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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4
votes
2answers
58 views

Finding the dimension and basis of an orthogonal space

I am trying to find the basis and dimensions of the the space orthogonal $S$ which is in $\mathbb R^3$. $$S = \begin{bmatrix}1\\2\\3\end{bmatrix}$$ So the dimension would be two because it is $3 - ...
0
votes
1answer
104 views

Given its pseudo-inverse, is there a fast way to measure the degree of full-rankness of a nonsquare matrix?

update: I realized the core of question is about ill-conditioning of the matrix (aka Multicollinearity). In a computer, with floating point arithmetic, it is impossible to talk about full-rankness. We ...
2
votes
2answers
49 views

How to find a matrix $C$ such that $C^{-1}AC$ is in Jordan block form.

$A:=\begin{bmatrix} 6 & -1\\ 4 & 2 \end{bmatrix}$ Now, just to show I've done some working, at least to find $A$'s eigenvalues and deduced that it's not diagonalisable: Any hints/advice? ...
0
votes
1answer
22 views

Checking validity of eigenvectors

I've written a program that finds the first $K$ eigenvectors of a matrix and would like to figure out if my solutions are truly valid eigenvectors. What is a good way of doing this?
1
vote
2answers
325 views

Matrix Exponential of Identity Matrix

I was just wondering what would the sum be of $e^{I_n}$ where $I_n$ is the identity matrix. I know the maclaurin series for $e^x$ is $1+\frac x{1!}+\frac {x^2}{2!}+...$. I know that $e^0$ is 1 right? ...
1
vote
0answers
24 views

Regression factors and covariance matrix

I am trying to follow some notes. They have two matrices. One is called comfact (company factors). This is a $580 \times 5$ matrix. The $580$ rows represent $580$ different companies. The $5$ ...
1
vote
1answer
35 views

Show $||A||_2=(\sum\limits_{i,j=1…n} a^2_{ij})^{(1/2)}$ defines a Matrix Norm

Show $$\def\norm#1{\left\lVert{#1}\right\rVert_2{}}\norm A ={\left(\sum_{i,j=1\ldots n} a^2_{ij}\right)}^{1/2}$$ defines a Matrix Norm for $A\in\mathbb R^{n\times n}$ to show: $\norm{ AB}\le ...
0
votes
2answers
99 views

Solution to $n$ by $n$ game of lights out

How can I solve an $n$ by $n$ game of Lights Out?
1
vote
0answers
155 views

Condition for a block matrix to be positive semi-definite

Let's say we have two positive semi-definite matrices $A$ and $B$ and a negative real $\lambda$. What would be the conditions for the matrix $M$, defined as follows, to be positive semi-definite too ? ...
0
votes
2answers
58 views

Difficulty in calculation of gradient in this specific case

I have tried but I am not able to calculate the gradient $\bigtriangledown_{\theta}J(\theta)$ of a function $J(\theta)$. $J(\theta) = (\| \sum_i \sum_j G_{i,j}G_{i,j}^T \|^2_{F})^{-1}$ Here $ ...
0
votes
1answer
63 views

Projection Matrix between two Vectors

Given a two normal vectors v1 = [a1;b1;c1] and v2 = [a2;b2;c2] as given in Fig1. How I can derive the projection matrix that ...
1
vote
1answer
28 views

Information content of an unlabelled matrix

I'm trying to get an idea of the amount of information that is "stored" in an "unlabelled" matrix. I assume that the vector $(x,y,z)$ contains more information than the set $\{x,y,z\}$. But purposely ...
1
vote
0answers
128 views

Inverse of positive definite matrix plus diagonal matrix

Let $C$ be a positive definite matrix, $D$ be a diagonal matrix with all elements being positive and $A=C+D$. By Woodbury matrix identity, we have $A^{-1}=C^{-1}-C^{-1}(D^{-1}+C^{-1})^{-1}C^{-1}$ or ...
0
votes
3answers
46 views

Non-Surjective Function

I'm reading an introductory text on abstract algebra, without the benefit of any recent experience with matrices or linear algebra. The text includes a statement that a map from $\Bbb{R}$ to ...
0
votes
3answers
352 views

zero matrix to the power of 0

Why $0^0=I$? I'd tried prove that considering $N^0$ where N is a Nilpotent matrix and then using the Cayley -Hamilton theorem Thanks in advance.
4
votes
2answers
3k views

For all square matrices A and B of the same size, it is true that (A+B)^2 = A^2 + 2AB + B^2

The below statement is a true/false exercise. Statement: For all square matrices A and B of the same size, it is true that (A + B)2 = A^2 + 2AB + B^2. My thought process: Since it is not a proof, I ...
2
votes
1answer
59 views

Submultiplicativity stronger than triangle inequality?

