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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1answer
56 views

On determinants computation

How can be proved this identity between determinants? $$\left|\begin{array}{cccc} 1&a&c&ac\\ 1&b&c&bc\\ 1&a&d&ad\\ 1&b&d&bd \end{array}\right|=\left| ...
0
votes
1answer
2k views

Consistency of a System of linear equations

Test the consistency of the system of linear equations $$\begin{align} 4x-5y+z & =2 \\ 3x+y-2z& = 9 \\ x+4y+z& =5\end{align}$$
4
votes
2answers
116 views

Determinants of Block Matricies

I read on wikipedia that $Det \begin{pmatrix} A & B\\B& A\end{pmatrix}$ is equal to $ Det(A+B)Det(A-B) $ if $A$ and $B$ commute. Does this hold true even if $ A $ and $ B$ are not ...
1
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0answers
39 views

Is there a special name for matrices with $M[j,i] = M[i,i] - k, i \neq j$?

Backgroud: I am working on a computer science problem and arrives at a matrix $M$ with the following property: The size of Matrix $M$ is $n\times n$. For each row $j$, we have $M[j,i] = ...
2
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0answers
100 views

Matrix of integers to boolean matrix

My Question is about converting a matrix of numbers, say each row is an item and each column is a feature of the item. The features are currently integers but I want to convert the feature ...
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1answer
154 views

Clarification on matrix notation subscript and superscript notation

If a matrix C exists in integers $\mathcal{Z}_q^{mxl}$ what does this mean?
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1answer
165 views

homework - Find a basis for the space of all vectors in R6 with x1 + x2 = x3+ x4 = x5+ x6

a) Find a basis for the space of all vectors in $\mathbb{R}^6 $ with $x_1 + x_2 = x_3 + x_4 = x_5 + x_6$. b) Find a matrix with that subspace as null space. c) Find a matrix with that subspace as ...
3
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0answers
66 views

The eigenvector of Laplacian matrix plus a rank one matrix

Denote $L=\left[\begin{array}{ccc} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{array}\right]$ and $M=\left[\begin{array}{ccc} 1\\ & 0\\ & & 0 \end{array}\right]$. ...
2
votes
0answers
81 views

Determining the matrix of a linear transformation

Let $A$ be an $n\times n$ matrix, and let $V$ denote the space of $n$-dimensional row vectors. What is the matrix of the linear operator ‘‘right multiplication by $A$’’ with respect to the standard ...
0
votes
1answer
53 views

Matrix inversion for a $3\times3$ matix

How does one show that a $3\times3$ matrix is invertible? The matrix is: $$ \left(\begin{array}{ccc} \cos X & -\sin X & 0 \\ \sin X & \cos X &0 \\ 0 & 0 & 1 ...
1
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1answer
278 views

the identity matrix is unique

let A be an $m\times n$ matrix. Prove that there are unique matrices $I_m$ and $I_n$ such that : $$I_mA=AI_n=A$$ Actually I can't prove the uniqueness here,any help is appreciated. Thanks
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0answers
48 views

Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
0
votes
2answers
67 views

Linearly independent matrix proof

Suppose that $S=\{u_1,u_2,…,u_n\}$ is a set of vector from $\mathbb{R}^m$. Show that $S$ is linearly independent if and only if the set $S'=\left\{u_1,\ \sum_{i=1}^2 u_i,\ \sum_{i=1}^3 u_i,\ \ldots,\ ...
1
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3answers
56 views

Why the columns of $A$ are linearly dependent set?

If $A_{m\times n}$ is a matrix such that $\sum_{j=1}^n a_{ij}=0$ for each $i=1,2,…,m,$ then why the columns of $A$ are linearly dependent set, and hence $\operatorname {rank}(A)<n$?
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0answers
56 views

Given an n x n matrix A, is there an n x n matrix E such that $A\odot E=A$ and $A \odot F=E$?

For two matrices of dimensions $m \times n$ and $n \times k$, define $C=A\odot B$ to be the matrix with entries $$C_{ij}=\max_{k=1}^n A_{ik} + B_{kj}$$. Given an $n \times n$ matrix $A$, is there ...
25
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2answers
1k views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : ...
2
votes
2answers
76 views

about Fourier transformation on zero-padded vector

I have a vector $x$ of n elements. I did a fft on it and return another vector of n elements also (i.e.$X = \text{fft}(x)$). Now I am trying to pad the $x$ vector by n zeros so to get $y$ $$ y = [x ...
0
votes
1answer
78 views

about diagonal matrix and eigenvalues

I am reading the introduce of linear system and eigenvalues. There I read if there is a matrix $A$ and vector $x$, it could find a eigenvalue $\lambda$ such that $$Ax = \lambda x$$ I have a really ...
0
votes
1answer
41 views

