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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0answers
5 views

nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right). $$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
1
vote
0answers
17 views

Order of $\mathrm{SL}(n,\mathbb{F}_p)$ (Constructive proof)

Most proofs of $$ \vert ~\mathrm{GL}(n,\mathbb{F}_p) ~\vert = \prod_{k=0}^{n-1} (p^n-p^k) $$ I have seen so far, are done by counting the possibilities to build up invertible matrices i.e. counting ...
0
votes
1answer
40 views

Find $B(B^{T}B)^{-1}B^{T}$.

To find: $$B(B^{T}B)^{-1}B^{T}$$ for $B=[0,1,-1]^T$ I have $$\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \left ([0,1,-1]\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \right )^{-1}[0,1,-1]$$ but ...
6
votes
4answers
184 views
+50

Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
2
votes
1answer
31 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
2
votes
0answers
30 views

Taylor series of a square matrix

Let $A$ be a constant square matrix, $\Delta t$ is a scalar. I would imagine a taylor series of its exponential like this: $$e ^{A\,\Delta t} = I + \sum_{n=1}^\infty \frac{(A\,\Delta t)^n}{n!} $$ ...
0
votes
1answer
11 views

Write summation of vector outer products into matrix form

My question is as follows: Given the weighted summation of vector outer products $\sum_i\sum_jh_{ij}{\bf v_i}{\bf u_j}^T$, where $h_{ij}$ is the weight, and ${\bf v_i,u_j}$ are column vectors, I was ...
0
votes
0answers
9 views

Non symmetric, zero column sum, positive diagonal and negative off diagonal entries matrix bound?

Under which conditions can the A-inner product of a non symmetric, zero column sum, positive diagonal and negative off diagonal entries matrix be bounded by the L2-inner product? $A \in \mathbb{R}^{n ...
0
votes
2answers
32 views

Mathematical calculation

I encountered during my reading to ridge regression that $$(X^TX+\lambda I)^{-1}X^TX = I-\lambda(X^TX+\lambda I)^{-1}$$ What mathematical manipulation has been done here? Thanks in advance
1
vote
2answers
42 views

Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
0
votes
2answers
36 views

Getting characteristic polynomial from a small matrix

Sorry I don't know how to format matrices, but if I have this matrix $\pmatrix{1& 1& 0\\ 0& 0& 1\\ 1 &0& 1\\}$ How is the characteristic polynomial $λ^3 − 2λ^2 + λ − ...
1
vote
2answers
72 views

An inequality on the rank of a block matrix

Let $\mathbb F$ be a field, and let $r_1, r_2, s_1, s_2$ be positive integers. Consider the matrix $$X:=\left[\begin{array}{cc} A & B \\ C & D \end{array} \right],$$ where $A \in \mathbb F^...
0
votes
2answers
446 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
0
votes
1answer
11 views

Matrix for a recurrence

The matrix for a recurrence of the form $a_{k+2} = ka_{k+1}+a_{k}$ where $a_0 = 0$ and $a_1 = 1$ is given by $$\begin{bmatrix}k & 1\\ 1 & 0 \end{bmatrix}^n = \begin{bmatrix} a_{k+1} & a_k \...
3
votes
1answer
316 views

$\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$

Let $A \in M_{n \times m}\left(F\right)$ and $B\in M_{m \times p}\left(F\right)$ for a field $F$. Prove: $\dim C(AB)=\dim C(B)-\dim(\operatorname{Null}(A)\cap C(B))$, where $C(X)$ denotes the column ...
0
votes
0answers
12 views

Calculating the null space and the column space

Suppose I have matrices $A_{n\times n},B_{n\times n}$ and the appended matrix $[A \hspace{0.25cm} B]_{n\times 2n}$, suppose both $A,B$ are of rank $n-1$, could anyone tell me how the followings are ...
5
votes
4answers
118 views

Not understanding derivative of a matrix-matrix product.

I am trying to figure out a the derivative of a matrix-matrix multiplication, but to no avail. This document seems to show me the answer, but I am having a hard time parsing it and understanding it. ...
0
votes
1answer
48 views

Trace zero means matrix is nilpotent?

