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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2
votes
2answers
69 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): $A\in\mathbb{R}_{nxn}$ $B\in\...
0
votes
1answer
49 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 ...
1
vote
0answers
49 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
44 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
vote
2answers
72 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
vote
2answers
53 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
42 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
47 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
3answers
30 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
45 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
110 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
76 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
1k 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
85 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 a_{ij} ...
3
votes
0answers
63 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.
3
votes
2answers
139 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 semi-...
0
votes
1answer
43 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
1answer
19 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
88 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
288 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 $m_A(\lambda)=\lambda^k+b_{k-1}\lambda^{k-1}+...+...
2
votes
1answer
66 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
18 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
72 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 $AB=BCB=...
3
votes
1answer
45 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 ...
-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) $ I'...
1
vote
1answer
91 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
54 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
95 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 $||T|| = inf\{M ≥ 0: ||T(x)|| ≤ M ||x|| \space \forall x \in \mathbb{K}^n\}$...
3
votes
2answers
72 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 $f(A,...
2
votes
0answers
52 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
182 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 that ...
1
vote
1answer
31 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
64 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
27 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
70 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
44 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
23 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
47 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, ...
0
votes
1answer
59 views

what represents this strange matrix [closed]

But I just encountered a very strange type of matrix. What is the name of this type of matrix? How can it be solved? The final result should be a $4\times 4$ matrix, but I only see $5$ elements on ...
1
vote
2answers
47 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 ...
2
votes
1answer
83 views

Absolute value of 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 $w$ is a cube root of unity (other than 1). Let $\lambda_1,\...
3
votes
2answers
58 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
25 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 ...
16
votes
0answers
276 views

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 $...
1
vote
1answer
50 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)) := \{f(\...
1
vote
3answers
116 views

Show that matrices are not similar

I have to show that the following matrices are not similar: $$A = \left[\begin{matrix} 1 & 3 & -3 \\ -3 & 7 & -3 \\ -6 & 6 & -2\end{matrix}\right]$$ and $$A' = \left[\begin{...
4
votes
3answers
358 views

Prove that if $B$ is similar to $A$, then $B^T$ is similar to $A^T$ .

If two matrix ($A$ and $B$) are similar if there exists an invertible matrix $P$, such that: $$ B=P^{-1} A P $$ I'm thinking if I can prove that $A$, $B$ , $A^T$ and $B^T$ have the same characteristic ...
0
votes
1answer
39 views

Can the matrix expression $D-A^{-1}DA$ be simplified (for diagonal $D$ and symmetric $A$)?

I can't seem to find a way to extract the $D$ from the inside or let $A$ and its inverse interact in some way. Is this the simplest form of this expression?
0
votes
1answer
53 views

solving equation with linear span (using row reduction)

We've got the following span: $$U = Sp\{(2, 5, -4, -10), (1, 1, 1, 1), (1, 0,3,5) , (0,2,-4,-8)\}$$ We need to find the values of the number $a$ where the vector $$v = (a, a-6, 4a-3, 6a-1)$$ belongs ...
2
votes
3answers
236 views

Can a Matrix with positive entries have a negative eigenvalue?

It seems intuitive to me that the answer is no, but I can't prove it.