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
136 views

Covariance matrix always positive semidefinite?

I actually was perusing here right now to see if anything could explain a result I have been getting - of a covariance matrix which has a negative eigenvalue, yet the correlation matrix does not - but ...
1
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1answer
110 views

No. of positive eigen values of $3\times 3$ real matrix [duplicate]

Suppose $A$ is a $3\times 3$ symmetric matrix such that : $$\begin{pmatrix} a & b & 1 \end{pmatrix} A \begin{pmatrix} a \\ b\\ 1\end{pmatrix} = ab -1$$ for all $a,b\in \mathbb{R}$ ...
0
votes
2answers
4k views

square matrix is not invertible if at least one row or column is zero

How to show that a square matrix is not invertible if at least one row or column is zero ? I can show if a row is zero, the result C of $AB=C$ can not be the identity matrix because there is a zero ...
-1
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1answer
155 views

$A^TA=B^TB$ implies $\exists P \in M_{m\times m}(\Bbb{R})$. such that $A=PB$,where $P$ is an orthogonal matrix [closed]

Assume $A,B \in M_{n\times m}(\Bbb{R})$,and $A^TA=B^TB$,show that there exists an orthogonal matrix $P$, such that $A=PB$.
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2answers
79 views

Gaussian elimination on matrix.. Is there a better way to extract the solution?

So, I used Gaussian elimination on this matrix $$\left( \begin{array}{c} -1 & 3 & 5 & 13 \\ 3 & -2 & 2 & 16 \end{array}\right)$$ to turn it to this: $$\left( ...
0
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2answers
97 views

Can you transform any coordinate from any “space” to another “space” that's defined?

This question pertains to Matrix Transformations. So to provide an example, if I have 3D coordinates where $X = -1$ to $1$, $y = -1$ to $1$, $z = -1$ to $1$. They are "normalized" in my mind. Can I ...
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1answer
1k views

The eigenvalues of the product of a positive definite and a symmetric matrix.

A fellow student posed the following question and I'd like to stop thinking about it so I can get back to work on my own research! Suppose that $A>0$, i.e. $A$ is a real symmetric positive ...
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1answer
68 views

Matrices with eigenvalues in geometric progression

Given $a\in\Bbb F$ are there any natural $k\times k$ matrices in $\Bbb F^{k\times k}$ with $a^{0},a^1,a^2,\dots, a^{k-1}$ as eigenvalues where $\Bbb F$ is any char $0$ field? The characteristic ...
2
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1answer
106 views

Possible determinant relation for PSD matrices.

Is $$\det(I+ABC)=\det(I+ACB),$$ when $A,B,C$ are symmetric positive semi/definite and $I$ is the identity matrix. I am mostly interested in the case when the matrices are in complex field. I know ...
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1answer
345 views

Solving systems of linear equations with an unknown 'a' using matrices and elementary row operations

Came across this one the other day... while I can narrow 'a' down I can't seem to find an exact/ optimised figure. For example 'a' cannot equal 1/3, 'a' must be less than 0.5... Anyway, here's the ...
2
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1answer
68 views

Producing a set of matrices in MAGMA

As part of my work I need to define the set of all $n\times n$ matrices with entries in $\{0,1,\ldots,n\}$ (considered as a subset of $\mathbb{Z}$), such that each row and column of the matrix sums to ...
1
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1answer
59 views

Right multiplication with matrices

If $X=(x_1,x_2,\dots)$ is an infinite real row vector and $A=(a_{ij}),0<i,j< \infty$ is an infinite real matrix, one may or may not be able to define the matrix product $XA$. For which A can one ...
2
votes
2answers
50 views

Positive semi/definite matrix claim.

If $A$, $B$ is positive semidefinite (PSD) and $C$ is positive definite (PD), all are Hermitian, complex valued. I want to claim that $$(B+C)^{-1/2}A(B+C)^{-1/2}$$ is PD. (I am sure it is PSD but ...
1
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1answer
216 views

Theoretical basis behind calculation of steady state probability distribution of 2-state Markov chain from its transition matrix

I am studying stochastic processes and have stumbled on a result that is puzzling me. I have searched elsewhere for an answer without luck so hoping some proper mathematicians here can explain the ...
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1answer
60 views

Derivative of a function of vector parameter. Problem with notation.

I have an error function $$err=\frac{1}{N}\left[\textbf{y}^T\ln{\textbf{x}}+(\textbf{1}-\textbf{y})^T\ln{(\textbf{1}-\textbf{x})}\right]$$ I need to find the gradient $\bigtriangledown_x{}err$, such ...
3
votes
2answers
135 views

Writing real invertible matrices as exponential of real matrices

Every invertible square matrix with complex entries can be written as the exponential of a complex matrix. I wish to ask if it is true that Every invertible real matrix with positive determinant ...
1
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1answer
45 views

Matrix of transformation

I have just finished understanding the topic of matrix of the transformation, and we just started T-invariant subspaces. Can anyone please help me with these questions, because I don't know how to ...
1
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1answer
119 views

Unique of eigenbasis of self-adjoint operator.

