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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7
votes
0answers
457 views

Condition of an eigenvector problem

Please, somebody help me with this problem. [Ciarlet 2.3-5] Let ${A}$ and ${B} = {A} + \delta{A}$ be two symmetric matrices with eigenvalues $$\alpha_1\ \leq\ \alpha_2\ \leq\ \ldots\ \leq\ ...
3
votes
4answers
240 views

If $X$ is an orthogonal matrix, why does $X^TX = I$?

It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of ...
1
vote
1answer
103 views

Trace Constraints and rank-one positive semi-definite matrices.

Let $C_1$,$C_2$...$C_N$ be $M\times M$ hermitian matrices and $c$ be a given positive constant. Let $W$ be a positive (possibly semi) definite matrix such that \begin{align} \text{trace}\{WC_1\}\geq ...
1
vote
1answer
27 views

distributed FFT matrix solvers

i am working in a small project where i code a distributed matrix solver where each matrix dots is represented by a function itself and the final computation of each threaded solver is queued to the ...
4
votes
2answers
102 views

$A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$

I found the following problem interesting but do not know how to tackle it. If $A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$.Then find $\det (A^2+B^2)=?.$ Can ...
1
vote
0answers
28 views

Prove that the Rank of a complex skew-symmetric matrix is even [duplicate]

I ran across this problem in a book a while back, and it has been on my mind lately. Can anyone think of a proof? Prove that the rank of a complex skew-symmetric matrix is even. I have thought of ...
0
votes
1answer
213 views

Simple Probability Matrix

Consider a simple model that predicts whether you pass you next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your ...
0
votes
1answer
36 views

condition number quasi-metric

Let $A, B,C$ be invertible matrices, and $$d(A,B)\ :=\ 1-\frac{1}{\mbox{cond}(A^{-1}B)},$$ where $\mbox{cond}(E)=\|E\|\cdot\|E^{-1}\|$ and $\|\cdot\|$ a subordinate matrix norm. Somebody knos how do I ...
0
votes
1answer
237 views

linear transformation of real symmetric square positive semi-definite matrix

I am trying to find properties or constraints on a $(p \times n) $matrix $U$ such that upon left multiplying a real symmetric square positive semi-definite matrix $V$ with $U$ the resulting matrix $W$ ...
4
votes
1answer
101 views

Inverse of matrices with 3 parts!

I just wonder if there is any closed form solution for the inverse of matrices with following form, or if it's possible to decompose them. $ \left[\begin{array}{cccccccccc} {\color{red}1} & ...
1
vote
0answers
57 views

A Has characteristic polynomial that can be reduced to linear products $\Rightarrow$ A similar to upper triangular Matrix

Prove that if $A\in M_{n}\left(\mathbb{F}\right)$ matrix with a characteristic polynomial that can be written as a product of linear elements (?) ...
1
vote
1answer
821 views

Real and complex canonical forms of quadratic form

How do I find the canonical form of $$q_1(x,y,z)= 4x^2 +4xz+2yz$$ Now I have put it in matrix form as: $$\left( \begin{matrix} 4 & 0 & 2 \\ 0 & 0 & 1 \\ ...
2
votes
2answers
526 views

Basis of complex matrix vector space over $\Bbb{R}$

I understand that the basis of the vector space $$Mat_2(\Bbb{R}) = \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$$ over $\Bbb{R}$ is $$e = \left\{ \begin{pmatrix}1 & 0\\ 0 ...
1
vote
1answer
77 views

Condition number of matrix order 2

I am trying to prove that $$\mbox{cond}_2(A)\ =\ \inf_{E\in\mathscr{E}}\mbox{cond}_2(E),\;\;\; \mbox{ where }\;\; A = \left(\begin{array}{cc} 100 & 99\\99 & 98 \end{array}\right).$$ And ...
4
votes
3answers
253 views

If $Ax=B$ has two solution, then there must be a third one?

How do I prove this conjecture? Let $A$ be a matrix, and $B$ be a column vectore. If $Ax=B$ has two solutions, then there must be a third one. Thanks in a advance!
2
votes
1answer
55 views

Mutiplication of matrices by multiplying blocks of entries.

Let $$\mathbf{A}= \begin{bmatrix} A_{11} & A_{12}&\dots &A_{1r} \\ A_{21} &A_{22} &\dots &A_{2r} \\ \vdots & \vdots &\ddots &\vdots \\ A_{s1} &A_{s2} ...
1
vote
1answer
34 views

Is this necessarily a basis? [duplicate]

Let $T$ be a linear endomorphism of $\mathbb{R}^{3}$, suppose there exists non-zero vectors $u,v,w$ with $T(u)=u, T(v)=2v, T(w)=3w$. Is $\{u,v,w\}$ necessarily a basis of $\mathbb{R}^{3}$?
1
vote
0answers
57 views

When will a random matrix with real entries have all real eigenvalues?

