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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1answer
29 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
0answers
13 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 ...
0
votes
1answer
21 views

Check for basis of a matrix

Given the matrices in $M_{3,3}$. ...
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 ...
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 ...
2
votes
2answers
27 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
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
votes
1answer
24 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
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 ...
1
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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
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
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
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 ...
0
votes
1answer
39 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
vote
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
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 ...
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
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 ...
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 ...
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
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
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 $$f: A \mapsto A^{-1}$$ ...
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 ...
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 ...
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
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 ...
0
votes
0answers
25 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
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) $ ...
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
16 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
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 ...
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
29 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
14 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
20 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 ...