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
48 views

What exactly is meant by linearity of a transformation?

I understand that formula for proving that it is linear, however, I don't understand what makes the transformation itself linear. This example from my notes has completely confused me. How do I even ...
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3answers
86 views

What is meant by a transformation being linear?

Im doing a calculus course and just started linear algebra and matrices. I understand most matrix rules like multiplication, reduction and so on. When it comes to linear mapping I'm completely ...
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1answer
51 views

Singular matrix proof

Let $A,B$ are real orthogonal matrices of an odd order $n\in\mathbb{N}$. Prove that at least one matrix of the form $A+B$ and $A−B$ has to be singular. For $n=3$ we can choose $A=B= \begin{...
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0answers
30 views

Matrix family : is it a generating family?

I have the following family of matrices: $$ F = \left\{ \begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix} ; \begin{pmatrix} -1 & 4 \\ 0 & 1 \end{pmatrix} ; \begin{pmatrix} -1 & 6 \\ ...
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2answers
48 views

Eigenvectors span linear space of dimension 1

I need to show whether or not eigenvectors of the matrix below span a linear space of dimension 1. $$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ Looking at the matrix you can see that it ...
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0answers
38 views

Using Gershgorin's theorem to show tridiagonal matrix is nonsingular

I need to use Gershgorin's theorem to show that the following matrix is non-singular \begin{array}{ccccc} 3i & i & \\ i & 3i & i \\ & . & . & . \\ && . & . &...
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1answer
62 views

Standardizing a two-dimensional random vector

Suppose that we have a two-dimensional random vector $$ \left(\begin{array}{c}X \\Y\end{array}\right) $$ with random variables $X$ and $Y$ such that $\operatorname EX=\operatorname EY=0$, $\...
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0answers
14 views

Loss functions in vector style or not?

Suppose I use bold lower case for vectors, and I calculate the loss function as the square function the vectors: $(\mathbf{\hat{y}}-\mathbf{y})^{2}$ Now if I call this the loss L (or Cost E or ...
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1answer
27 views

$A$ Hermitian and $0<P\leq I$ is projector. Is $Tr (PA)\leq Tr(A)$ correct? [closed]

Suppose $A$ is a Hermitian positive semi-definite matrix and $0<P\leq I$ is a projector matrix. Is this true that we always have: $$Tr (PA)\leq Tr(A)$$ If it is not true in general, what is the ...
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0answers
34 views

How can it be shown that a matrix multiplied by it's adjoint has positive definite eigenvalues?

Essentially show that $ AA^\dagger $ has positive eigenvalues. It's obvious that it's hermitian, so it will have real eigenvalus, so what i've tried so far is: $ Au = λu$ $ u^\dagger A^\dagger = λ ...
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1answer
32 views

For $n \times n$ matrix, is zero row able to be interpreted as $n$-leading zeros? [closed]

From the definition of $k$-leading zeros, the first $k$ elements of the row are all zeros and the $(k+1)$th element of the row is not zero. From the above definition, can zero row be interpreted as $...
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0answers
18 views

Matrix Vector Multiply Notation

I just want to ask a question about a notation used in one paper. There is something like: $\ldots = \mathbf C * \mathbf x$ where the $\mathbf C$ is a matrix $\mathbf C \in \mathbb R^{1000*1000}$ ...
2
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2answers
54 views

A smoothly varying family of positive-definite matrices

Consider a smoothly varying family of matrices $(g_{ij}^{t})$ where $0\leq t\leq1$. For all $0\leq t\leq1$, $\det(g_{ij}^{t})>0$ and for $t=0$, the matrix $(g_{ij}^{t})$ is positive definite. How ...
2
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3answers
82 views

Trace inequality on positive matrices

Let $A,B,C\geq 0$ be self-adjoint matrices. Assume $A\leq B$. Is it true that $$\mathrm{tr}(ACAC) \leq \mathrm{tr}(BCBC)?$$ How to prove this?
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1answer
30 views

Logarithm of complex matrix

For invertible matrix $A$, we have $\log(\det A) = \mathrm{tr}(\log A)$ due to a corollary of Jacobi's formula. What if we had the argument $iA$ instead? Would the above relation still hold? Edit: ...
2
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2answers
92 views

Application of $(Ax)^Ty = x^T(A^Ty)$

I am working on this exercise: Wires go between Boston, Chicago, and Seattle. Those cities are at voltages $x_B$, $x_C$, $x_S$. With unit resistances between cities, the three currents are in $y$: ...
2
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0answers
33 views

Real points in a matrix interval

Given $A$ and $B$, two $n \times n$ complex Hermitian positive semidefinite matrices such that $A< B$. I want to show existence (or non existence) of a real symmetric positive matrix $X$ such that \...
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2answers
24 views

Relation of change of basis matrix from basis $A$ to basis $B$ and from $B$ to $A$

Given the basis $A$ and $B$, and given $M_1$ a change of basis matrix from $A$ to $B$, and $M_2$ from $B$ to $A$. How do I find the relation between $M_1$ and $M_2$? I know that $M_2 = M_1^{-1}$ but ...
0
votes
1answer
29 views

How do I find optimal ω for SOR method?

