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
1answer
256 views

Sorting Matrix to Block structure

I have a symmetric matrix and I want it to be as block-like as possible. I don't have a clear definition. I want the smallest number of groups of non zero elements or maybe the most non-zero elements ...
1
vote
2answers
439 views

Why do we need a diagonal matrix?

Apart from simplifying matrix powers, why do want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices. Thanks for any answers.
0
votes
1answer
89 views

Matrix norm $|||A|||_{2}=\max_{\lVert x \rVert_{2}=\lVert y \rVert_{2}=1}\lVert y^{*}Ax\rVert_{2}$

We know that $|||A|||_{2}=\max_{\lVert x \rVert_{2}=1}\lVert x^{*}Ax\rVert_{2}$. I have two questions about it. How can we prove $$|||A|||_{2}=\max_{\lVert x \rVert_{2}=\lVert y \rVert_{2}=1}\lVert ...
0
votes
1answer
93 views

Question regarding the diagonalizability of certain matrices

Let $A\in \mathbb{C}^{n^2}$ such that $A^m=I_n$, for some $m,n\in \mathbb{N}$. Please prove that $A$ is diagonalizable. Now let $B\in \mathbb{C}^{n^2}$ such that $B^m=B$, for some $m\in \mathbb{N}$ ...
1
vote
2answers
76 views

Math question using matrices

I have the following system: $$\left\{\begin{array}{cccccccc} 2x&+&3y&+&z&-&3v&=&2 \\ x&-&y&+&2z&+&v&=&0\\ ...
4
votes
3answers
155 views

Prove an identity including determinant

Prove that: $$\begin{equation} \begin{vmatrix} x_0^{2n+1}&x_0^{2n}&\cdots&x_0&1\\ x_1^{2n+1}&x_1^{2n}&\cdots&x_1&1\\ ...
0
votes
1answer
50 views

Solve system using matrices? Is this correct?

So I have the system of equations: $$ \begin{align*} 2a+3b+c-6d&= 1 \\ a-b+c+2d&=0\\ 3a+2b+3c-4d&=-1 \end{align*} $$ I have to prove that this system has no solutions. So, first I prove ...
0
votes
1answer
124 views

Orthogonally diagonalizable or symmetric real matrix - are all eigenvalues distinct?

For all orthogonally diagonalizable real matrices, or symmetric real matrices, are all eigenvalues distinct? What would be the proof it is so, or if not, what would be the proof?
1
vote
1answer
206 views

How do you construct a lattice from its basis or its Gram Matrix?

I'm really having trouble trying to understand this. A few weeks back, I got pretty interested in sphere packing and I'm trying to grasp the idea of using a matrix to represent the basis of a lattice. ...
3
votes
0answers
207 views

What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
3
votes
0answers
58 views

When can the commutator of two matrices be neglected in series expansions?

Under what conditions can the higher order commutators in the Baker–Campbell–Hausdorff formula be neglected when the commutators does not vanish exactly and there is no small parameter in the ...
1
vote
1answer
1k views

From a vector to a skew symmetric matrix

Is there an existing linear mapping that maps a 3-dimensional vector: $$\mathbf{v}=\begin{pmatrix} v_1\\v_2\\v_3 \end{pmatrix}$$ to a corresponding skew-symmetric matrix: $$\mathbf{V}=\begin{pmatrix} ...
18
votes
0answers
634 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
0
votes
1answer
99 views

biggest eigenvalue of inverse matrix dependent on original matrix?

I heard somewhere that the biggest eigenvalue of $A^{-1}$ is reciprocal to the biggest eigenvalue of $A$. In which theorem is this stated? Also, what would be the proof of it? Also, what happens if ...
4
votes
3answers
151 views

Matrix P to the power of 4, i.e $P^4$, is this the same as $P^2 \cdot P^2$?

Basically, what it says in the title. I have a $5 \times 5$ matrix and I need to work out $P^4$, is it possible to just do $P^2$ and multiply this with itself?
1
vote
1answer
277 views

Splitting Matrix Into Sub-Matrices With Constraints

I have a question regarding matrices for a personal project of mine. I have a large matrix that needs to be split into smaller matrices. I know its dimensions are X and Y. I know that the max amount ...
3
votes
0answers
49 views

finding the largest $p$ components of $x$

Given an $n \times n$ matrix $A$, and an $n \times 1$ vector $b$, the conventional way of computing an $n \times 1$ vector $x$ such that $x=Ax+b$ is to use the following iterations: ...
0
votes
1answer
98 views

Condition number of the product of a diagonal and a triangular matrix

Given a triangular matrix L and a diagonal matrix D, what can be said about the singular values of the product D*L ? Precisely, is it possible to express the singular values of D*L as function of ...
1
vote
1answer
41 views

Stabilization property of an operator in a finite-dimensional vector space?

