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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4
votes
5answers
132 views

Calculating the matrix $M^{2006}$

Say you have the matrix $M$: $$\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$$ How do you find $M^{2006}$? My thinking was that you ...
0
votes
0answers
11 views

Proof of the rotation matrix is an extreme point of $\text{conv } SO(n)$

Define the set of rotation matrices: \begin{equation} \begin{aligned} SO(n) := \{X\in \textbf{R}^{n\times n}: X^TX=I, \text{det}(X)=1\} \end{aligned} \end{equation} I want to prove that if $X\in SO(...
1
vote
1answer
48 views

What seems to be the minors of the Adjugate matrix $\text{adj}(A)$ of a square matrix $A$?

It is by definition that entries of the adjugate matrix $\text{adj}(A)$ are the corresponding $(n-1)$-minors of $A$ (up to a sign). What can we say about the $k$-minor of $\text{adj}(A)$ in relation ...
-1
votes
1answer
31 views

MAthematical notation for sorting submatrix and replacing it back

I need help in expressing the following paragraph in mathematical form as much as possible. I have a matrix $A$ which is $N\times M$. For each element of $A$, $A(i,j)$, I consider a submatrix of $A$ ...
1
vote
0answers
38 views

Trace of the product of a Lie algebra and Lie group element

Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What is known about \begin{align} \text{Tr} (UX) \end{align} In particular Are there any useful identities that apply here? When does $\text{Tr} (...
0
votes
1answer
19 views

Find a,b,c to match the linear transformation matrix?

P.S. Sorry for my bad explanation of the task, it was really hard to translate this into meaningful english For the given linear-transformation $A$ find all possible combinations of a,b,c for which ...
13
votes
1answer
435 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
0
votes
0answers
10 views

Pearson Correlation

I have two matrices, which are square but of different size. I want to find correlation between data which is stored in these two matrices. It seems Pearson Correlation Coefficient is applicable for ...
1
vote
1answer
108 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in \mathcal{M}_n(...
1
vote
1answer
28 views

Which is true about $Q$ where $Q=I+2P$

Let ${a_{1},a_{2},...a_{n}}$ and ${b_{1},b_{2},...b_{n}}$ be two bases of $\mathbb{R}^{n}.$ Let P be an $n \times n$ matrix with real entries such that $Pa_{i}=b_{i}$ for $i=1,2, ...,n.$ Suppose that ...
0
votes
0answers
14 views

Let $A^*$ denote the matrix whose $(ij)$-th entry is $A_{ij}$, $1 ≤ i, j ≤ 5.$

Let $A \in M_5(\Bbb R)$. If $A = (a_{ij})$, let $A_{ij}$ denote the co-factor of the entry $a_{ij}, 1 ≤ i, j ≤ 5.$ Let $A^*$ denote the matrix whose $(ij)$-th entry is $A_{ij}$, $1 ≤ i, j ≤ 5.$ a. ...
2
votes
0answers
10 views

Properties of matrix stable (numerical) rank

I happened to notice that there is concept "stable rank" that people used a lot in matrix computation theories, such as the work of Rudelson & Vershynin (2005). It is defined to be the ratio ...
5
votes
0answers
30 views

Hyperdeterminant of 4x4x4 hypermatrix

If given the hypermatrix (which I've written here in bracket notation since I'm not all too sure how to display this) { {{1,1,1,1},{1,1,-1,-1},{1,-1,-1,1},{1,-1,1,-1}}, {{1,1,1,1},{1,1,-1,-1},{1,-1,1,-...
0
votes
0answers
8 views

Problem in finding introductory material (matrix spectra)

I am looking for introductory material on: 1) matrix eigenvalue spectra and useful matrix algebra theorems that can be applied in the field. 2) Statistics of random matrices (i.e. ensembles, ...
1
vote
0answers
32 views

Condition number of a $2\times 2$ square block matrix

Is there a general rule to relate the condition number of the $2\times2$ square block matrix $ \left(\begin{array}{cc} A & B\\ C & D\\ \end{array}\right), $ where the matrices have the ...
4
votes
1answer
60 views

Nonsingular matrices with bounded coefficients

I can show that there exists $n^2$ positive integers $a_1,\ldots ,a_{n^2}$, such that each $n\times n$ matrix with coefficients $a_i$ (used once and only once) is nonsingular. Two questions: Could ...
1
vote
3answers
80 views

how can I create a random matrix with specific entries

I would like to create/generate a random square $n \times n$ matrix with the following specifications: the first and the last row of the matrix are nonzeros (i.e all the elements in the first and ...
0
votes
0answers
17 views

Name of a type of Hankel Matrix

Let's define the Hankel Matrix $H\in\mathbb{R}^{m\times m}$, such that $H_{i,j}=\rho^{2m-i-j}$, i.e., $$ H=\left( ...
0
votes
0answers
41 views

What is name of a matrix which construct from a identity matrix?

