ij})$ be a real $n \times n$ matrix satisfying, $a_{ii} > 0 \space (1 \leq i \leq n) ,$ $a_{ij} \leq 0 \space (i \ne j, 1 \leq i,j \leq n) ,$ $\sum_{i=1}^ {i=n} \space ...
2
votes
0answers
24 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
-1
votes
0answers
13 views

how to convert eigenvectors & eigenvalue to rotation matrix?

I would like to know how to convert an eigenvector and an eigenvalue(if needed) to a rotation matrix. I am in charge with writing software to calculate the attitude of a satellite in space. K is a 4 ...
0
votes
0answers
25 views

Basis of square matrices

Find a basis of the space of complex $n \times n$ matrices, all the elements of which are invertible matrices. I suggest the following: using transvections for $i\neq j$ $T_{i,j}(1) := ...
3
votes
2answers
57 views

problem about symmetric positive semi-definite matrix

Let $A,B$ be symmetric positive semi-definite matrix with real entries I have to show that $ Im(A) \subset Im(A+B)$ if $tr(AB)=0$ then $ AB=O $ I know that a symmetric matrix A is positive ...
0
votes
1answer
30 views

System of linear equations: and a small perturbation

If $Ax=b$ and $Ax'=b'$ where $x'$ and $b'$ are $x$ and $b$ with a small perturbation, the following inequality will always hold: $ (\left\lVert x-x' \right\rVert/\left / \lVert x \right\rVert) \le ...
0
votes
0answers
20 views

A smart way to bound this function and get rid of covariance matrix

I have the following function which I am trying to bound as follows $$A({\bf h},\Sigma)= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - \rho_1 \rho_2^* ...
0
votes
1answer
13 views

Form matrix and calculate it's determinant

I need help with this problem: For every $i,j \in \{1,2,...,n\}$ is $d_{i,j}=min\{i,j\}$. Calculate determinant of a matrix $[d_{i,j}]_{n_Xn}$. Is it right that all the elements of this squared ...
2
votes
2answers
51 views

Homeomorphism between the set of invertible matrices and itself

Consider the set of invertible $n \times n$-matrices $GL_n(\mathbb{R}) = \{A \in M_{n \times n}(R) \mid A\text{ is invertible}\}$. I now want to prove that the transformation $$f: A \mapsto A^{-1}$$ ...
2
votes
4answers
147 views

Minimal polynomial for an invertible matrix and its determinant

So here's one that I can't quite crack: Let $A\in M_n(\mathbb{F})$ be an invertible matrix with integer eigenvalues. Its minimal polynomial is ...
2
votes
1answer
39 views

How do I differentiate a Kronecker product with respect to a vector?

I am trying to differentiate $[\mathbf{I} \otimes \mathbf{t}^*\mathbf{t}^T]$ with respect to $\mathbf{t}$. I did the following $\mathbf{I} \otimes \mathbf{t}^*\mathbf{t}^T = (\mathbf{I} \otimes ...
2
votes
0answers
53 views

Inverse of two matrices multiplied

I've been asked to find the inverse of $AB$ where $A$ and $B$ are: $$A=\begin{bmatrix}5 & 3 \\4 & 2\end{bmatrix}$$ $$B=\begin{bmatrix}2 & -3 \\1 & 3\end{bmatrix}$$ My answer: What I ...
2
votes
0answers
9 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if $A$ and $B$ are two $n \times n$ Hermitian matrices, and $[A,B]=C$. I'd like a ...
4
votes
2answers
52 views

Let $n$ be a positive integer. If $A∈\mathscr{M}_{n×n}(\mathbb{C})$, show that $A$ and $A^T$ are similar.

Let $n$ be a positive integer. If $A∈\mathscr{M}_{n×n}(\mathbb{C})$, show that $A$ and $A^T$ are similar. I have that $A=BC$ where $B,C$ are symmetric, then $A^T=(BC)^T=C^TB^T=CB$ and then ...
3
votes
1answer
29 views

Matrix with prime entries and largest possible determinant

Let $n\ge 1$ be a natural number. Arrange the first $n^2$ primes in a $n\times n$-matrix, such that the determinant becomes as large as possible. What is the largest possible determinant and which ...
0
votes
0answers
26 views

when the spectral radius of a matrix product is equal to the product of spectral radius?

