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
151 views

How can you tell how many invariant factors a matrix has?

In my notes, I have a $4\times 4$ complex matrix $A$ with the following properties. The characteristic polynomial of $A$ is $(x-1)(x+1)^3$, and the geometric multiplicity of $-1$ is $2$. That is all ...
1
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0answers
88 views

Regression with a Vandermonde matrix.

From what I understand, when doing a least squares regression with a Vandermonde matrix, you're essentially solving the equation $y=Xa$ Where y is a vector of y-values, X is the Vandermonde matrix, ...
1
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2answers
143 views

Matrix (correct) notation

Say I have a real $m \times n$ matrix $\mathbf{M}$. Shall I write $\mathbf{M} \in \mathbb{R}^{m \times n}$ or $\mathbf{M} \in \mathbb{R}^{m,n}$? What is commonly accepted and most beautiful and ...
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4answers
839 views

Reduced row-echelon form of a matrix with variables

I have been staring at this for an hour. How would you reduce such a matrix? \begin{bmatrix} p & 0 & a \\ b & 0 & 0 \\ q & c & r \end{bmatrix} $abc\neq0$
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1answer
40 views

Unicity of identity

I've two doubts: 1) If $A,B$ are square matrices and $AB=I_n$, but not necessary $BA=I_n$, is true that $A$ is invertible and $A^{-1}=B$? 2) If $AB=B$, then $A=I_n$. Well, I know that the ...
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2answers
76 views

Multiplying matrices

I'm solving a exercise and the following doubt has arisen: Let $A,B,C$ be non-zero matrices, if $AB=CB$ then $A=C?$ I already have a partial solution when $A,B,C$ are square matrices and $B$ is ...
3
votes
3answers
481 views

Given a vector equation with $n$ vectors, how can we determine if the span of the vectors is equal to $\mathbb{R}^n$

So I've recently started taking Linear Algebra, and I've been thinking about how to determine if the linear combinations of any n vectors can represent any vector in $\mathbb{R^n}$. More formally put, ...
1
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1answer
74 views

Are there any mathematical/physical concepts or theories for dealing with a matrix in which the values are changing in a certain way?

As a matter of fact, my application scenario is a recommender system in which the interests/preferences of the users change. I have such a global user-interest matrix: the rows are the records of many ...
2
votes
4answers
222 views

another way to find inverse matrix

The most common way to find inverse matrix is $M^{-1}=\frac1{\det(M)}\mathrm{adj}(M)$. However it is very trouble to find when the matrix is large. I found a very interesting way to get inverse ...
0
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1answer
118 views

Can This Matrix Proof Be Done Without the Definition? [duplicate]

The following was taken from Stephen Friedberg's Linear Algebra, 2nd Edition. Let $A$ be a $m \times n$ matrix, $B$ and $C$ be $n \times p$ matrices. Show that $A(B+C) = AB + BC$ and, for any $k \in ...
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2answers
151 views

Formula for Nth Derivative of Matrix Inverse

I was looking for an equation for the nth derivative of a matrix inverse, ie $\frac{d^n \bf{A}^{-1}}{dx^n}$ I know that the first derivative $\frac{\text{d} \bf{A}^{-1}}{\text{d}x} = -\bf{A}^{-1} ...
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1answer
158 views

Invariant subspaces and minimal polynomial

I wanted to know whether every invariant subspace $U$ of an endomorphism $A$ with minimal polynomial $m_A= \Pi_{i=1}^n p_i$, where the $p_i$ are mutually coprime polynomials, can be written in the ...
2
votes
1answer
145 views

Does same characteristic polynomial and same rank imply similar?

Are two matrices with the same characteristic polynomial and the same rank necessarily similar? Where can I find the proof for such a thing?
4
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1answer
1k views

Eigenvalues of matrix with entries that are continuous functions

For each $t \in [0,b]$, let $M(t)$ be an $n \times n$ matrix with entries $m_{ij}(t).$ The matrix $M(t)$ is invertible and positive-definite, so the eigenvalues of $M(t)$ exist and are positive for ...
1
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1answer
46 views

About a matrix, its powers, and a particular value…

Is it possible to find two matrices, $A$ an $B$ such that: $$A B^0 x = \begin{pmatrix} c_0 \\ ? \\ ? \\ \end{pmatrix}, $$ $$A B^1 x = \begin{pmatrix} c_1 \\ ...
3
votes
1answer
161 views

$A^TA+I$ is always invertible?

How to prove a general matrix invertible given by as below? How to prove that $A^TA+I$ is always invertible for $\forall A \in \mathbb{R}^{n\times n}$?
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2answers
59 views

How do I see that for every self-adjoint and positive-definite $A$ there is an $S$ such that $A=S^*S$?