Let $L: V\to V$ be an operator in a finite-dimensional vector space $V$ over $R$. For any $n \geq 0$, let $K_n = \ker (L^n)$, $I_n = \mathrm{Im}(L^n)$. (a) How do we prove the stabilization ...
2
votes
2answers
97 views

How to do $\frac{ \partial { \mathrm{tr}(XX^TXX^T)}}{\partial X}$

How to do the derivative \begin{equation} \frac{ \partial {\mathrm{tr}(XX^TXX^T)}}{\partial X}\quad ? \end{equation} I have no idea where to start.
0
votes
1answer
114 views

Finding the eigenvalues for a $3\times 3$ matrix

With the matrix $A$ given by $$\left( \begin{array}{ccc} 0 & -1 & 0 \\ 0 & 1 & a \\ 1 & 0 & 1 \end{array} \right)$$ the solution to the initial value problem $x'=Ax$, $x(0) = ...
3
votes
1answer
288 views

Eigenvalues of symmetric matrix in real inner product space

I got the following exercise to solve: Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix and let $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$ be its eigenvalues sorted in a ...
3
votes
1answer
493 views

Derivatives of a the Matrix diagonal function

If A is a not diagonal but symetric matrix and diag() is a function such that returns the diagonal, i.e. diag(A) is a matrix of zeros except on the diagonal. Im interested in the derivative of ...
0
votes
1answer
580 views

'H' symbol as a power of matrix

Here is capture of problem That is what I saw while I was studying maximum ratio combining (MRC) in communication. It is probably very simple and stupid thing to ask. I guess it is Hermitian of a ...
2
votes
1answer
156 views

Consider a matrix $A$ with integer entries such that $a_{ij}=0$ for $i>j$ and $a_{ii}=1$ . then which of the followings are true?

Consider a matrix $A=(a_{ij})_{ n ×n }$ with integer entries such that $a_{ij}=0$ for $i>j$ and $a_{ii}=1$ for $i=1,…,n$. then which of the followings are true? $A^{-1}$ exists and it has ...
0
votes
1answer
238 views

Similar Matrices in Subfields

This is a exercise question from Denis Serre, Matrices: Theory and Applications. Let $M$ and $N$ be two similar matrices in the field $K$. Let $k$ be the subfield spanned by the entries of $M$ ...
1
vote
2answers
777 views

derivative of a determinant of a matrix with respect to an element that appears many times in the matrix

I've been trying to find material on matrix calculus but it seems hard to find ones with understandable proofs. I'm doing research work and I am trying to verify some computation. Suppose that I have ...
1
vote
1answer
31 views

If $A\sigma=\sigma\tau$, does $[\tau]_B=A$ for some basis $B$?

I was trying to figure out the following at work today. Suppose $\tau\in\mathscr{L}(V)$, for $V$ an $n$-dimensional vector space over a field $F$. Let $A\in M_n(F)$, and $\sigma\colon V\to F^n$ be an ...
2
votes
3answers
96 views

norm of a product of matrices

Suppose for every $x \in \mathbb{R}$ and $y \in [0,1]$, $M(x,y)$ is an $n$ by $n$ matrix and suppose that for every $y \in [0,1]$, $M(x,y) \to M_\infty$ as $|x| \to \infty$, where $M_\infty$ is a ...
0
votes
1answer
55 views

What does “the Gershgorin discs $C_j$ corresponding to the columns of $A$” mean?

It says in Wikipedia: Corollary: The eigenvalues of $A$ must also lie within the Gershgorin discs $C_j$ corresponding to the columns of $A$. Before that, a Gershgorin disc is defined as. ...
2
votes
1answer
80 views

The derivative of characterestic polynomial?

Let $A\in M_{n}(R)$ and $f(x)$ be the characterestic polynomial of $A$. Is it true that $f'(x)=\sum_{i=1}^{^{n}}\sum_{j=1}^{n}\det(xI-A(i\mid j))$ which $A(i\mid j)$ is a submatrix of $A$ obtained by ...
3
votes
1answer
139 views

Why does this covariance matrix have additional symmetry along the anti-diagonals?

In my self study of a statistics book, I came across a page that has confused me somewhat. I am already familiar with covariance matricies, (or maybe not!), and the author's explanation leaves me a ...
1
vote
1answer
112 views

interesting matrix

Let be $a(k,m),k,m\geq 0$ an infinite matrix then the set $$T_k=\{(a(k,0),a(k,1),...,a(k,i),...),(a(k,0),a(k+1,1),...,a(k+i,i),...)\}$$is called angle of matrix $a(k,0)$ is edge of $T_k$ ...
1
vote
4answers
688 views

how does addition of identity matrix to a square matrix changes determinant?