In linear algebra, the identity matrix ,$I$, or sometimes ambiguously called a unit matrix, of size $n$ is the $n \times n$ square matrix with ones on the main diagonal and zeros elsewhere, for ...
0
votes
1answer
46 views

rank of the matrices of the form $A+\lambda B$

Could anyone help me for the following problems? $1. $ If $A+\lambda B\in M_{m\times n}(\mathbb{R})$ where we can vary $\lambda\in \mathbb{R}$, has rank $r<n$ then what can we say about the rank ...
5
votes
2answers
299 views

Geometric understanding of the Cross Product

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
1
vote
1answer
46 views

Does $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({AY+YA}^T) = \{0\} $ imply $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({CAY}) = \{0\} $?

Given $ \mathbf{Y}=\mathbf{Y}^T \in \mathbb{R}^{n\times n} >0, \mathbf{A} \in \mathbb{R}^{n\times n} $ Hurwitz, $ \mathbf{C} \in \mathbb{R}^{m\times n}, \mathrm{rank}(\mathbf{C})=m,\ m \le n $, I ...
0
votes
2answers
26 views

What's wrong with this formula for the dot product of a vector and a matrix acting on that vector?

Suppose we have an $n \times n$ matrix $M$ and a vector $v$. I want to find an explicit formula for $v \cdot Mv$. I begin by saying $$v \cdot Mv = \sum_{i=1}^{n} v_i(Mv)_i$$ and since $Mv$ is given by ...
-1
votes
2answers
35 views

Example of a degenerate bilinear map?

I seek an example of a nonzero $\Bbb{R}$-bilinear map $f:V\times V\rightarrow W$ on a vector space $V$ (s.t: $\dim V<\infty$, $\dim W<\infty$) such that it is degenerate map, where $V$ and $W$ ...
2
votes
0answers
28 views

Resolvent Inequality

Let $H$ be a Hermitian matrix and $h$ some vector of the same length. The resolvent of $H$ at $z\in\mathbb C$ shall be denoted by $$G(z):=(H-z\cdot1)^{-1}.$$ Is it true that $$\frac{(\Im z)\left(1+\...
5
votes
3answers
740 views

Find diagonal of inverse matrix

I have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the cholesky ...
3
votes
1answer
50 views

Is there an $\alpha\in\mathbb{R}^m$, such that $\alpha_i > 0$ and $A\alpha\in S$?

$A$ is a real $n\times m$ matrix and set $S\subseteq \mathbb{R}^n$ is defined as $$S = \{(x_1,\dots, x_n)\in \mathbb{R}^n\mid \forall(i,j)\in I.\; x_i< x_j\}\text{,}$$ where $I$ is a possibly empty ...
2
votes
1answer
44 views

A non-strict inequality on skew symmetric matrices

As we know that skew-symmetricity means $A=-A^\top$ where $A\in\mathbb{R}^{n\times n }$. But recently I came across an inequality that states, $A+A^\top\preceq0$ can also be considered as an ...
1
vote
1answer
47 views

Matrix with irreducible minimal polynomial gives rise to a field

For a field $K$, $A\in Mat_n(K)$ with minimal polynomial (irreducible) $\mu_A(T)\in K[T]$ with $d=\deg\mu_A(T)$. Let $$E=\left\{\sum_{i=0}^{d-1} a_iA^i: a_i\in K\right\}\subset Mat_n(K).$$ Prove that $...
6
votes
0answers
93 views
+50

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
2
votes
2answers
1k views

Invertability of Singular 2x2 Matrix with all same real values.

Question: Let set G = { matrix [{a a},{a a}] such that a is real but not 0 } represent the set of 2x2 matrices with same elements of the reals excluding a = 0, show that G is a group under matrix ...
0
votes
4answers
42 views

If $Y = X\beta$ are a system of linear equations and that $X$ is NOT full rank. Is this system under or over determined?

Suppose I have a system of linear equations, $Y = X\beta$, where $Y$ is a $n$ by $1$ matrix, $X$ an $n$ by $n$ matrix, and $\beta$ a $n$ by $1$ matrix. Suppose that I know what $Y$ and $X$ are, and ...
0
votes
1answer
52 views

What exactly are operations involving tensors… In terms of their indices

So I have heard that tensor operations involve the faces of the rectangular prism. These are matrices right, and different properties of those matrices say things about the tensor? Could someone ...
0
votes
0answers
8 views

HMM Training: Testing convergence

What is the best way to test for convergence while training an HMM? I understand that we need to iterate till the change in parameters ( transition matrix, emission matrix ) is less than the threshold....
5
votes
1answer
80 views

Creating a tight frame of $\mathbb{R}^{n}$ when already knowing some of its vectors.

