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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1answer
20 views

minimal polynomial of linear transformation

Let V and W are finite dimensional vector space over R.$ T_1:V\to V$ and $ T_2:W\to W$ be linear transformation whose minimal polynomials are given by $ f_1(x)=x^3+x^2+x+1 , f_2(x)=x^4-x^2-2$. Let $ ...
2
votes
2answers
26 views

Solution to $A = BX + YC$ where $A$ is a square matrix of rank $n$, $B,C$ known, rank $m<n$

I hope this isn't too trivial of a problem. I'm really struggling with it and I feel like it shouldn't be that difficult. As stated in the title: Given (full rank): $A\in\mathbb{R}_{nxn}$ ...
0
votes
1answer
385 views

Diagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)

Prove or disprove: If a matrix has the property $0 \neq |a_{ii}| \geq \sum_{\substack{j=1 \\ j \neq i}} |a_{ij}| $ then Gaussian Elimination (without pivoting) will preserve this property. I assume ...
0
votes
1answer
13 views

Determinant and eigenvalues of Gram matrix lower bounds

I'm trying to find a non-zero lower bound on the determinant of the Gram matrix $\Gamma$ assigned to linearly independent set of vectors (is there such a lower bound?). But that is not my question ...
2
votes
0answers
11 views

convert the inverse of sum of two hermitian matrices into sum of two or more matrices.

I want to convert the inverse of sum of two hermitian matrices into sum of two or more matrices. I mean I want to simplify the bellow equation in a way that not to have inverse of sum of matrices any ...
2
votes
2answers
21 views

Do the spaces spanned by the columns of the given matrices coincide?

Reviewing linear algebra here. Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \qquad ...
0
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1answer
19 views
1
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5answers
84 views

What does $T:V\to W$ mean in vector spaces?

What does the sign $\to $ mean in contexts like: "show $T:V\to W$ is an isomorphism" or "if $T:V\to W$ is a linear transformation"...
1
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0answers
11 views

Time complexity for the multiplication of three rectangular matrix

For the multiplication of two matrix, one can use the classical methods, such as strassen algorithm, to reduce the time complexity. Here, I just wonder if there is any efficent algorithm aiming at the ...
1
vote
2answers
15 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 ...
0
votes
0answers
16 views

Norm of operator matrix

I'm having trouble with the following: suppose H is a Hilbert space and $f_{i, j}, g_{i, j} : H \rightarrow H$, $1 \leq i, j \leq n$ are bounded operators. Then we have operators $(f_{i, j}) , (g_{i, ...
0
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2answers
155 views

Does an overdetermined system always have no solutions? [closed]

What is the problem with over-determined systems in linear algebra? Do they always have no solution? Is there a proof of that?
0
votes
1answer
19 views

How do you find the 4x4 matrix corresponding to the transformation T with respect to the basis?

If the transformation $T$ acting on the vector space $A \in Mat_{2,2}$ is given by $T(A)=CA$, where $ C= \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) $ how would you find the ...
1
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2answers
35 views

Evaluate determinant of an $n \times n$ matrix, help

I need help with this problem: $D_{n}= \begin{vmatrix} 1 & 1 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 1 & 1 & \cdots & 0 & 0 & 0 \\ 0 ...
2
votes
0answers
34 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
1answer
23 views

determinants of large and infinite matrices

Given a square n x n matrix A, is it possible to find the determinant of the matrix for large values of n easily, and thereby as n goes to infinity? I know that the number of components of the ...
0
votes
0answers
10 views

what's the potential application of low rank approximation of stochastic matrices

Suppose we have a stochastic matrix $P$ for a Markov chain, and we can compute a low rank approximation of $P$, say $P_k$, or we can find the nonnegative matrix factorization of $P$, i.e., $P=AW$ ...
0
votes
1answer
38 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
2answers
27 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 ...
0
votes
1answer
45 views

Matrices - Inverse of the principal square root of a covariance matrix (^-1/2)

Say you have a square (variance)covariance matrix S How would one go about working S^-1/2 (inverse of the principle square)? Bearing in mind, I'm trying to understand a paper which states: I've ...
1
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3answers
42 views

Volume of a parallelepiped when not given values for three vectors

There is a parallelepiped determined by three dimensional vectors x, y, and z. The volume of this parallelepiped is $11$. What is the volume of the parallelepiped determined by the three dimension ...
2
votes
0answers
28 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
votes
2answers
44 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) $ ...
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
vote
1answer
13 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 ...
1
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0answers
27 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
3answers
12k views

Image and Kernel of a Matrix Transformation

So I had a couple of questions about a matrix problem. What I'm given is... Consider a linear transformation $T: \mathbb R^5 \to \mathbb R^4$ defined by $T( \overrightarrow{x} )=A\overrightarrow{x}$, ...
0
votes
2answers
65 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 ...
4
votes
0answers
55 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
2
votes
0answers
38 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
43 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
0answers
24 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? $\rho(AB)=\rho(A)\rho(B)$.
0
votes
1answer
18 views

Find set of solutions $S_z:=\{y \in\mathbb{R}^{N}: y'z=\iota_N ' z\}$, $ z\in\mathbb{R}^N$.

How to characterize $S_z:=\{y \in\mathbb{R}^{N}: y'z=\iota_N ' z\}$, $ z\in\mathbb{R}^N$? Is there also a general way for more complex equations $y'\beta(z)=\iota'z$ where $\beta(z)\in\mathbb{R}^N$ ...
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
12 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
22 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
56 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 ...
2
votes
2answers
50 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 $$f: A \mapsto A^{-1}$$ ...
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
0answers
22 views

What do the ellipses mean in $x^TW_{\dots ij}$

I cam across this notation: $x^TW_{\dots ij}$. I don't understand the notation. Why are there 3 dots? Source: http://jmlr.csail.mit.edu/proceedings/papers/v28/goodfellow13.pdf, top of second page, ...
3
votes
3answers
137 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 ...
0
votes
1answer
12 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
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
34 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 ...
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)) := ...
4
votes
2answers
49 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
2answers
52 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 ...
3
votes
1answer
28 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 ...