Suppose there is $n \times n$ matrix $A$. If we form matrix $B = A+I$ where $I$ is $n \times n$ identity matrix, how does $|B|$ - determinant of $B$ - change compared to $|A|$? And what about the case ...
1
vote
0answers
51 views

Smith normal form when vector has an entry with zero

Based on the tags, I'm assuming that the $x_i$ and $a_i$ are supposed to be integers. The procedure below generalizes to the case where you want simultaneous solutions to multiple equations. ...
2
votes
1answer
129 views

For square matrices $A$, $B$, is $AB=I$ sufficient that $A$ and $B$ are inverse of each other? [duplicate]

Possible Duplicate: If $AB = I$ then $BA = I$ If $A$ and $B$ are two square matrices, and we know $AB=I$ where $I$ is the identity matrix. Is it sufficient that $BA=I$ as well so that $A$ ...
1
vote
2answers
780 views

Find a matrix $B$ such that $B^2=A$

$A= \begin{pmatrix} 1 & 2+3i \\ 2-3i & -1 \end{pmatrix}$ What is matrix $B$ such that $B^2=A$? Its eigenvalues are $\sqrt{14}, -\sqrt{14}$ and I tried to use formula ...
1
vote
1answer
722 views

Commuting matrices - unclear on steps

I would like to find all matrices that commute with matrix $$ A =\begin{pmatrix}1 & -1 \\ 0 & 1 \end{pmatrix}$$ Proposed solution $\begin{pmatrix}a&b \\c ...
1
vote
2answers
104 views

does multiplication of singular matrix with some matrix result in singular matrix?

Suppose that there is singular (non-invertible) matrix $A$. If it gets multiplied by any square matrix $B$, would $AB$ be singular?
1
vote
1answer
160 views

Input-output economics and stability of general equilibrium

Here, I will start with a simple expression for an input–output system with $x(t)$ representing the vector of outputs and $A$ the input–output matrix. Then, the simplest possible linear ...
2
votes
1answer
195 views

Reconstructing a Matrix in $\Bbb{R}^3$ space with $3$ eigenvalues, from matrices in $\Bbb{R}^2$

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three ...
1
vote
1answer
64 views

Power of a matrix and its symmetricity

Let $A$ be a real $N\times N$ matrix. If $A^k$ is symmetric for some $k>0$, does that give away something about $A$.
1
vote
4answers
73 views

Number of possible ways to form matrix so that $Ax =Ix$

Suppose that a vector $x$ is given. We want to find out the total number of possible ways to form a matrix $A$ so that $Ax = Ix$ where $I$ is $n \times n$ identity matrix and $A$ is some $n \times n$ ...
2
votes
1answer
97 views

Second Order Matrix derivative $\frac{\partial Tr(XBX^TA)}{\partial X}$

Let $F(X) = Tr(XBX^TA)$, where A,B are matrix. What is the derivative $\frac{\partial F(X)}{\partial X}$? Is it something like $A(XB^T+XB)$?
2
votes
1answer
255 views

Decompose a complex symmetric matrix to retain positive definitness

I have a complex symmetric matrix $A$, (i.e. non-Hermitian and obeying $A=A^T$), which is positive definite, in the sense that: $$\Re({z^HAz}) > 0$$ for any $z$. I am able to verify this ...
1
vote
1answer
88 views

What is the necessary condition for a matrix to have eigenvalue 1?

What would be the necessary condition for a matrix of any $n \times n$ to have eigenvalue 1? I know that it must have a corresponding eigenvector - that is obvious - I want to know things like how ...
2
votes
2answers
196 views

Matrix permanent evaluation problem

I am asked: Let $n$ be a positive integer, let $I = \{1, 2, 3, . . . , n\}$ and let $A$ be the $n \times n$ matrix whose entry in row $i$ and column $j$ for each $i, j \in I$ is equal to ...
1
vote
0answers
122 views

Probability of a matrix having determinant zero

What would be the probability of matrix having determinant zero out of all matrices with all entries being positive? How does one calculate such? Edit: Restriction to natural numbers and size of $n ...
2
votes
4answers
1k views

Eigenvectors of similar matrices

If $A$ and $B$ are similar matrices then every eigenvector of $A$ is an eigenvector of $B$. Is the above statement is true? I know that similar matrices have same eigenvalue but I'm not sure about ...
1
vote
2answers
64 views

Uniqueness of inverse matrix and possibility of $P=PX$

Suppose that there is non-zero vector $P$ of size $1 \times n$. 1) Does there exist some $P$ that $P=PX$ without $X$ being identity matrix? 2) When $AB = BA = I$ and $A$ given, can there be several ...