I would like to ask a question about matrix norm. Is the submiltiplicativity property always stronger than the triangle inequality? So, if i prove for a matrix norm that it's submultiplicative, i ...
1
vote
2answers
98 views

Laplacian matrix eigenvalues

Let $L$ be the laplacian of a connected graph. Is the maximum eigenvalue of $A=\begin{bmatrix} I & 0\\0&0\end{bmatrix} -L$ different than $1$??
3
votes
1answer
247 views

Eigenvalues of a submatrix

A self-adjoint matrix $A$ in a complex linear space has real eigenvalues bounded between $0$ and $1$. If $A$ is projected onto an arbitrary two-dimensional subspace, what would be the bounds on the ...
3
votes
1answer
88 views

Clever way to square a matrix

How do you square a matrix $A$? Do you use any clever way to do it (i.e not using the standard matrix multiplication)? It can be useful 'considering' $A$ like a linear application?
0
votes
1answer
51 views

Basic concepts about matrices and their decompositions

I am studying the basics of linear algebra and I have some questions that I can not conclude them by my own. Let $A \in \Bbb R^{m \times n} $ $A$ can always be expressed as a LU decomposition? ...
1
vote
0answers
37 views

Quick relatively sharp upper bound for the largest singular value of $m \times n$ matrix $X$

Is there anything analogous to the Gershgorin Circle Theorem but for the singular values of an $m \times n$ matrix $X$? I'm interested in a relatively sharp upper bound for the largest singular value ...
1
vote
0answers
47 views

Gershgorin's Theorem for a matrix X'X when X'X cannot be computed

Is there any way to estimate or approximate the largest eigenvalue of X'X using X alone? I know that this can be done by approximating the singular values of X, but I'm looking for another approach. ...
0
votes
2answers
86 views

Rigorous proof that any linear map between vector spaces can be represented by a matrix

I searching the internet in hope of finding a proof. However, most of what I have seen this relationship is defined informally and/or gloss off this. Would you kindly point me in the direction of a ...
3
votes
2answers
80 views

Prove $A^{2}=0$ iff $C\left(A\right)=R^{0}\left(A\right) $

I'm learning some Linear Algebra through a University Textbook and I've come across this question which I have a hard time solving: Let There be a square Matrix A. Prove that $A^{2}=0$ iff ...
2
votes
1answer
99 views

How can I find a matrix $\bf B$, with positive eigenvalues, such that its square $\bf B^2$ is another matrix $\bf A$?

I've been given a 2x2 matrix $\mathbf A$, its eigenvalues $\lambda_1$ and $\lambda_2$, its eigenvectors $\mathbf v_1$ and $\mathbf v_2$, and a diagonal matrix $\mathbf D = \text{diag}(\lambda _1, ...
0
votes
2answers
63 views

Proof matrices and their eigenvalues

Let $C=A-B$ where $A=\begin{bmatrix}I &0\\ 0&0\end{bmatrix}$ and $B$ is a Laplacian matrix of a connected graph, so sum for rows is null and it doesn't have any zero row(or column). ...
0
votes
1answer
35 views

Hermitian matrices and their eigenvalues

Let $C=A+B$ where $A$ and $B$ are two hermitian matrices can I prove that $\lambda_{i,C}=\lambda_{i,A}+\lambda_{i,B}$ iff $x_{i,A}=x_{i,B}$? Where $x_i$ is the eigenvector related to eigenvalue ...
0
votes
2answers
30 views

How to solve a system of linear equation when the first column is all $0$s?

I want to solve this: $$\left[ \begin{array} {cc} 0& 1 \\ 0& 0 \end{array}\right] \left[ \begin{array} {c} x_1 \\ x_2 \end{array} \right]= \left[ \begin{array} {c} 0 \\ 0 \end{array} ...
1
vote
0answers
72 views

Eigenvalues of sum of two particular matrices

Let $A$ be a matrix with real eigenvalues, its maximum eigenvalue is $0$ and it has sum for rows equals to zero. Let $B$ be a matrix $\mathrm{diag}([1\,0\, ...\, 0])$ and let $I$ be the identity ...
0
votes
2answers
408 views

orthogonal projection and Cauchy Schwarz inequality

Show that if P is an orthogonal projection matrix, then $||Px||\le||x||$ for every x. Use this inequality to prove the Cauchy Schwarz inequality. I know that if P is an orthogonal projection matrix, ...
2
votes
1answer
935 views

Proof of Vandermonde Matrix Inverse Formula

I'm working through Exercise 40 from section 1.2.3 of Knuth's The Art of Computer Programming volume 1, but am finding myself unable to produce a rigorous proof, and the one here is suspect and not ...
0
votes
2answers
46 views

Proving a Simple equation

I have a not so smart question; but I just cannot figure it out ! Suppose that I have a real $2 \times 2 $ matrix $(a_{ij})$ of non-zero determinant, and let $z \in \mathbb{C} $ be such that $ ...
0
votes
1answer
76 views

If A = BC and B is invertible, then how does reducing “B to I” also reduce “A to C”?