Matrix column addition

Suppose you have a matrix in the form of: $$\left[\begin{array}{c} a\\ b\end{array}\right]$$ How can this be represented be a two by two matrix?
3
votes
0answers
39 views

The singularity of a family of matrices

Let $l\ge2$ be an even integer, $\zeta$ be a primitive $l$th root of unity in $\mathbb{C}$. Is it true for any $\alpha=(\alpha_1,\dots,\alpha_l)$ and $\beta=(\beta_1,\dots,\beta_l)$ such that ...
1
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2answers
73 views

Help with calculating the determinant

Does anyone know how to go about answering the following? Any help is appreciated! Calculate the determinant of $D = \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$ and use it to find ...
1
vote
1answer
54 views

Does $\mathbf{W}^H\mathbf{W}=\mathbf{I}$ imply $\mathbf{WW}^H=\mathbf{I}$?

Does $\mathbf{W}^H\mathbf{W}=\mathbf{I}$ imply $\mathbf{WW}^H=\mathbf{I}$? Note: $\mathbf{W}$ is a square complex constant matrix, $\mathbf{W}^H$ is the conjugate transpose of $\mathbf{W}$, and ...
1
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1answer
124 views

Finding a mapping such that its kernel equals the image of another non bijective mapping

For an $a \in \mathbb{R}$ let $\phi_a: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear mapping such that $\phi_a(x) := \begin{pmatrix} 1 & 2 & 2 \\1 & 3 & 5 \\ 1 & -1 & a ...
16
votes
8answers
1k views

How can I show that $\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$?

Well, the original task was to figure out what the following expression evaluates to for any $n$. $$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^{\large n}$$ By trying out different values of ...
2
votes
2answers
130 views

How to solve this system of linear equations

$$M = \left(\begin{smallmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4\\ a_1 & c_2 & b_2 & c_4\\ a_4 & d_2 & b_3 & c_4\\ b_1 & c_2 & a_2 ...
8
votes
1answer
165 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
2
votes
1answer
554 views

How to solve the matrix equation $ABA^{-1}=C$ with $\operatorname{Tr}(A)=a$

I have the following matrix equation: $$ABA^{-1}=C$$ with $B$ and $C$ given and $A$ unknown. The constraint on $A$ is $\operatorname{Tr}(A)=a$ with $a\in\mathbb{R}$. The matrices are $N\times N$.
0
votes
3answers
779 views

If a triangular matrix $A$ is similar to diagonal matrix, then $A$ is already diagonal matrix

Question is to Prove/Disprove : If a triangular matrix $A$ is similar to diagonal matrix, then $A$ is already diagonal matrix. I guess the answer is Yes. I am not able to write the solution ...
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0answers
67 views

How to solve this nonlinear matrix equation

I would like to solve for $A$ is the following equation. $$ A^{-1} = B + \sum_{k=1}^d\sum_{l=1}^d A_{kl} C^{(kl)} $$ where $A$ is a $d\times d$ positive semi-definite matrix $B$ is a $d\times d$ ...
0
votes
1answer
90 views

If A invertible, is $A + A^{T}$ invertible ?

I'm trying to solve the following problem about matrices: If $A$ is invertible is $A + A^{T}$ invertible? This is what I have done so far: $A + A^{T}$ $A(A^{-1}) + A^{T}(A^{T})^{-1} = 2I$ $I + ...
2
votes
1answer
192 views

Gradient of a scalar function with respect to a matrix

I need to calculate $\dfrac{\partial}{\partial K}f(K)$, with: $$ f(K)=-\frac{1}{2}(u-Kx)^T\Sigma^{-1}(u-Kx)$$ $K$ and $\Sigma$ are $n\times n$ matrices, $\Sigma$ is symmetric, $u$ and $x$ are column ...
0
votes
2answers
49 views

linear map question

Prove that $f: \Bbb{R}^2 \to \Bbb{R}^2$ is a linear map if and only if there exists $a,b,c,d \in \Bbb{R}$ such that $f$ acts like multiplication by the matrix $$\left( \begin{array}{cc} a & b \\ ...
0
votes
2answers
35 views

What is the term to make one matrix from two or more?

I am looking for the proper term for the operation of creating one block matrix from two or more for example $[AB]$ from $A$, $B$. And what is the correct notation to denote such a matrix. Do we use a ...
1
vote
1answer
80 views

Inverse a matrix $B+\lambda C$, where $\lambda$ is variable.

