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), ...
7
votes
5answers
547 views
Showing determinants using trace in a 2x2 matrix
I am confused about this homework question.
It says "Show that :
$\det(A) = \frac 12 \begin{vmatrix}\operatorname{tr}(A)&1\\\operatorname{tr}(A^2)& \operatorname{tr}(A)\end{vmatrix}$
for ...
7
votes
7answers
620 views
Proving A is Invertible if $A + A^2 = I$
I'm trying to prove A is invertible by proving there is an $A'$, for $AA' = I$
So I got to this stage $A(I + A) = I$
Now I determine that $A' = I + A$,
and from that I get $AA' = I$, I wanted to ...
7
votes
6answers
407 views
$\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)$ not diagonalizable
I would like to ask you about this problem, that I encountered:
Show that there exists no matrix T such that $$T^{-1}\cdot
\left( \begin{array}{cc}
1 & 1 \\
0 & 1 \\
\end{array} ...
7
votes
4answers
669 views
Show that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$
I know about the fact that $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$, where $A$ and $B$ are $m \times n$ matrices.
But somehow, I don't find this as intuitive as ...
7
votes
5answers
3k views
When is matrix multiplication commutative?
I know that matrix multiplication in general is not commutative. So, in general:
$A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$
But for some matrices, this equations holds, e.g. A = ...
7
votes
2answers
267 views
Summing Matrix Series
I need to sum the series
$$I + A + A^2 + \ldots$$
for the matrix
$$A = \left(\begin{array}{rr}
0 & \epsilon \\
-\epsilon & 0
\end{array}\right)$$
and $\epsilon$ small. The goal is to ...
7
votes
2answers
472 views
Algebraic proof of a trig matrix identity?
I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question:
I have a geometric proof, but is there an elegant algebraic proof ...
7
votes
1answer
246 views
If $ \mathrm{Tr}(M^k) = \mathrm{Tr}(N^k)$ for all $1\leq k \leq n$ then how do we show the $M$ and $N$ have the same eigenvalues?
Let $M,N$ be $n \times n$ square matrices over an algebraically closed field with the properties that the trace of both matrices coincides along with all powers of the matrix. More specifically, ...
7
votes
3answers
1k views
Rank of skew-symmetric matrix
Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all ...
7
votes
3answers
160 views
What kind of matrix $A$ satisfies $Ax\geq 0\Rightarrow x\geq 0$?
$A\in \mathbb{R}^{n\times n}$ is an n-by-n matrix. $x=(x_1,x_2,\ldots ,x_n)\in \mathbb{R}^n$ is a vector. $x\geq 0$ means $x_i\geq 0,\forall i$.
Q1: When $A$ satisfies what conditions, $\forall ...
7
votes
3answers
120 views
Problem involving permutation matrices from Michael Artin's book.
Let $p$ be the permutation $(3 4 2 1)$ of the four indices. The permutation matrix associated with it is
$$ P =
\begin{bmatrix}
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 ...
7
votes
3answers
350 views
Derivative of determinant of a matrix
Good morning everyone,
I would like to know how to calculate:
$\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big)$
Help me please.
Thank you
7
votes
2answers
386 views
Use of determinants
I have been teaching myself maths (primarily calculus) throughout this and last year, and was stumped with the use of determinants. In the math textbooks I have, they simply show how to compute a ...
7
votes
2answers
332 views
Stirling Numbers and inverse matrices
Let $s(m,n)$ be the Stirling Numbers of the first kind, $S(m,n)$ be the Stirling Numbers of the second kind.
The matrices $$\mathcal{S}_N := (S(m,n))_{N \geq m,n \geq 0} \text { and } \mathcal{s}_N ...
7
votes
3answers
266 views
Two introductory algebraic geometry problem
I remember when I was in Moscow one of my homework questions was:
Is there a $2\times 4$ matrix whose $2\times 2$ minors are:
a): (2,3,4,5,6,7)
b): (3,4,5,6,7,8)
c): (5,6,7,8,9,10)
This problem ...
7
votes
2answers
4k views
Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks
Let $A$ be a block upper triangular matrix:
$A = \left( \begin{matrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{matrix} \right)$ where $A_{1,1} ∈ C^{p×p}$, $A_{2,2} ∈ C^{(n-p)×(n-p)}$
Show ...
7
votes
2answers
258 views
$AB-BA$ is a nilpotent matrix if it commutes with $A$
I saw this in a MathOverflow post and am putting it here for posterity.