I have to prove or disprove: If $A$ is an $n \times n$ matrix in $\mathbb{Z}/p\mathbb{Z}$ for any prime number $p$ and the trace of any power of $A$ is $0$, then the matrix is nilpotent: $A^k = 0$ ...
3
votes
1answer
129 views

Inverse of a matrix and its transpose

I'm trying to figure out why the calculation below works. I do know that $(A^T)^{-1} = (A^{-1})^T$. The matrix A = $\begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1 ...
0
votes
0answers
20 views

sending basissen

Lets say we have this $3\times3$ matrix: $$ \begin{bmatrix} 4 &−4 &12\\ 1& -1& 3\\ −1& 1 &−1 \end{bmatrix} $$ What is the algorithm to find a basis of $\Bbb R^3$ for which ...
4
votes
2answers
95 views

What are the rules for taking derivatives in linear algebra?

I was reading through a paper on beamforming and came across an equation whose derivative I don't fully understand. A cost function is given as: $$ J(\mathbf{w}) = \mathbf{w}^HR\mathbf{w} +\lambda^*[...
1
vote
2answers
45 views

Compute $R^{2016}$ of a given counterclockwise rotation.

Write out the matrix $R$ of counterclockwise rotation by 30$^{\circ}$ in $\mathbb{R}^2$. Compute ${R}^{2016}$. Now this is an easy question to answer overall; 30 goes into 360 12 times and one twelfth ...
2
votes
1answer
78 views

Does $\mathrm A \mathrm A^T \succeq x^2 \mathrm I$ imply $\frac{\mathrm A + \mathrm A^T}{2} \succeq x \mathrm I$?

Let $A $ be an $n \times n $ matrix such that $AA^T \geq x^2I, x\geq 0 $, which means that the matrix $AA^T-x^2I$ is positive semidefinite. Can we show that $(A+A^T)/2 \geq xI$? Thanks
3
votes
3answers
539 views

Space spanned by matrices

I have a set of $5 \times 5$ matrices, $M_1, M_2,\dots, M_{19}, M_{20}$. I want to try to find a basis from this set and also to find relationships between these matrices. This is how I think I ...
1
vote
0answers
22 views

Element-wise derivative of matrix logarithm

$E = ln(C) = -\sum_{a=1}^{\infty}\frac{1}{a}(I-C)^a$ I want to find a simple formula for $\frac{\partial E_{ij}}{\partial C_{pq}}$ $\frac{\partial C_{ij}}{\partial C_{pq}} = \delta_{ip}\delta_{...
0
votes
1answer
47 views

$A^2$ is bounded $\implies$ $A$ is bounded?

Let $A_n$ be a sequence of $k \times k$ real matrices. Assume $A_n^2$ is bounded w.r.t some norm. Is $A_n$ also bounded? I was able to show this is true if $A_n$ are symmetric matrices (using SVD). ...
1
vote
3answers
45 views

Matrix and scalar multiplication

Say we have the following variables: A, a matrix that is nxn in size containing complex numbers B, a matrix that is also nxn in size containing complex numbers x, a scalar If you multiply, does it ...
0
votes
1answer
35 views

Time Derivative of a Positive Definite Matrix

Suppose we have a positive definite symmetric matrix $\mathbf V(0) \in \mathbb S^{n}_{++}$, which changes with time according to the following equation, $\dot{\mathbf V}(t) = \mathbf A \mathbf V(t) + ...
0
votes
1answer
52 views

Is there an effect for the eigenvalues on vectors other than the Eigenvectors?

Does having an eigenvalue greater than one mean that the magnitude of any vector multiplied by the matrix will be increased?
0
votes
2answers
37 views

Regarding element-wise derivative of matrices

Let $X=(x_{ij})$ be a real n-by-n matrix where $x_{ij}$ are in a range such that $X$ is insvertible. Let $Y=X^{-1}=(y_{pq})$, and regard $y_{pq}=y_{pq} (x_{11},x_{12},\ldots,x_{nn})$ as a function of ...
7
votes
3answers
109 views

Inverse of the Pascal Matrix

Let $P_n$ be the $(n+1) \times (n+1)$ matrix that contains the numbers of Pascal's triangle in the upper triangle. For example in the case of $n=3$ $$ P_3 = \begin{pmatrix} 1 & 1 & 1 & 1 \...
5
votes
1answer
12k views

Expressing the determinant of a sum of two matrices?