How do I see that for every self-adjoint and positive-definite $A \in \mathbb{C}^{n \times n}$ there is an $S \in \mathbb{C}^{n \times n}$ such that $A=S^*S$? Is something that I am trying to ...
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0answers
45 views

How to show that entries of this matrix are in $L^\infty(0,T)$?

I have a problem. Let $A(t)$ be a $n \times n$ matrix for each $t \in [0,b]$ with the property for all vectors $x$ that $$x^TA(t)x \geq C|x|^2$$ where $C$ doesn't depend on $t$. Can I use this fact ...
6
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2answers
1k views

Lower bound on smallest eigenvalue of (symmetric positive-definite) matrix

If $M$ is a positive-definite symmetric matrix, is it possible to get a positive lower bound on the smallest eigenvalue of $M$ in terms of a matrix norm of $M$ or elements of $M$? Eg. I want ...
2
votes
1answer
113 views

Show the following matrix recursion is symmetric

I need to show the following matrix equation is symmetric and I'm not sure where to start: $A_i=\sum_{j_1=1}^{i-2}2(i-j_1-1){i-2 \choose j_1-1}A_{j_1}\Big(\sum_{j_2=1}^{i-j_1-1}{i-j_1-2 \choose ...
0
votes
0answers
81 views

Eigenvalues of time-dependent matrix bounded?

For each $t \in [0,b]$, let $A(t)$ be a $n$ by $n$ matrix with entries $a_{ij}(t) \in L^\infty(0,b).$ Suppose I know that the eigenvalues of $A(t)$ for each $t$ are positive. Can I conclude that the ...
2
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1answer
49 views

Constant in the positive-definite condition of matrices

Suppose I have a positive definite matrix $A$, so $$x^TAx \geq C_A|x|^2$$ holds for all $x$. $A^{-1}$ is positive definite too: $$y^TA^{-1}y \geq C_B|y|^2,$$ is there any way I find the constant ...
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0answers
45 views

Matrix with elements in $L^\infty(0,T)$ means inverse matrix also has elements in $L^\infty(0,T)$?

Let $A(t)$ be a matrix with entries $a_{ij} \in L^\infty(0,T)$. Suppose $A(t)^{-1}$ exists, and is positive-definite. Does it follow that $A(t)^{-1}$ has elements in $L^\infty(0,T)$ too?
2
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0answers
85 views

Lower bound on Gram determinant

Is there a lower bound for the determinant of the Gram matrix when the elements are linearly independent? I am talking about Hilbert space setting not just vectors. Thanks.
2
votes
1answer
96 views

How to represent a Matrix Space?

How can we represent the space of matrices? E.g. A vector $z\in{}R^m$ in the column-space of matrix $A\in{}R^{m\times{}n}$ can be represented as $$ z=Ax $$ for some $x$. Context: In the following ...
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2answers
85 views

Basis means determinant of matrix of inner products is non-zero

Let $x_i$ be a basis of Hilbert space $X$ (NOT necessarily orthogonal) How do I show that $\text{det}((x_i,x_j)_H)_{ij} \neq 0$ for $i,j=1,...,n$? I see this fact used in Galerkin approximation ...
0
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0answers
77 views

Matrix differentiation.

Given $A$ is square matrix depends on a scalar $x$, is there rules to find $df(a)/dx$? For example, indices rule or chain rule? I found this kind of information is not much on the internet. Thanks.
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1answer
71 views

Linear Algebra :: Multiplying by Matrices on both sides of equation

I was wondering, for scalar equations, it is true that $A=B$ implies $PA=PB$ where P is also any scalar. Are these true for matrices as well? Is $PAx=Pb$ true when $Ax=b$? Assume that dimensions ...
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0answers
82 views

Closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$

I need a closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$. Here $A, X$ are $n\times n$ matrices, $\log$ is the matrix logarithm. ${d \operatorname{Tr}(\log(X))\over dX}=X^{-1}$, ...
2
votes
1answer
97 views

Am I right about this exercise of Matrices?

In my book of Linear Algebra asks me the following exercise: For any square matrix $T$, over $\mathbb{R}$, it defines $p(T)$ to be the matrix $c_nT^n+\cdots + c_1T+I$ with $c_i \in \mathbb{R}$ for all ...
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3answers
84 views

Name for this matrix operation?

