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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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 ...
0
votes
2answers
96 views
1
vote
1answer
34 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 ...
1
vote
0answers
145 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 ? ...
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
1answer
261 views

Bareiss Algorithm

I need to find the row Echelon form of a large, (sparse,) integer matrix. It seems that the Bareiss algorithm is a prime candidate, but I can't find any resources beyond the Wikipedia page that ...
1
vote
0answers
124 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
44 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
296 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.
3
votes
2answers
65 views

Matrix which when multiplied, gives a maximal minimum of elements of result.

I'm working on an optimization problem and am stuck at this particular step. Let $\bf{A}$ be a matrix with 4 columns and a finite number of rows, consisting of elements which are either 0 or 1. Let ...
4
votes
1answer
2k views

Why is the operator norm of a diagonal matrix it's largest value?

I read this in my textbook have tried working through it - I keep getting max 2-norm(Ax), which is just the magnitude of Ax. How should I do this proof? (note, this is not for homework, I'm just ...
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
56 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 ...
0
votes
4answers
4k views

How to prove $A+ A^T$ symmetric, $A-A^T$ skew-symmetric.

Prove that if $A$ is a square matrix, then: a) $A+ A^T$ is symmetric. b) $A-A^T$ is skew-symmetric. c) Use part (a) and (b) to show $A$ can be written as the sum of a symmetric matrix $B$ and a ...
1
vote
0answers
69 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 ...
1
vote
0answers
33 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
44 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. ...
3
votes
1answer
87 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?
2
votes
1answer
95 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
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
178 views

Example of matrix $M\in GL_3(\mathbb{Z}/7\mathbb{Z})$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is 3

I would want to make an example of a matrix $M \in GL_3(\mathbb{Z}_7)$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is $3$. Any hints how to do that ...
0
votes
3answers
820 views

Reflection across the plane

Let $T: \Bbb R^3 \rightarrow \Bbb R^3$ be the linear transformation given by reflecting across the plane $S=\{x:-x_1+x_2+x_3=0\}$ (...) Then, $S=\gen[(1,1,0),(1,0,1)]\$ But how can I get the matrix ...
5
votes
3answers
6k views

Diagonalizable matrix $A$ invertible also?

If a matrix $A$ is diagonalizable, is $A$ invertible? I know that $P^{-1}AP = \text{some diagonal matrix}$ and therefore $P$ is invertible, but not sure of $A$ itself.
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} ...
0
votes
2answers
385 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
0answers
872 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 ...
3
votes
1answer
100 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 ...
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
74 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.
1
vote
1answer
112 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 ...
1
vote
0answers
58 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 ...
2
votes
2answers
79 views

When can we write the square of a matrix as the product of the matrix and its transpose?

I often see something like $(A - B)^2$ being written as $(A - B)(A - B)^T$ . Here $A$ and $B$ are two matrices. I can see that this is possible when $A$ and $B$ are scalars (i.e) single element ...
2
votes
1answer
854 views

How to solve a 3x4 matrix has no solution, a unique solution, and infinite solutions??

The system is : $$ \begin{matrix} 1 & -4 & 6 & a & | & 0 \\ -2 & 5 & -4 & -1 & | & b \\ 1 & -10 & 22 & 8 & | & c \end{matrix} $$ After ...
4
votes
2answers
261 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& ...
-2
votes
2answers
75 views

Vector derivative $\frac{d(Ax)}{d(x)}$ [closed]

I just need to know that whether it is $A$ or $A^T$ . I need it for an homework . Please be quick in telling me . Thanks !
2
votes
2answers
52 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
2answers
595 views

Show that A is invertible and that it is Lower Triangular.

Does anybody have a solution to the given word problem below? Let A be a lower triangular n x n matrix with nonzero entries on the diagonal. Show that A is invertible and and that A-inverse is lower ...
2
votes
1answer
889 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
2answers
225 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 ...
1
vote
1answer
30 views

$|Av||A^{-1}v|$, $A$ non-singular, $|v|=1$.

Let $A$ a non-singular $n \times n$ matrix, $v \in \mathbb{R}^{n}$ a variable vector. The operator norm of $A$ is defined to be $|| A ||=\max_{|v|=1} |Av|$ where $|Av|$ is the standard Euclidean norm ...
4
votes
1answer
94 views

Commutator subgroup of $GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$ is $SL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$

How would I go about showing this, where $p$ is an odd prime? The inclusion $[GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z}),GL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})] \subseteq SL_{2}(\mathbb{Z}/p^{2}\mathbb{Z})$ is ...
4
votes
1answer
61 views

Online tools for generating the NULL SPACE of the matrix over Finite Field of size 2

Is there any online tool where I just enter the values in (0,1) Finite Field of size 2 and it's give me the NULL SPACE matrix ? I have 25x25 , 36x36 , 25x36 , 36x25 matrix. Below is my 25 x 25 ...
0
votes
1answer
122 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.
3
votes
3answers
92 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 ...
31
votes
3answers
2k views

How to find the determinant of this matrix?

Today at my linear algebra exam, there was this question that I couldn't solve. There was a matrix $A$ $$A=\begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & ...
1
vote
2answers
74 views

Relation between condition number and perturbed matrix

Prove that if $A\vec{x} = \vec{b}$ and $(A+\delta{}A)(\vec{x}+\delta\vec{x}) = \vec{b}$, then $\dfrac{\|\delta\vec{x}\|/\|\vec{x}+\delta\vec{x}\|}{\|\delta{}A\|/\|A\|} \le \kappa{(A)}$, where ...
2
votes
2answers
564 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?
0
votes
1answer
29 views

Is the spectrum of a product of two operators, $AB$, invariant under $UAU^{\dagger}$ for unitary $U$?

This question is about linear operators on a Hilbert space. If necessary, the Hilbert space can be assumed to be finite dimensional. I have two Hermitian operators, $A$ and $B$. Do we have $$ ...