I'm wondering whether or not there's an optimal way for adding rows to a given matrix $S\in\mathbb{R}^{m\times mn}$, $m\leq n$, so that the columns of the resulting matrix form an orthogonal system of ...
2
votes
1answer
18 views

Can we get $\|A^\dagger x-B^{-1}x\|_2\leq \epsilon \|B^{-1}x\|_2$?

In the question: the $A\in R^{d\times d}$ is positive semi-definite, $B\in R^{d\times d}$ is positive definite, $x\in R^d$ is a vector, and $\epsilon$ is a variable that may depend on $A$, $B$ and ...
0
votes
1answer
43 views

Matrices that are invariant under a change of basis

Is there a special name or something for when a matrix is invariant under a change of basis, i.e. $$XAX^{T} = A$$ I'm trying to find what properties $A$ or $X$ have but it's a little hard to search ...
2
votes
2answers
659 views

Linear Algebra: Change of Basis

Let $A[a,b,c]$ and $B[d,e,f]$ be two non-standard bases. I have to find the $3\times3$ matrix that will convert a vector defined in terms of $A$ to $B$. My solution is: Let's assume a standard basis ...
2
votes
4answers
111 views

How to prove $I-BA$ is invertible [duplicate]

Show that $I-BA$ is invertible if $I-AB$ is invertible. And also, we have to prove that eigenvalues are same for $AB$ and $BA$ Till now, I used the equation $(I-AB)(I-AB)^{-1}=I$ which gives $(I-AB)...
0
votes
0answers
22 views

Show that this vector is not a function of $\tau$

I have a variance matrix given by: $\boldsymbol{\Sigma}\boldsymbol{\Sigma}^{'}+\Omega$ where $\Omega=\left(\begin{array}{cccc} \sigma_{\varepsilon}^{2}\psi\left(\tau_{1}\right) & 0 & \...
0
votes
0answers
24 views

How to estimate the product of the $k$ largest eigenvalues of a matrix

Now I have a question which let me to prove that the product of the largest $k$ singular values of a real matrix is always larger than the one of $k$ largest eigenvalues. For $k=1$, I use the ...
9
votes
3answers
365 views

Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements

I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is neither diagonal nor triangular. Is there a term for such matrices, and have they been researched?
0
votes
0answers
10 views

Spectral norm of the matrix derivation

I understand one possible way how to derive induced norm of symmetrix matrix M, i.e. $sup |M \tilde{x} |$, s.t. $|\tilde{x}|=\tilde{x}^T\tilde{x}=1$ (i.e. $\tilde{x}$ is lie in unit sphere) Here is ...
2
votes
2answers
37 views

is the trace of inverse of positive, positive definite matrix decreasing?

Let $A, B$ be non-negative, and symmetric positive definite matrices. If $A\le B$, i.e., all the entries of $B-A$ are non-negative, is it true that $\mbox{trace}(A^{-1}) \ge \mbox{trace}(B^{-1})$?
3
votes
1answer
56 views

The maximal rotation matrix

Let's consider two numbers calculated for a rotation matrix which are: $s_e=$ the sum of all entries of a matrix $s_a=$ the sum of absolute values of all entries for a given matrix. It ...
1
vote
2answers
41 views

If A is positive definite (but not necessarily symmetric) can you decompose it?

If A is a $2 \times 2$ matrix that is positive definite but may or may not be symmetric, does there exist another matrix B such that $A=B^TB$?
3
votes
2answers
33 views

How may I use a 3x3 matrix to simulate a larger square matrix?

I am using a game engine where the library only provides 3x3 matrices with the multiplication and inverse operation. I could build my own matrix library to provide larger matrices, but it would be ...
4
votes
2answers
77 views

Prove that if $y=(y_1, \ldots, y_n)$ is such that $y_1a_1 + \cdots + y_na_n = 0$, then $∀x ∈ \mathbb{R}^n$, $Ax · y = 0$

I have no idea how to start the following question. Any help will be greatly appreciated. (a) Let $A$ be a $n\times n$ matrix and let $a_1,\ldots,a_n$ be the rows of $A.$ Suppose $y=(y_1, \ldots, ...
2
votes
1answer
51 views

Black are berries and maroon are cherries. Place 8 more cherries removing berries 1 from each row and each column. No of ways?

I tried to see it as a matrix where for a position (i,j) , i+j = 8, 9, 16 means you can't change that position. Any help?
0
votes
1answer
22 views

Matrix Rotations and Enlargements Help

Matrix $M$ is given as \begin{bmatrix}3&-{\sqrt 7}\\{\sqrt 7}&3\end{bmatrix} I then am asked to describe the transformation, you are also told dis an enlargement followed by a rotation and you ...