I'm playing around generating random matrices with either sparse gaussian entries or sparse 0-1 entries. In both cases, I find that the power method often fails to find an eigenvector (from an ...
2
votes
1answer
74 views

Condition number question

Please, help me with this problem: Let $A$ a matrix of orden $100$, $$A\ =\ \left(\begin{array}{ccccc} 1 & 2 & & & \\ & 1 & 2 & & \\ ...
2
votes
2answers
75 views

Reversing the Gram Matrix

Let $A$ be a $M\times N$ real matrix, then $B=A^TA$ is the gramian of $A$. Suppose $B$ is given, is $A$ unique? Can I say something on it depending on $M$ and $N$.
1
vote
2answers
561 views

form of symmetric matrix of rank one

The question is: Let $C$ be a symmetric matrix of rank one. Prove that $C$ must have the form $C=aww^T$, where $a$ is a scalar and $w$ is a vector of norm one. I think we can easily prove that if ...
0
votes
1answer
69 views

Calculate $|X|$ , where $X=(A+A^2B^2+A^3+A^4B^4\dots _{100 \ terms})$

$$A = \left[ \begin{array}{rrr} 2 & -2 & -4 \\\ -1 & 3 & 4 \\\ 1 & -2 & -3 \end{array} \right]$$ $$B = \left[ \begin{array}{rrr} -4 & -3 & -3 \\\ 1 ...
1
vote
2answers
69 views

a simple question

I have an inequality: $$2ax^2+by^2\geq0$$that $x^2\geq y^2$. Actually $x^2$ is $c.c$ that is $(\sum c_{ij}^2)$ and $y^2$ is $(tr(c))^2$, where $c$ is $2\times 2$ matrix. Now, I want to show that ...
6
votes
2answers
231 views

Besides Vandermonde matrix, is there any other $m$ by $n (m>n)$ matrix in which any $n$ rows has a full rank?

I want to find some $m \times n(m>n)$ matrices that have the property that any $n$ rows has a full rank. Vandermonde matrix and Cauchy matrix are the only two matrices I know, can you guys give me ...
1
vote
1answer
51 views

Find the smallest integer parameter $a$ for which the following matrix is the Gram matrix

I have the following matrix: $$ \begin{pmatrix} 13 & 5 & 1 \\ 5 & 2 & 2 \\ 1 & 2 & a \\ \end{pmatrix} $$ I need find the smallest ...
0
votes
1answer
3k views

Inverse of upper triangular matrix

I have an upper triangular matrix that I want to solve the inverse for. I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
2
votes
1answer
243 views

Proving linear independence

Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy: $$ Av_1 = v_1 \\ Av_2 = 2v_2 \\ Av_3 = 3v_3 $$ Prove that $\{v_1, v_2, v_3\}$ is ...
0
votes
1answer
52 views

Kernels of Adjoints

Let $A$ be an $m \times n$ matrix. Show that $\mbox{Ker} A = \mbox{Ker} (A^*A)$. To do that you need to prove 2 inclusions, $\mbox{Ker} (A^*A)$ is a subset of $\mbox{Ker} A$ and $\mbox{Ker} A$ is a ...
0
votes
2answers
47 views

v(1, 0, 0) -> v(0, -1, 0) with rotation matrix?

How can v(1, 0, 0) be changed to v(0, -1, 0) with a single rotation matrix? I suppose a 2D vector of v(1, 0) -> v (0, -1) is fine as well.
4
votes
2answers
56 views

Nondiagonal $3 \times 3$ matrix

Can someone give an example of a nondiagonal, $3 \times 3$ matrix that is diagonalizable but is not invertible? Explanation would be appreciated.
-2
votes
1answer
54 views

Determine Span of vectors?

I did not see question like this before? What is span of $(1,1+x,1+x+x^2,....,1+x+x^2+...+x^n)$ ? The question also says to let $V=P_n(X)$ be the space of all polynomials whose degrees are less than ...
0
votes
1answer
104 views

Solve: This System of equations for $X$ (does a real solution, exist?)

How can I solve $AX + diag(X)[I-c]=0$ for $X$? All matrices have real entries, $diag(X)$ is a diagonal matrix with the diagonal entries being the diagonal entries of $X$, and $c$ is a constant, real ...
0
votes
2answers
165 views

Multiplying Out Inner Products

If I have a product of the form $(x-s)^tA(x-s)$ where $x$ and $s$ are vectors and $A$ is a matrix, how would I go about multiplying this out? Further, how would I go about taking its derivative?
5
votes
1answer
183 views

Condition of the eigenvalue problem

[Ciarlet, 2.3-1] I know this result: Let $A$ a diagonalisable matrix, $P$ a matrix such that $$P^{-1}AP\ =\ \mbox{diag}(\lambda_i)\ =\ D,$$ and $\|\cdot\|$ a matrix norm satisfying ...
0
votes
1answer
333 views