In my research I need to calculate $\operatorname{Trace}(A^{-1} C)$ where $A$ is given by two large, but sparse, matrices $B$ and $C$ by $A=B+\lambda C$. I need to do this inversion many times, so ...
2
votes
1answer
42 views

Summing infinite numbers of matrices

Let $\mathbf{I}$ be a identity matrix, and $\mathbf{A}$ be a symmetric matrix in which every entry $a_{ij}$ follows $0 \le a_{ij} < 1$. I want to get $\mathbf{S} = \mathbf{I} + \mathbf{A} + ...
0
votes
1answer
83 views

Extremum of a multidimensional quadratic function

I have the following function: $$ g(h) = h'\Sigma\Sigma'h-h'm-r, $$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
0
votes
1answer
534 views

Simultaneous Equations That Should Be Inconsistent Has a Unique Solution

Find the values of $k$ for which the simultaneous equations do not have a unique solution for $x, y$ and $z$. Also show that when $k = -2$ the equations are inconsistent $$kx + 2y +z =0$$ $$3x + 0y ...
1
vote
2answers
134 views

Describe the set of all matrices C such that $C^{-1}AC = A$

Given a square matrix A, can we find all invertible matrices C such that $A = C^{-1}AC$ ? In other words, can we find a set of all bases such that the matrix of an endomorphism $f$ in those bases is ...
2
votes
1answer
146 views

Variance Covariance Matrix, positive definiteness

Suppose we have a variance covariance matrix $\Sigma$. Under what conditions on the variance covariance matrix, $\Sigma$ is positive definite, that is $\forall w \neq 0, w^T \Sigma w>0$. In fact, ...
0
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2answers
265 views

Given a matrix factored into a product, how do you determine the determinant?

I'm preseneted with the question: Suppose that a 3x3 matrix A factors into the product of the two matrices below: \begin{matrix} 1 & 0 & 0 \\ I21 & 1 & 0 \\ I32 & I32 & ...
2
votes
0answers
115 views

What are the real world uses of Eigenbasis

The title pretty much says it all, I am wondering what the real world application (especially pertaining to electrical engineering) of an Eigenbasis is. I am also having some trouble understanding ...
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1answer
230 views

Questions about unipotent matrices

I'm reading Lang's Algebra. There is an example on page 19. Let $k$ be a field. Let $I$ be the unit $n\times n$ matrix, let $N$ be the additive group of matrices which are zero on and below the ...
5
votes
0answers
107 views

Complex Numbers vs. Matrix

I have a line starting at the origin, and i extend it to a point $(a,b)$ in the plane. This thing can be called a vector and be represented as $(a,b), [a\text{ }b]^T$ (column vector) or by ...
1
vote
1answer
49 views

Consequences of the properties of B on the results of a Generalized Eigenvalue $\lambda$ B v=Av

I'm trying to find a good source for the consequences of the properties of the matrix $B$ for the generalized eigenvalue problem: $\lambda B v = A v \Leftrightarrow \textrm{eig}(B^{-1}A)$ For ...
0
votes
1answer
36 views

About invariant spaces

Let $A\in M_{n}$ and $W\subseteq\mathbb{C}^n$ be a subspace, such that $\textrm{dim}(W)\geq 1$. If $W$ is $A$-invariant, then $A$ has an eigenvector in $W$. I don't know how to prove it.
2
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1answer
82 views

Proof about diagonal matrices

Suppose that $A\in M_{n\times n}(\mathbb{R})$ such that their eigenvalues are $\{\lambda_1,\cdots, \lambda_n\}$, i.e. $\sigma(A)=\{\lambda_1,\cdots,\lambda_n\}$, then if the geometric multiplicity ...
1
vote
0answers
467 views

Frobenius Inequality Rank

I was looking for an answer for this problem in terms of matrices, but I really don't know how to prove this result. The proposition says that: Let $A\in M_{m\times k}(\mathbb{C})$, $B\in M_{k\times ...
4
votes
1answer
106 views

Matrices $B$ that commute with every matrix $C$ commuting with $A$

There have been many questions in the vein of this one, but I can't find one that answers it specifically. Suppose $A,B\in M_n(\mathbb C)$ are two matrices such that, for any other $C\in M_n(\mathbb ...
2
votes
1answer
1k views

inverse of diagonal plus sum of rank one matrices

Is there formula for the inverse of a matrix which is diagonal plus a sum of rank one matrices? $$S=\alpha I + \sum_1^N u_iu_i^T$$ $$S^{-1} = ?$$ Is there any decomposition or special trick that I ...
0
votes
1answer
36 views

Transform matrix to zero diagonals

Given a Hermitian positive semidefinite matrix $A$, is it possible to find a unitary matrix $U$ such that $UAU^H$ has zeros along the diagonal?