Suppose two matrices have the same number of rows. I want to perform an operation of element-wise product between all possible column pairs between the two matrices. For example, if $A = ...
1
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1answer
96 views

Why is it true that there exists an $n$ such that $V = \operatorname{im}(A-\lambda \operatorname{Id})^n \oplus \ker(A-\lambda\operatorname{Id})^n$

Let $A : V \rightarrow V$ be an endomorphism. Then we have that $$V = \operatorname{im}(A-\lambda \operatorname{Id})^n \oplus \ker(A-\lambda\operatorname{Id})^n$$ for some $n \in \mathbb{N}$. I ...
0
votes
1answer
77 views

Mapping a plane in $\Bbb R^3$ to $\Bbb R^2$

I have three points that represent a rigid body. The rigid body undergoes a planar transformation in $\Bbb R^3$ due to rotation and translation. I am working with angular velocity with nonzero $\vec ...
2
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0answers
69 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. ...
-1
votes
4answers
114 views

Find the trace of the matrix? [duplicate]

Let $A$ be a $227\times227$ matrix with entries in $\mathbb{Z}_{227}$, such that all the eigenvalues are distinct. What is the trace of $A$?
1
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4answers
203 views

Calculate the Determinant?

$$D=\begin{bmatrix} 246 & 427 & 327 \\ 1014 & 543 & 443 \\ -342 & 721 & 621 \\ \end{bmatrix}$$ What's the trick? Hints? Of course I know calculate by definition... Please ...
3
votes
2answers
165 views

Is there meaning for $\bf uv^T$?

Let $\bf{u,v}$ be two column vector in $\mathbb R^n$, which can be represented by $n\times1$ matrix. $\bf u^T v$ is the inner product of $\bf u,v$, then is there meaning for $\bf uv^T$, which is a ...
0
votes
1answer
176 views

What does it mean by “the origin is moved by the transformation” in linear transformations?

Linear transformations have the special property that the origin is not moved by the transformation. I don't really understand what this means. The example I'm given is that the following ...
2
votes
2answers
629 views

A property of positive definite matrices

Assume we have 2 positive definite matrices A and B . Show that there exists a non-singular matrix S such that - SAS' = I SBS' = L Here I is the Identity matrix and L is a diagonal matrix. S' is ...
0
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0answers
122 views

Dimension of a subspace (Hermitian matrices)

I have the following exercice from T. Tao's blog which I want to solve: Suppose that $n \geq2$. Show that the space of Hermitian matrices with at least one repeated eigenvalue has codimension 3 in ...
8
votes
4answers
599 views

When will $AB=BA$?

Given two square matrices $A,B$ with same dimension, what conditions will lead to this result? Or what result will this condition lead to? I thought this is a quite simple question, but I can find ...
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1answer
1k views

Simultaneous diagonalization of two positive semi-definite matrices

Let matrices $A, B$ be two $n \times n $ positive semi-definite matrices and they can be represent in the following form $$A=\sum_{i=1}^{n} \psi_{i}p_{i}p_{i}^{T}=P\Psi P^{T}, \quad B=\sum_{i=1}^{n} ...
1
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2answers
99 views

Inverse of a Particular $n\times n$ Matrix

Let $A$ be a square matrix of order $n$ where the elements are defined as $a(i,i)=2 ,a(i+1,i)=a(i,i+1)=-1$ for all $i=1 \ldots n-1$.Also a(i,j)=0 otherwise. Find the inverse of the matrix. Is there ...
5
votes
1answer
140 views

Proving a well-known formula regarding adj(A)

The adjugate of a matrix $A$ is defined as $$ (\mathrm{adj}(A))_{ij} = (-1)^{i+j}M_{ji}(A) $$ where $M_{ji}(A)$ is the determinant of the matrix $A$ after row $j$ and column $i$ have been removed. It ...
8
votes
1answer
654 views

Is the zero matrix the only symmetric, nilpotent matrix with real values?

My intuition tells me that the zero matrix is the only matrix that is symmetric and nilpotent with real values, but I'm having trouble proving it (or finding a counterexample.) I have searched for ...
2
votes
1answer
174 views

If $P$ is an invertible transition probability matrix, does $P^{-1}[i,j]$ have any interesting meaning?

Suppose we have a Markov chain transition probability matrix $P$ that is invertible, i.e., $P^{-1}$ exists. Question: Does there exist a meaningful interpretation of the $(i,j)$ entry in $P^{-1}$? ...
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0answers
53 views

Understanding a proof by R.C Lyndon and J.L Ullman

Here in this article I have difficulties understanding the theorem on page 162. Theorem. Let $A, B$ and $C=AB$ be an elements of group $GL_2(\mathbb{Z})$, all with real fixed points. Suppose that ...
1
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0answers
49 views

Understanding a proof by R.C Lyndon and J.L Ullman.

Here in this article I have difficulties understanding the theorem on page 162. Theorem. Let $A, B$ and $C=AB$ be an elements of group $GL_2(\mathbb{Z})$, all with real fixed points. Suppose that ...
0
votes
1answer
82 views

Fixed point of matrix

Suppose that $a$ is a fixed point of matrix $A$, what that means? What is a fixed point of matrix? Thank you!
0
votes
1answer
87 views

Definition of parabolic matrix.

Suppose that $A$ is unimodular $2\times 2$ matrix, what is the meaning of saying "If $A$ is parabolic..."? Is that when characteristic polynomial of $A$ is with degree of $2$? Thank you.