The question is simply as follows, when do we have the following equality? $\rho(AB)=\rho(A)\rho(B)$.
-1
votes
2answers
45 views

Why $ (A\vec{x})'A \vec{x} = \vec{0}$ implies that $A\vec{x} = \vec{0}$

A is a symmetric matrix. And $\vec{x} \neq \vec{0} $ where $\vec{x} \in Nul(A^2) $ Since A is symmetric we know that this relation holds: $A^T = A$ So $A^2 = A^TA = AA$ And $ Nul(A^2) = Nul(A) $ ...
1
vote
0answers
12 views

Change of Basis Matrix: Cartesian to Spherical Laplacian

I was looking at how a change of basis matrix, $[P_{\beta\leftarrow\alpha}]$, is made. While this is a bit more advanced that than what was taught at the course, I wonder what would be the change of ...
1
vote
3answers
34 views

Why is this matrix invertible [duplicate]

I was wondering if there is a way to see why $(1+A)$ invertible, if $A$ is a skew symmetric matrix. and I know that all eigenvalues of $A$ have zero real part and $A$ is unitarily diagonalisable.
0
votes
1answer
17 views

operator norm of a linear transformation, given by the transformation matrix

Consider $\mathbb{K}^n$, $\mathbb{K}^m$, both with the $||.||_1$-norm, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. Let $||T|| = inf\{M ≥ 0: ||T(x)|| ≤ M ||x|| \space \forall x \in ...
3
votes
2answers
54 views

Find all complex matrices $A$ such that $n\operatorname{Tr}(AB) = \operatorname{Tr}(A)\operatorname{Tr}(B)$ for all $B$. [duplicate]

Consider a bilinear form $f(A,B) = n\operatorname{Tr}(AB) - \operatorname{Tr}(A)\operatorname{Tr}(B)$ defined on $M_n(\mathbb{C})$. I need to find the set $U^\perp$ of all matrices $A$ such that ...
1
vote
0answers
26 views

When do a Regular graph has an odd eigenvalue?

Merely looking at adjacency matrix of a regular graph, without explicit calculation, can we decide that graph will have an odd eigenvalue or not? If regularity is odd, we are sure that it will be an ...
2
votes
2answers
32 views

How to find general inverse of a matrix

Find the general inverse (G) of the matrix $$A=\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6\end{bmatrix}$$ Also check that $AGA=A$ I am new in G- inverse calculation. I understand ...
1
vote
1answer
24 views

Upperbound a logarithmic expression that has a covariance matrix

Let $\Sigma$ be a $2\times 2$ covariance matrix and ${\bf h}$ a vector of complex values entries. $$A= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - ...
0
votes
1answer
15 views

Summation notation for the “ij'th” entry of matrix $(AB)^t$.

I'm just trying to figure out how to write out a formula to find the ij'th entry of the transpose of a matrix product. We have an $l \times m$ matrix $B$ and an $m \times n$ matrix $A$. We have $B = ...
2
votes
0answers
22 views

bilinear forms on $M_{n, n}(K)$

Let $K$ be a field and $V = M_{n, n}(K)$ the ring of $n \times n$ matrices over $K$. For any $f \in V^*$ (the dual space of $V$), we set: $\gamma_f: V \times V \to K, (A, B) \mapsto f(A B^t)$. I now ...
2
votes
2answers
69 views

Each eigenvalue of $A$ is equal to $\pm 1$. Why is $A$ similar to $A^{-1}$? [duplicate]

$A$ is a non-singular matrix ($n \times n$) and each eigenvalue of $A$ is equal to $\pm 1$. Why is $A$ similar to $A^{-1}$? (by Jordan form)
-1
votes
1answer
26 views

In an augmented matrix representing a system of equations, why is it a contradiction when the LHS isn't zero and RHS is zero but not when flipped?