If $A = B*C$, where $B$ is an inverse, use row-ops to reduces "$B$ to $I$" also shows that it will reduce "$A$ .. $C$". Big-Hint: Represent the row operations by a sequence of elementary matrices.
3
votes
1answer
101 views

eigenvalues of sum of matrices: $A$+block matrix are strictly less than eigenvalues of $A$+identity

Let $A$ whose sum for rows is 0, can I prove that the $ \lambda_i \left ( \begin{bmatrix} I & 0\\ 0 & 0 \end{bmatrix} +A\right ) $ are strictly less than $\lambda_i \left ( \begin{bmatrix} I ...
1
vote
0answers
60 views

I want to generate(or count) all possible binary matrix that satisfy certain Condition

I want to generate(or count) all possible binary matrix that satisfy below Condition. let A be arbitrary binary matrix 4*4 ...
0
votes
5answers
101 views

multiplying a matrix by a row vector

Is multiplying a matrix by a row vector the same as multiplying it by a column vector? Or are there any differences between the two?
4
votes
2answers
262 views

Why and When is a determinant of a larger matrix equal to a determinant of a smaller matrix?

The following is written in the solution of my textbook. $$|A|= \left| \begin{array} {cccc} 1 & 2& -1& 4 \\ 0& 5& -1& 6 \\ 0& -3& 3& -6 \\ 0& 2& 2& ...
1
vote
4answers
102 views

Reducing the System of linear equations

\begin{align*} x+2y-3z&=4 \\ 3x-y+5z&=2 \\ 4x+y+(k^2-14)z&=k+2 \end{align*} I started doing the matrix of the system: $$ \begin{pmatrix} 1 & 2 & -3 & 4 \\ 3 & -1 & 5 ...
3
votes
2answers
54 views

Determinant algebra

If $A$ and $B$ are $4 \times 4$ matrices with $\det(A) = −2$, $\det(B) = 3$, what is $\det(A+B)$? At first I approached the problem that $\det(A+B) = \det(A) + \det(B)$ but this general rule would ...
1
vote
1answer
120 views

Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
2
votes
1answer
915 views

Show that if $A$ is invertible and $AB = AC$, then $B = C$.

Question: Show that if $A$ is invertible and $AB = AC$, then $B = C$. My work: My thought process: If I can find the inverse of $A$, then I can show A is invertible. I will prove by example. $A$ is ...
1
vote
1answer
187 views

How many ways are there to represent a monomial order, defined by $>$, by term order via matrices?

During the lecture, my professor brought up the list of project ideas to work on. One of the ideas I am interested and currently working on is term order via matrices. That is: I need to find the ...
1
vote
3answers
833 views

Show square matrix, then matrix is invertible

Question: Show that if a square matrix A satisfies the equation A^2 + 2A + I = 0, then A must be invertible. My work: Based on the section I read, I will treat I to be an identity matrix, which is a ...
1
vote
2answers
236 views

(generalized) eigenvectors

$\DeclareMathOperator{\rank}{rank}$ First off I'm sorry I'm still not able to make of use the built in formula expressions, I don't have time to learn it now, I'll do it before my next question. I ...
0
votes
0answers
256 views

Block Diagonalisation of 4x4 Matrix

I'm attempting to find a 4x4 matrix, P, that will convert my matrices, $A = \begin{bmatrix}1&1&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$ and, ...
3
votes
3answers
93 views

Prove that for every vector $V$, $||V||_{\infty} \leq ||V||_2 \leq || V||_1$

$\newcommand{\inf}{||V||_\infty}$ $\newcommand{\two}{||V||_2}$ $\newcommand{\one}{||V||_1}$ Prove that for every vector $V$, $\inf \leq \two \leq \one$ I have tried to look online for a solution to ...
0
votes
1answer
131 views

proof for matrix norms

How do I prove these two inequalities on matrix norms: $\Vert A \Vert_1 \leq n\Vert A \Vert_\infty,$ $\Vert A \Vert_1 \leq \sqrt{n}\cdot\Vert A\Vert_F$ , where A is $m$-by-$n$ real matrix.
2
votes
2answers
612 views

Show linear system have no, only one or many solutions

Under what conditions on a and b will the following linear system have no solutions, one solution, infinitely many solutions? first row: 2x − 3y = a second row: 4x − 6y = b My work: I wrote the ...
0
votes
2answers
53 views

$AB=I_{n \times n}$ and $CA=I_{m \times m}$ prove that $m=n$

Let $A$ be an $m \times n$, $AB=I_{n \times n}$, $CA=I_{m \times m}$, prove that $n=m$. Is using inverse matrix is a valid solution?