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), ...

learn more… | top users | synonyms (2)

5
votes
3answers
7k 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
387 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
875 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
856 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
890 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
227 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 $$ ...
0
votes
2answers
150 views

Matrix norm proof

Given is $\left | \left | A \right | \right |_{2} =\left ( \sum_{i,j=1}^{n} a_{ij}^{2} \right )^\frac{1}{2}$ for $A\in \mathbb{R}^{n\times n}$. Show that this defines a matrix norm. I remember i've ...
0
votes
2answers
20 views

If T, S, linear operators, does TS upper triangular imply that both T and S are upper triangular?

I think this is false, but am struggling to find a counterexample. Can anyone come up with a counterexample or proof?
0
votes
2answers
26 views

Find a real matrix with eigen vectors v and v's complex conjugate so that they have different eigenvalues.

I need to find a real matrix with eigenvector v, and eigenvector v's complex conjugate, such that they will have different eigenvalue. any hints please?
0
votes
2answers
406 views

Matrix challenge: How many submatrices in a matrix

I have came across a challenging question I'd to solve. I am a self-taught man, so be indulgent. Caution: the following is a translation from a non English language. Sorry if it contains syntax ...
1
vote
1answer
61 views

Determinant Of Matrix (A) - Confusion about wording of the question.

Okay, So I'm a bit confused on what to do for this question. I figured out that Det(B) is just the determinant of matrix A and that matrix B is just the upper-triangular version of Matrix A. But how ...
0
votes
1answer
68 views

Eigenvector of the A

So I have the following matrix $\begin{bmatrix}4 & 2 & 3\\-1 & 1 & -3\\2 &4 &9 \end{bmatrix}$ for which I found $\lambda=8, 3, 3$. Now when I try to row reduce the matrix with ...
2
votes
2answers
192 views

Eigenvalues of vectors with irrational entries

I have been trying to find eigenvalues and eigenvectors of this matrix: $\begin{bmatrix}3 & -2\\1 & -1\end{bmatrix}$. So far I have got $\lambda_1=1+\sqrt2$ and $\lambda_2=1-\sqrt2$. I am ...
0
votes
2answers
69 views

Using QR algorithm to compute the SVD of a matrix

How to use the QR algorithm to compute the SVD of a matrix $X\in R^{m\times n}$? Is there any algorithm for doing that?
-1
votes
1answer
89 views

Differentiation of the transpose of a vector? [closed]

Suppose $s$ is a scalar, and $x$ is a vector, how would I calculate $$ \left(\frac \delta {\delta x} (x^T s)\right) $$Basically I couldn't find any reliable source letting me know how to ...
0
votes
2answers
156 views

Norm of a matrix and lower bound for its determinant

Assume that $M$ is a positive constant, $A=[a_{ij}]$ is a matrix, and $\vert a_{ij}\vert \geq M $ for all $1\leq i,j \leq n$. Also, assume that $\det(A) \neq 0$ .Can we conclude that there exists a ...
0
votes
1answer
188 views

How do I calculate the Expectation value?

http://en.wikipedia.org/wiki/FastICA In the above wiki page, how do I calculate the expectation values in the second step of the algorithm for "Single Component Extraction"? My interpretation is as ...
0
votes
1answer
38 views

What do these terms actually mean about linear transformation?

In a book it is written " Let A be a fixed $m\times n $ matrix with entries in the field $\mathbb F$. The function $T$ defined by $ T(X)=AX$ is a linear transformation from $\mathbb F^{n\times 1 }$ ...
2
votes
0answers
864 views

Natural matrix norm of an inverse matrix

Let $\left\|\cdot\right\| : \text{GL}(n,\mathbb{R})\to\mathbb{R}_{\ge 0}$ denote the natural matrix norm, i.e. $$\left\|A\right\|:=\max_{x\ne ...
1
vote
0answers
59 views

Linear Algebra Matrix Transformation Question

Can someone please help me out with this question. If a nonzero matrix $A$ is transformed from $\mathbb{R}^3$ to $\mathbb{R}^2$, then the null space of $A$ must be a one dimensional (sub)space of ...
1
vote
0answers
38 views

find error for $AX = B$ for many right hand sides

In order to find error for $AX = B$ for many right hand sides,is the below the right way? Let's say the product $$ AX = \begin{pmatrix} 1 &2\\ 3 &4\\ 5 &6 ...
3
votes
2answers
250 views

Is the determinant of a matrix Lipschitz continuous?

I want to know if the determinant of a matrix is Lipschitz continuous or not. To be precise, does there exist a constant $K$ such that $|\det(A)-\det(B)|\leq K||A-B||_F$, for all matrices $A,B\in ...
0
votes
2answers
55 views

How to expand $(x+ty)^{T}A(x+ty)$

I am trying to expand the expression: $$(x+ty)^{T}A(x+ty),$$ with $x,y$ being vectors and $A$ a matrix. All I know is the distributed law, $(A+B)C = AC+BC$. Can someone explain how to arrive to the ...
0
votes
3answers
90 views

Derivative of matrix and vector in $\mathbf {v^TMv}$

Suppose I have a ($n\times 1$) vector $\mathbf v$ and a ($n\times n$) matrix $\mathbf M$ and I want to compute the derivative w.r.t. some $x$. Both $\mathbf v$ and $\mathbf M$ depend on the scalar ...
0
votes
1answer
68 views

linear independent or dependent set - linear algebra

I have the following set: $\{ [1; -1; -2], [-1;0;1], [1;2;1] \}$ and I need to find out whether the set is independent or dependent. My answer and the book's answer contradict. I thought it was ...
0
votes
1answer
82 views

A simple optimization problem

$$f = x^Tx$$ $$g = Ax-b $$ The constraint is $Ax-b = 0$ I calculated $J' = f'+\lambda g'$ which is $2x^T+\lambda A^T = 0 $ and $Ax-b=0$ . I dont know what to do next please help me out .
2
votes
2answers
98 views

Understanding that $GL_n(\mathbb{R})$ has two connected components

I am trying to understand the proof of the theorem: $GL_n(\mathbb{R})$ has two components. The proof says that The group of matrices with positive and negative determinant, ...
0
votes
1answer
86 views

Help understanding a theorem on transitivity of a relation

The theorem states this: The relation R on a set A is transitive if and only if $R^n \subseteq R$ for n = 1, 2, 3,... What I'm reading is that the nth power of that set is transitive if the set ...