Problem:
Let $A$ and $B$ by square matrices and set $C=AB-BA$. If $AC=CA$, prove $C$ is nilpotent.
7
votes
1answer
380 views
Matrix raised to a matrix
Good evening,
I was wondering if there is such a valid operation as raising a matrix to the power of a matrix, e.g. vaguely, if $M$ is a matrix, is
$$
M^M
$$
valid, or is there at least something ...
7
votes
2answers
419 views
Square Root of a Matrix
From a problem set I'm working on: (Edit 04/11 - I fudged a sign in my matrix...)
Let $A(t) \in M_3(\mathbb{R})$ be defined: $$ A(t) =
\left( \begin{array}{crc} 1 & 2 & 0 \\
0 & -1 ...
7
votes
1answer
2k views
Is it faster to multiply a matrix by its transpose than ordinary matrix multiplication?
I'm writing a program that multiples a matrix by its transpose, and was trying to find efficiency hacks I could exploit considering that the two matrices being multiplied are related. Any ideas?
7
votes
1answer
96 views
Calculating the eigenvalues of a matrix
How to find the eigenvalues of
$$\begin{bmatrix}
0 & 1 & & &\\
k & 0 & 2 & &\\
& k-1 & 0 & 3 &\\
& ...
7
votes
3answers
196 views
Solving matrix equation $XA=AY$ with known $X$ and $Y$
I am having problem in solving set of matrices multiplication.
There are three matrices $A,X$ and $Y$, all are non-singular $2\times 2$ matrices. Where matrix $X$ and $Y$ are known and $A$ is unknown. ...
7
votes
3answers
106 views
Matrix Determinant Identity
I have come across an observation about the determinant of a matrix, but I don't know how to prove it in general. Let me demonstrate it through an example.
$$
\begin{align}
\left|
\begin{matrix}
1 ...
7
votes
2answers
388 views
A question on Gram matrix
The entries of Gram matrix is defined by $ \langle x_i,x_j\rangle$ in the $(i,j)^{\text{th}}$ position. It is known that Gram matrix is positive semidefinite.
Is it still positive semidefinite if $ ...
7
votes
3answers
845 views
Computing the largest Eigenvalue of a very large sparse matrix?
I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter w, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue of ...
7
votes
1answer
111 views
Probability that a $3\times 3$ matrix with entries in $\{0,1,2,3\}$ is invertible.
Let $A$ be a $3\times 3$ matrix, and each of its entries takes value from $\{0, 1, 2, 3\}$ with probability $1/4$ for each value. What is the probability that A is invertible?
I have tried to list ...
7
votes
2answers
87 views
Given a square matrix A of order n, prove $\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$
Given $A\in F^{n \times n}$ prove:
$$\operatorname{rank}(A^n) = \operatorname{rank}(A^{n+1})$$
$\operatorname{rank}(A^{n+1}) \leq \operatorname{rank}(A^n)$ is easy, just from:
How to prove ...
7
votes
3answers
4k views
Calculate Rotation Matrix to align Vector A to Vector B in 3d?
I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
7
votes
1answer
385 views
On matrix norm equivalence
For finite dimensional spaces, all norms are equivalent, i.e. there exist constants say $A,B$ such that for all matrices from the $\mathbf M \in R^{d\times d}$ (let $d$ be a fixed positive integer) ...
7
votes
1answer
269 views
Is there a basis-independent proof of Abel's identity?
Abel's identity states that if $X(t)$ and $A(t)$ are $n\times n$ matrix-valued functions such that $X'(t)=A(t)X(t)$, then $\frac{d}{dt}(\det X(t)) = \mathrm{tr}\,A(t) \cdot \det X(t)$.
The ...
7
votes
1answer
218 views
Matrix Norm set
I need help with this problem:
Let $\|\cdot\|$ and $\|\cdot\|^{\prime}$ two matrix norms, and consider the relation
$$\|\cdot\| \leq \|\cdot\|^{\prime}\ \Leftrightarrow\ \|A\| \leq \|A\|^{\prime},$$
...
7
votes
1answer
110 views
Why would $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ be close to $A^{\dagger}$ when $A$ is with rank deficiency?
In many applications that is not with high requirements, it is common to use $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ or $A^{\text T}(AA^{\text T}+\lambda I)^{-1}$ ($\lambda$ is small) to ...
7
votes
1answer
60 views
*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$
I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form
$A\in ...