Today I am reading an article: Eigenvalues and sums of Hermitian matrices there is an exercise copy from that article: Exercise 1 Suppose that the eigenvalues $\lambda_1(A)>\cdots>\lambda_n(A)$ ...
0
votes
1answer
56 views

What's the derivative of F over R?

$F(R) = \|A-RY\|^2_F$ what should $\partial F \over \partial R$ looks like? Since $\partial \|X\|^2_F \over \partial X$ is $2X$, so $\partial \|A-RY\|_F^2$ should be $2(A-RY){ \partial RY \over ...
2
votes
1answer
127 views

Rank of a matrix with structure

Let $P \in [0,1]^{n \times n}$ ($n > 1$) be a matrix such that the diagonal entries $P_{ii} ~~\forall i$ are $0$ and upper diagonal entries $P_{ij} ~~\forall i < j$ $\in (0.5,1)$ and lower ...
3
votes
3answers
1k views

Writing u as a linear combination of the vectors in S.

Write vector u = $$\left[\begin{array}{ccc|c}2 \\10 \\1\end{array}\right]$$ as a linear combination of the vectors in S. Use elementary row operations on an augmented matrix to find the necessary ...
2
votes
1answer
327 views

standard deviation calculation using covariance?

i require a formula to calculate the standard deviation using variances of three or more variables (lets call them a,b,c) and the covariances between them. To complicate matters more i only need a ...
0
votes
1answer
129 views

Jacobian of a matrix equation $X^2-BX-C$?

I need to find the critical points of a matrix equation $X^2-BX-C$, all matrices are square, $X$ is the solution to be found and $B,C$ are constant matrices. I have seen that I should find first the ...
3
votes
4answers
184 views

How to solve matrix equation $AXH+AHX−BH=0$

How to solve matrix equation $AXH+AHX−BH=0$? All matrices are square, $A$, $B$ known constant matrices and invertible, $H$ can take any value, $X$ represent the solution to be found. I have seen ...
0
votes
1answer
69 views

Right multiplication on $\mathbb{R}^{\infty}$

If $X=(x_1,x_2,\dots)$ is an infinite real row vector and $A=(a_{ij}),0<i,j< \infty$ is an infinite real matrix, one may or may not be able to define the matrix product $XA$. For which A can one ...
3
votes
4answers
135 views

Looking for an elegant proof of $\det(A) = \det(A^t)$ without Schur decomposition

Looking for an elegant proof of $\det(\textbf{A}) = \det(\textbf{A}^{t})$ without Schur decomposition. Proof 1 with Schur decomposition $$\textbf{A} = \textbf{P}^{t}\Delta\textbf{P} ...
3
votes
2answers
993 views

Symmetric Tridiagonal Matrix has distinct eigenvalues.

Show that the rank of $ n\times n$ symmetric tridiagonal matrix is at least $n-1$, and prove that it has $n$ distinct eigenvalues.
0
votes
1answer
22 views

will the rank of a projector matrix be equal to the dimension of vector space it projects to?

Let the projector be the $N \times N$ matrix $A$. Let its rank be $r$. Let the dimension of the space for which it is the projector be $m$. Is $m==r$?
4
votes
3answers
134 views

What am I doing wrong when trying to find a determinant of this 4x4

I have to find the determinant of this 4x4 matrix: $ \begin{bmatrix} 5 & -7 & 2 & 2 \\ 0 & 3 & 0 & -4 \\ -5 & -8 & 0 & 3 \\ 0 & -5 & 0 & -6 \\ ...
1
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1answer
138 views

exponential and Lie bracket

I've been reading that one can compute a form of the Baker-Campbell formula with direct computations through the power series of $\exp(X)$, $\exp(Y)$ and $\log(1+W)$, where $X$, $Y$ are non commuting ...
2
votes
1answer
38 views

If $a^HUa=0$ for all $a$, can we conclude that $U=0$?

I have the following equation: $a^HUa=0$ where '$a$' can be any arbitrary vector and $U$ is a matrix ($H$ means Hermitian). Can we conclude that $U=0$? Any reference? Thanks.
0
votes
2answers
235 views

Rank and the column space of matrix product

If $A$ and $B$ be matrices for which the product $AB$ makes sense, and the rank of $AB$ equals the rank of $A$. Is the column space of $AB$ equals the column space of $A$?
1
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1answer
90 views

Linearly independent vectors and matrix

If $\{v_{1},v_{2},\cdots,v_{n}\}$ is $n$ linearly independent vectors in $\mathbb{R}^{n}$, what would be necessary and sufficient condition of $A$ ($n\times n$ matrix) $A$ so that the vectors ...
1
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1answer
27 views

Ask a question about the definition of trace norm.