Following is the example from this book. My question is, what value of λ did he put in? He did not explain that, can anybody explain how did he get 1.24?
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2answers
84 views

Find the nth power of a matrix

Let matrix A is $$A=\left (\begin{array}{rrr} 1&0&0 \\ 1&1&0\\ 0&0&1 \end{array}\right)$$ How can I find the 30 th power of A.. Is diagonalization possible? I found the ...
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0answers
31 views

Rectangular Matrix, Non Rectangular Matrix and square Matrix

It is well know that a square real matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$. Then it is also know that a rectangular real matrix $\mathbf{A} \in \mathbb{R}^{m\times n} \;\;\; , \;\;\;m \neq n$. ...
1
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0answers
45 views

Invertibility, inverse, and line weight of big circulant matrices

I am generating a random square sparse binary circulant matrix, defined by its first row. The length of the matrix is 9857 bits, and each line contains 71 ones, the rest are zeroes. I need to ensure ...
0
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1answer
27 views

If $P$ be permutation matrix then $({P^T}AP)x\mathop = \limits^? \left( {\begin{array}{*{20}{c}} * \\ 0 \\ \end{array}} \right)$

Let $A\in M_n$ is nonnegative(all $a_{ij}\ge0$). $x\in C^n$ be eigenvector of $A$ with $r ≥ 1$ positive entries and $n − r$ zero entries . There is $P\in M_n$(permutation matrix) such that $Px = \...
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0answers
37 views

Matrix splitting procedures - is there equivalent of Helmholtz decomposition?

My post consists of two separate questions: I am interested in different ways to split a matrix in a form: A = B + C where both B and C would have some specific, useful properties. I am familiar ...
0
votes
1answer
43 views

For which $a$ and $b$ has $P = \begin{bmatrix} 1 & 1 & 1 \\ 0 & a & b \\ a&0&b \end{bmatrix}$ the eigenvalues $0$ and $3$?

If the matrix $$ P = \begin{bmatrix} 1 & 1 & 1 \\ 0 & a & b \\ a & 0 & b \end{bmatrix} $$ has eigenvalues $0$ and $3$ then to determine values of $a$ and $b$. I know ...
1
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1answer
33 views

Product of diagonal and positive definite matrix.

Let $C$ be an $n \times n$ diagonal matrix with positive diagonal entries, and let $G$ be an $n \times n$ ymmetric positive definite matrix. What can we say about $CG + GC$? For example, is it non-...
2
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0answers
46 views

Null Space Equals to Column Space?

Can someone please explain to me why the statement "if $n$ is even, then there exists an $n\times n$ matrix $A$ such that $\text{Null}(A) = \text{Col}(A)$" is true?
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2answers
112 views

$A^2X = X$ show that AX = X, if $A_{ij} > 0$ and $x_{i} > 0$

Problem: Let A and X be matrices n x n and n x 1, respectively, with all entries real and strictly positive. Assume that $A^2 X = X$. Show that A X = X. What I thought: (i) $A^2 X = X \Rightarrow A (...
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0answers
26 views

Finding the inverse of $H-G E^{-1}F$

What's the inverse of $H-G E^{-1}F$ where $H$ is an invertible matrix? Is there a systematic way to find it or should I play with the expression until I get lucky? edit: Maybe there's no way and I'...
1
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0answers
72 views

A general form of a matrix $A$ raised to a natural power, $A^k$.

Suppose I have the $2\times 2$ matrix: $$A = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$$ I want to find the general matrix $A^k$ where $k\in \mathbb{N}$. I.e I want to ...
0
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2answers
32 views

Proof of Anti-Linearity of Hermitian Conjugate

How can I prove that the adjoint operation/ Hermitian conjugate in anti-linear i.e $(\sum_{i} a_i A_i)^\dagger = \sum_{i} a_i^* A_i^\dagger$, where $A$ is any linear operator on a Hilbert space $V$. ...
0
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2answers
41 views

Let $P$ be permutation matrix. Can we say that $PA=AP$? [closed]

Let $A,P\in M_n$ and $P$ be permutation matrix. Can we say that $PA=AP$?
0
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2answers
40 views

Find the dimensions of two given matrices

Matrix $A$ has $x$ rows and $x+5$ colums, and matrix $B$ has $y$ rows and $11-y$ columns. If $AB$ and $BA$ are both defined, then find the orders of $ A$ and $B$. No any idea to solve
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1answer
33 views

Simultaneous similarity of symmetric and antisimmetric matrices

It's proved here that every real antisymmetric matrix is orthogonally similar to its transpose? Now let $A,B$ a pair of symmetric and antisymmetric matrices $(A^T=- A,B^T=B)$. Is it true that $A,B$ ...
1
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1answer
29 views

Proof that $A^{-1}=adj(A)/|A|$

I know that inverse of a matrix is given by $adj(A)/|A|$ but I cannot prove it.Nor did I find the proof in my books.Can you guide me?
0
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2answers
95 views

Kronecker product decomposition

How can I decompose a matrix Z into two matrices X and Y as below: $$ Z=X\otimes Y $$ in which $\otimes$ is the kroncker product. Is there any function in matlab or any other library which ...
0
votes
2answers
46 views

Let $A=[a_{ij}]$ and find $\operatorname{tr}(A)$.