Can $$\det(A + B)$$ be expressed in terms of $$\det(A), \det(B), n$$ where $A,B$ are $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
1
vote
1answer
56 views

Linear independence of standard basis vectors from Vandermonde style vectors

Is it true a statement that all $n$ dimensional vectors of the standard basis (e.g. $[1 \ 0 \ 0 \ ...]^T$, $[0 \ 1 \ 0 \ ...]^T$ etc ..) are linearly independent from the set of the $n-1$ vectors $...
2
votes
1answer
2k views

How to remove linearly dependent rows/cols

How would one remove linearly dependent rows/columns from a rank-deficient matrix. For example, (from wikipedia): $$ A = \begin{bmatrix} 2 & 4 & 1 & 3 \\ -1 & -2 & 1 &...
1
vote
1answer
51 views

How to prove that the vectors of the Krylov space of A are linearly independent if A is nonsingular.

$\mathbf{K}$ is a Krylov matrix. \begin{align} \mathbf{K}&= \left[ \begin{array}{ccccc} \mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots & \mathbf{A}^{N-1}\...
1
vote
1answer
111 views

Representing natural numbers as matrices by use of $\otimes$

What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number $n$, how can I represent this number as a matrix in which I can do matrix multiplication ...
0
votes
1answer
21 views

QR decomposition subcases

Is the full QR decomposition the most general, which includes the reduced QR, i.e, is it alright to always compute the full QR Decomposition for a given matrix blindly? What's the point of having two ...
1
vote
1answer
32 views

Condition for Linear Dependence

Let $\mathbf{x}\neq \mathbf{0}$ and $\mathbf{y}\neq \mathbf{0}$ be $n \times 1$ vectors, $\mathbf{A}\neq \mathbf{0}$ and $\mathbf{B}\neq \mathbf{0}$ be $m \times n$ matrices with $m>n$, and, for ...
1
vote
2answers
528 views

Similarity in two 2x2 Matrices and finding the S in A=SBS-1

I am doing something wrong here and I am not sure what. The object of the exercise is to find the S for similar matrices $A$ and $B$. $A=SBS^{-1}$ with $B=\begin{pmatrix}4& 1\\1& 2\end{...
5
votes
1answer
53 views

Rank of a lower triangular block matrix

For $$A= \begin{bmatrix}B&0\\C&D\end{bmatrix}$$ where $B, C, D$ are matrices that may be rectangular, is it true or false that $$\text{rank}(A)=\text{rank}(B)+\text{rank}(D)$$ I think that if ...
-1
votes
1answer
35 views

Range space of matrices over $\mathbb{Z}$

Let A and B be $m \times n$ matrices over $\mathbb{Z}$ such that $B=PAQ$ for some invertible matrices P and Q. Then can we tell that Range space of A is same as that of the range space of B when A ...
0
votes
0answers
24 views

Linearize Matrix Equation

I want to find a linearized formula for G in terms of A. $G = B^TC^{-1}T(I+BA)$ $G$ is 4x2 $B$ is a constant matrix 2x4 $A$ is a variable matrix 4x2 $C = I + A^TB^T + BA + BAA^TB^T$, so $C$ is ...
2
votes
1answer
27 views

Characterization of a square matrix.

I would like to see a proof to this fact. For a square matrix the following are equivalent: $A$ has a right inverse. $A$ has rank $n$, where $A$ is $n \times n$. $A$ is invertible.
1
vote
1answer
31 views

Right inverse matrix

I know that if $A, B$ and $C$ are square matrices such that $$ AC=I \quad \mbox{and} \quad BA=I, $$ then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \...
1
vote
2answers
47 views

A proof of the Continuity of the inverse matrix function

I would like to see a proof to this fact. If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there ...
2
votes
2answers
49 views

Given a symmetric matrix $A$, find $P$ such that $P^T A P$ is a diagonal matrix

Given $$A =\begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0\end{pmatrix}$$ find a matrix $P$ such that $P^T A P$ orthogonally diagonalizes $A$. Verify that $P^TAP$ is ...
0
votes
0answers
47 views

Simplify Series composed by Noncommutative Matrices

Problem I need to find a simpler formula for the following series: S = $\sum_{a=1}^{\infty} \frac{1}{a} \sum_{b=1}^{a} X^{b-1}MX^{a-b} = \sum_{a=1}^{\infty} \frac{1}{a} X^{a-1} \sum_{b=0}^{a-1} X^...
3
votes
1answer
91 views

Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...
15
votes
2answers
244 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose $\...
0
votes
0answers
19 views

Change eigenvalues of correlation matrix and transform into original basis

I use the Random Matrix Theory to filter out the information from the correlation matrix that is associated with noise - Marcenko Pastur band. That is straight forward. Then I follow Rosenow, Bernd, ...