Finding all $\alpha$ such that the linear system has infinite solutions

I have the linear system: $$ \begin{bmatrix} 2 & -1 & 3 & 5\\ 4 & 2 & 2 & 6\\ -2 & \alpha & 3 & 1 \end{bmatrix} $$ I want to find all $\alpha$ such that the system ...
2
votes
1answer
171 views

Proving that the sum of the errors of a least square linear approximation is $0$

Let $(x_1,y_1),\dots,(x_n,y_n)$ be points in $\mathbb{R^2}$ and $e=[\epsilon_1,\dots,\epsilon_n]^T$ the error vectors belonging to the least square solution of the linear approximation. Prove that ...
3
votes
1answer
189 views

Proof that $U(n)$ is connected

I'm trying to prove that $U(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I\}$ is connected, but most of the proof comes down to proving that $SU(n)= \{ X\in Mat_n(\mathbb{C})|X^T\bar{X}=I $ and $ ...
14
votes
5answers
879 views

A symmetric matrix whose square is zero

I was once asked in an oral exam whether there can be a symmetric non zero matrix whose square is zero. After some thought I replied that there couldn't be because the minimal polynomial of such a ...
0
votes
1answer
61 views

Benefit of Drazin inverse

What benefits gives Drazin Inverse? Physically what it corresponds to? Thanks much
5
votes
6answers
196 views

Show that every subspace of $\mathbb{R}^n$ is a kernel of a linear map.

Let $S$ be a subspace of $\mathbb R^n$. Show that there is an $n \times n$ matrix $A$ such that $$S= \{x \in \mathbb R^n : Ax=0\}.$$ How to proceed?
0
votes
2answers
576 views

$x^TAx=0$ for all $x$ when $A$ is a skew symmetric matrix

Let $A$ be an $n\times n$ skew symmetric matrix. Show that $x^TAx =0 \ \forall x \in \mathbb R^n$. How to prove this?
1
vote
1answer
59 views

Proving properties of triangular matrices

This is the question that I'm having trouble with: I understand what the alternating and multilinear properties are, and I know that the determinant of the matrix is an alternating, multilinear ...
3
votes
1answer
117 views

Is this a matrix norm?

In wikipedia, the operator norm of a matrix is given by (assume: real, $n$-dimensional) $$ ||A||= \max \left\{ \frac{|Ax|}{|x|}:x \in \mathbb{R}^n, x\neq 0 \right\}$$ (I'm not sure why it is not a ...
2
votes
3answers
87 views

How to show $\mathrm{Sym}_{n\times n}(\mathbb{R}) + \mathrm{Skew}_{n\times n}(\mathbb{R})= \mathrm{M}_{n\times n}(\mathbb{R})$?

$$\mathrm{Sym}_{n\times n}(\mathbb{R}) + \mathrm{Skew}_{n\times n}(\mathbb{R}) \stackrel{?}{=} \mathrm{M}_{n\times n}(\mathbb{R})$$ Using the simplest core ideas of linear algebra, could someone ...
1
vote
1answer
54 views

How to show that $\mathrm{Sym}_{n\times n}(\Bbb{R})$ and $\mathrm{Skew}_{n\times n}(\Bbb{R})$ are subspaces of $\mathrm{M}_{n\times n}(\Bbb{R})$

A matrix $M \in \mathrm{M}_{n\times n}(\mathbb{R})$ is called symmetric (respectively, skew-symmetric) if $M^t = M$ (respectively, $M^t = -M$). How does one prove that the sets $\mathrm{Sym}_{n\times ...
3
votes
1answer
69 views

Finding $A^n$ in terms of $P$ and $D$ (diagonalized)

My question is regarding the last two parts. I have Found $D$ and $P$, how can I obtain $A^{200}$ and det $(A^{200})$ form $PDP^{-1}?$ Thanks!
6
votes
1answer
299 views

Matrix Norm set

I need help with this problem: Let $\|\cdot\|$ and $\|\cdot\|^{\prime}$ two matrix norms, and consider the relation $$\|\cdot\| \leq \|\cdot\|^{\prime}\ \Leftrightarrow\ \|A\| \leq \|A\|^{\prime},$$ ...
1
vote
1answer
33 views

How to compute future's (tomorrow's) distribution given today's?

Marital status can be defined as single, married, separated, or divorced. Today's distribution is: Single = 49% Married = 25% Separated = 15% Divorced = 11% I ...
7
votes
3answers
284 views

Is $\;\det(A^n) =\left(\det (A)\right)^n\;$?

How can the value of $\;\det\left(A^{11}\right)\;$ be calculated from $\;\det(A)$? Generally how can $\;\det\left(A^n\right)\;$ be obtained from $\;\det(A)$?
0
votes
2answers
155 views

Positive semidefinite matrices and polynomials

If $A$ is positive semidefinite and if $p(t)$ is any polynomial such that $p(t) > 0$ for $t \geq 0$, show that $p(A)$ is positive semidefinite.