Let $A=[a_{ij}]$ where $a_{ij}=u_iv_j$ for all $1\leq i,j \leq n$ and $u_i,v_j$ are real numbers which satisfy $A^5=16A$ then find $\operatorname{tr}(A)$. I can't solve this. Help!
0
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1answer
51 views

$D=\operatorname{diag}(a_1,a_2,a_3,…,a_n)$, then show that $D^{-1}=\operatorname{diag}(a_1^{-1},a_2^{-1},…,a_n^{-1})$

If $D=\operatorname{diag}(a_1,a_2,a_3,...,a_n)$, where $a_i$ for all $i=1,2,3,...$ then show that $D^{-1}=\operatorname{diag}(a_1^{-1},a_2^{-1},...,a_n^{-1})$. First of all what on earth does $\...
1
vote
1answer
25 views

A test to ascertain that both equation lies on the same line

What tests can you devised to ascertain that two equations $ax+by=c$ and $rx+sy=t$ define the same line? (assume the coefficients a,b,r,s are all non-zero) Putting this into linear combinations of ...
1
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1answer
67 views

If $(A-I/2)$ and $(A+I/2)$ are orthogonal matrices then how to show $A$ is skew symmetric matrix of even order?

In this question I cannot understand the last step by the user.How is $|-\frac{3}{4} I|=(-3/4)^n$ ?Should'nt it be just $-\frac{3}{4}$ ? Moreover the OP did not prove that $A$ cannot be orthogonal....
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0answers
23 views

determining whether two matrices are row equivalent without any calculation

Is there a simple theorem that actually states that given two arbitrary matrices, one can determine whether the two matrices are row equivalent?
1
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1answer
60 views

How to find the possible square roots of the two rowed unit matrix I?

How to find the possible square roots of the two rowed unit matrix I ? I took a matrix like this $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ and wrote,$A.A=I$ and got a set of ...
2
votes
2answers
78 views

If $A,B,A+I,A+B$ are idempotent matrices how to show that $AB=BA$?

If $A,B,A+I,A+B$ are idempotent matrices how to show that $AB=BA$ ? MY ATTEMPT: $A\cdot A=A$ $B\cdot B=B$ $(A+I)\cdot (A+I)=A+I$ or, $A\cdot A+A\cdot I+I\cdot A+I\cdot I=A+I$ which implies $A\...
1
vote
1answer
41 views

Jordan measure and singular matrix

My question is, Define $V$ as the set of all singular $n\times n$ matrices, we can think V as a subset of Euclidean space $\mathbf{R}^{n^2}$. I need to prove that V can be written as, $$V=\bigcup^{\...
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0answers
51 views

Eigenvectors of Approximations to Infinite Stochastic Matrices

Given a function $[0,1]\to[0,1]\times[0,1]$ on the reals, such that the function is "stochastic" (probably an abuse of vocabulary: defined such that integrating along any vertical line gives $1$), ...
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1answer
38 views

Basis for Column Space

If the RREF of a matrix is the identity matrix, would the standard basis be a basis for its column space? And would the theorem that says a basis for the column space is the corresponding columns with ...
0
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2answers
57 views

Prove a bipartite graph is connected

I have the following Bipartite graph, how can I prove that it is connected? I have been searching the Internet for hours but couldn't find a solution, I need a mathematical method to prove it is ...
0
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0answers
41 views

Inverse of sum of matrices

Let $A,B$ be invertible positive definite matrices of the same size. My goal is to efficiently compute $(xA + yB + zI)^{-1}$ for many triplets of positive real numbers $(x,y,z) \in \mathbb{R}^3$. ...
8
votes
2answers
414 views

This theorem about matrices of linear maps doesn't look correct.

Consider the following theorem: Theorem. Let $f\colon L\to M$ be a linear mapping of finite-dimensional vector spaces. Then there exist bases in $L$ and $M$ and a natural number $r$ such that the ...
0
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1answer
112 views

Eigenvalues of triangular block matrix 2

I need a way for compute the eigenvalues of these block matrix \begin{equation}Acc=\begin{bmatrix} A & I \\ D & 0 \\ \end{bmatrix}\end{equation} Where: $A$ is a ...