In an augmented matrix representing a system of equations, say a $1\times 3$ matrix: $(a,b \mid c)$, why is it a contradiction when $a=b=0, c\neq 0$ but not when $a,b\neq 0, c=0$ ?
0
votes
2answers
22 views

Not quite similiarity

If I let $GL_r(\mathbb{R}) \times GL_s(\mathbb{R})$ act on the set of all $r \times s$ matrices by $(A,B) \cdot M = AMB^{-1}$, why am I able to reach a diagonal matrix with $0's$ and $1's$ along the ...
1
vote
1answer
37 views

Representing linear transformations as matrices. What benefit, if any, is there to *not* expressing them as diagonal matrices?

This question concerns something quite basic and I've done a questionable job of explaining my confusions, because I'm confused. I apologize. I was playing with linear transformations earlier today, ...
1
vote
1answer
33 views

space engineering - strange matrix - attitude [on hold]

I am in charge with developing software to determine the satellites attitude. But I just encountered a very strange type of matrix $K$ (from the $q$-method attitude determination algorithm) What is ...
-4
votes
0answers
21 views

Gauss elimination [on hold]

Why we change row in matrix ? a=2 0 1, 0 22 1, 0 -3 -23, this is matrix. ~ a=2 0 1, 0 -3 -23, 0 22 1 Here, in first matrix , why we change second row to third row .
0
votes
0answers
27 views

Origin of Matrix A when calculating Eigenvalues and Eigenvectors [on hold]

I understand how to calculate Eigenvalues and Eigenvectors ($Ax = \lambda x$), but what I don't understand is how the matrix $A$ originates. Is its origin from measurements or the like? Thanks in ...
0
votes
0answers
10 views

How can I define the order of rotation from one rotation matrix to another?

I am trying to rotate a koordinatesystem in the defined order of z-x-y. I have the matrix $rot_x$ describing the rotation around the x-Axis, the matrix $rot_y$ describing the rotation around the ...
0
votes
2answers
32 views

Decomposition for a Sum of Matrix Products

I need to find the following matrix decomposition: \begin{align} AB+BA-BAB=XX', \end{align} where $A$ is a $n\times n$ symmetrical matrix with full rank, $B$ is a $n\times n$ matrix of ones, so it has ...
3
votes
1answer
49 views

Easy Problem:Find eigenvalues of a $3 \times 3$matrix

Let $ A = \left[ {\begin{array}{cc} 1 & 1 & 1 \\ 1 & w^2& w \\ 1 & w & w^2 \end{array} } \right] $ where $w$(other than 1) is a cube root of unity.Let ...
3
votes
2answers
41 views

Solve linear system with variables?

I have a system of equations like the below: $$x + 3y - z = a \\ x + y + 2z = b \\ 2y - 3z = c$$ And have put it in an augmented matrix: $$\begin{bmatrix} 1 & 3 & -1 & a \\ 1 & 1 ...
0
votes
2answers
17 views

Uniformly Pick Row and Uniformly Pick Column == Uniformly Pick Matrix Entry??

Let $A$ be $M \times N$ matrix. I need to sample a entry uniformly from the matrix. If I sample a row (pick a number among integers $1,\dots,M$) and sample a column, both uniformly, is it equivalent ...
2
votes
0answers
39 views
+50

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
0
votes
0answers
25 views

Solve linear system; Gaussian vs. Gauss-Jordan elimination? Solutions?

I have a matrix of a size $m*n$ and am asked to solve it first using Gaussian elimination, and then once with Gauss-Jordan elimination. I know what to do for both: reduce via row operations, but what ...
1
vote
1answer
31 views

Maximum singular value of a matrix valued function

Let $f$ be an analytic matrix-valued function, $\Lambda(A)$ be the spectrum of $A$ and $\sigma_1(A)$ the maximum singular value of $A$. It is known that $$\Lambda(f(A)) = f(\Lambda(A)) := ...