7
votes
1answer
149 views
Can you find a minimal polynomial of $A^n$ if you know the minimal polynomial of $A$?
Can you find a minimal polynomial of $A^n$ if you know the minimal polynomial of $A$?
I'm talking about minimal polynomials of matrices.
I'm asking in the general sort of way, I know that in some ...
7
votes
4answers
456 views
Cool/Useful Examples of Characteristic and Minimal Polynomials?
I'm teaching a Linear Algebra II undergrad course and for the section on characteristic & minimal polynomials, I really don't want to just give the students a bunch of matrices that have no ...
7
votes
3answers
349 views
Reconciling 'intersecting planes' and 'linear transformation' interpretations of matrices
I've learned in linear algebra class that an $n \times m$ augmented matrix can be thought of as a collection of n planes in $\mathbb {R}^m$ . If the matrix is invertible, the planes all intersect at a ...
7
votes
3answers
118 views
Prove the inequality: $\prod_{j=1}^ka_{jj}\leq\left(\frac{1}{k}\sum_{j=1}^k\lambda_j\right)^k.$
To all those who are eagerly awaiting a new question, all those who love math, I give this challenge and I hope for you good moments of reflection.
Let $A=(a_{ij})_n$ a real nonnegative symmetric ...
7
votes
0answers
119 views
Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$
What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$?
$\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
6
votes
5answers
254 views
How to prove $I + t X$ is invertiable for small enough $ | t | ?$
Let $X \in \text{GL}_n(\mathbb{R})$ be an arbitrary real $n\times n$ matrix. How can we prove rigorously:
$$ \underset{b>0} {\exists} : \underset{|t|\le b} {\forall} : \det (I + t X) \neq 0 $$
If ...
6
votes
4answers
206 views
Given $A^2+cA+cI=0$, how to find inverse of $A+(c-1)I$?
Suppose a square matrix $A$ such that $A^2+cA+cI=0$ for all $c \in \mathbb{Z}$. How can I show that $A+(c-1)I$ is invertible and find its inverse?
I started off this way:
$A+(c-1)I = A+cI-I$
Then ...
6
votes
3answers
7k views
What is the most efficient way to determine if a matrix is invertible?
I'm learning Linear Algebra using MIT's Open Courseware Course 18.06
Quite often, the professor says "... assuming that the matrix is invertible ...".
Somewhere in the lecture he says that using a ...
6
votes
5answers
3k views
How to check if a symmetric $4\times4$ matrix is positive semi-definite?
How does one check whether symmetric $4\times4$ matrix is positive semi-definite?
What if this matrix has also rank deficiency: is it rank 3?
6
votes
2answers
711 views
Prove that $AB=BA=0$ for two idempotent matrices.
Suppose that $A, B$ are idempotent matrices ($A^2=A$), such that $A + B$ is idempotent, prove that $AB = BA = 0$
6
votes
4answers
4k views
Similar matrices have the same eigenvalues with the same geometric multiplicity
Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities.
Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices ...
6
votes
3answers
581 views
Is the product of two positive semidefinite matrices positive semidefinite?
If $X$ and $W$ are real, square, symmetric, positive semidefinite matrices of the same dimension, does $XW + WX$ have to be positive semidefinite?
This is not homework.
6
votes
4answers
206 views
Finding $A^n$ for a matrix
I have a matrix $$
A =
\left[ {\begin{array}{cc}
1 & c \\
0 & d \\
\end{array} } \right]
$$
with $c$ and $d$ constant. I need to find $A^n$ ($n$ positive) and then need to prove ...
6
votes
3answers
898 views
Integer Matrices with Inverse Integer Matrix
We know that if all the entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational.
Now suppose that all entries of an invertible matrix $A$ are integers. ...
6
votes
3answers
223 views
Characterization of the trace function
We know that the trace of a matrix is a linear map for all square matrices and that $\operatorname{tr}(AB)=\operatorname{tr}(BA)$ when the multiplication makes sense.
On the Wikipedia page for ...
6
votes
2answers
330 views
How did my professor do this?
I'm trying to figure out how my professor got to the step circled in red in the image below:
How did he get the values of the first row to become completely positive, and how did he derive the ...
6
votes
3answers
1k views
Eigenvalues and Eigenvectors of $2 \times 2$ Matrix
Lets say I have a $2 \times 2$ matrix (actually the structure tensor of a discrete image - I):
$$ \begin{bmatrix}
\frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial ...