Suppose $X\in \mathbb{R}^{M\times N}$ $\|X\|_*=\mathrm{trace}(\sqrt{X^*X})=\sum_i^{\min{M,N}}\sigma_i$ where $\sigma_i$ is the singular values of $X$. I know that ...
0
votes
1answer
154 views

Consider the quadratic form $q $ and $p$ given by

Problem:Consider the quadratic form $q $ and $p$ given by $q(x,y,z,w)=x^2+y^2+z^2+bw^2$ $p(x,y,z,w)=x^2+y^2+czw$ Which of the following is true $?$ $1)p,q$ are equivalent over ...
2
votes
1answer
86 views

Eigenvalues of Matrix.

Here is my Question. Prove or disprove : There is a real $n$ by $n$ matrix $A$ such that $A^2 + 2A + 5I = 0$ iff $n$ is even. What i have done is I started with polynomial representation of ...
1
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1answer
144 views

Eigenvalues of $AB$ and $BA$

Let $A$ and $B$ be $m \times n$ and $n \times m$ real matrices. I proved that if $\lambda$ is a nonzero eigenvalue of the $m \times m$ matrix $AB$ then it is also an eigenvalue of the $n \times n$ ...
7
votes
1answer
99 views

An inequality about positive inertia index of two symmetric matrices. $P_{A+B}\leq P_A +P_B$

Assume $A$ and $B$ are real symmetric matrices of order $n$. I denote the number of the positive eigenvalues of matrices $A,B,A+B$ by $P_A,P_B,P_{A+B}$ respectively. Show that: $$P_{A+B}\leq P_A ...
2
votes
1answer
88 views

Constructing invariant subspaces from scratch. An algorithm

So, basically i am trying to prove that there exist a basis w.r.t which there exists an upper triangular matrix in a complex field. Most of the books which i read incorporate induction as a method ...
2
votes
1answer
205 views

Determine the matrices that represent the following rotations of $\mathbb{R}^3$

I need to determine the matrix that represents the following rotation of $\mathbb{R}^3$. (a) angle $\theta$, the axis $e_2$ (b) angle $2\pi/3$, axis contains the vector $(1,1,1)^t$ ...
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0answers
34 views

Is the coefficient uniquely determined by the sign function?

Suppose $a\in R^p$, $b\in R^p$, and $||a||=||b||=1$, is it true that if $sign(a'x)=sign(b'x)$ for any $x\in R^p$, then $a=b$, where $sign(t)=1$ if $t\geq 0$ and $sign(t)=-1$ if $t<0$?
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2answers
792 views

If a linear system has no free variables, then it is consistent: Why is the statement false?

I recently got this question wrong on a test, and I have no clue why. How can a system be inconsistent if there are no free variables?
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0answers
50 views

How to solve linear system of equations in 1 and inf-norm?

I have the problem to find a linear program that is equivalent to solving the problem that finds a minimum for $||Ax-b||_1$ and $||Ax-b||_{\infty}$. We defined a linear program as follows: $min_{x} ...
1
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1answer
177 views

Matrix vector spaces isomorphic to column vector spaces?

my question is a basic linear algebra question, so hopefully someone can answer without too much trouble. My question was motivated by a problem I was doing about a linear transformation from the ...
2
votes
2answers
112 views

Square root of a diagonal matrix $\lambda I$

Could you help me prove that if $M \in \mathcal{M}_{2 \times 2}(\mathbb{R})$ satisfies $X^2=\lambda I$, $ \ \ \ \lambda \in \mathbb{R}, \ \ \lambda <0$, then there exist $y,z \in \mathbb{R}, \ \ ...
2
votes
0answers
116 views

Does a matrix represent a bijection

We have a square binary matrix that represents a connection from rows to columns. Is there a way to tell if a bijection exists (other than checking for all possible bijections and iterating through ...
4
votes
1answer
59 views

Projection and direct sum

I want to show that for every projection $A^2=A$ we have that there exists a subspace $U_1 \subset ker(A)$ and $U_2$ such that $A|_{U_2} = id$ such that $V = U_1 \oplus U_2$. Does anybody here have a ...
1
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1answer
80 views

If $A$ Hurwitz, $(A+A^*)$ is Hurwitz?

If I have $A$ Hurwitz matrix, is $(A+A^*)$, with $A*$ the transpose of $A$, still Hurwitz? Any reference or proof? Because if $(A+A^*)$ is still Hurwitz I can say that it is even negative definite ...
1
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1answer
165 views

Proof that the product of a positive semidefinite matrix and an orthogonal matrix could be an arbitrary matrix

Let $A$ be an arbitrary invertible $n\times n$ matrix. Prove that there exist a positive semidefinite matrix $R$ and an orthogonal matrix $B$ such that $A = BR$. Can anyone help me?
0
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2answers
326 views

Critical points of a matrix equation

The question I am trying to solve is: how to find the critical points of a matrix equation $$X^{2}-BX-C=0?$$ And then if I follow the method of http://www2.math.umd.edu/~jmr/241/crits.html, first I ...