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
0answers
49 views

Time complexity of inverting an $n \times n$ matrix which is the sum of a rank-$m$ matrix and a full-rank diagonal matrix

I want to know the time complexity of inverting $K$, where $K$ is an positive-definite $n\times n$ matrix: $$K=\Lambda+Q$$, where $\Lambda$ and $Q$ are both $n\times n$ matrix, $\Lambda$ is a full-...
1
vote
1answer
38 views

Operator norm and eigenvalue inequality

Can I say that $\|A\| < s$ where $A \in \mathbb{R}^{3 \times 3}$ is a symmetric, positive definite matrix and $s$ is the maximum eigenvalue of $A$. Here the norm used is operator norm.
2
votes
1answer
42 views

What is $\mbox{Tr}^2(A)-\mbox{Tr}(A^2)$ in terms of the eigenvalues of $A$?

I am looking for a way to relate the terms of the characteristic polynomial of a $3 \times 3$ matrix to its eigenvalues. The definition I start with (taken from Wolfram MathWorld) is $\\P_{3}(A)=x^{...
1
vote
1answer
37 views

Find SVD of $A$

How do I find the singular values? They somehow show that $\lambda_1 = 27, \lambda_2 = 6, \lambda_3 = 0$. I still can't see how they found them with the equations I made in my solution.
1
vote
2answers
102 views

$A^{n+1}=0\Rightarrow A^n=0$

A real $n\times n$-matrix $A$ satisfying $A^{n+1}=0$ must necessarily satisfy $A^n=0$. One way to see this is by looking at the Jordan Normal Form of $A$, another is by an argument involving the ...
0
votes
0answers
20 views

Can I do Gaussian elimination with a rectangular matrix

I'm writing some linear algebra scripts to understand the stuff I'm reading. I was wondering if I can only use square matrices as input for Gaussian Elimination, my guess is yes because permutation ...
0
votes
1answer
15 views

Significance of a capital R outside of brackets containing a matrix expression

I was trying to understand what a positive definite matrix is while reading a reinforcement learning paper today, and I came across this page: http://mathworld.wolfram.com/PositiveDefiniteMatrix.html ...
1
vote
1answer
62 views

Is this a correct way of proving that if $A^3-3A + I = 0$ then $A^{-1}=3I-A$?

$A^3-3A+I = 0$, I multiply both sides by $A^{-1}$ $$(A^3-3A+I)*A^{-1} = 0$$ $$A^3*A^{-1} - 3A*A^{-1} + I*A^{-1} = 0$$ $$A^3*A^{-1} - 3I + A^{-1} = 0$$ $A(A^2*A^{-1})-3I+A^{-1} = 0$ Here I ...
3
votes
1answer
37 views

additivity of rank

we know that for all $A,B\in M_n(\mathbb{C})$ : $$ rank (A+B)\leq rank(A)+rank (B) $$ see here for a simple proof, but for which condition on the coefficients of $A$ and $B$ we can obtain a perfect ...
4
votes
5answers
108 views

Calculating the matrix $M^{2006}$

Say you have the matrix $M$: $$\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$$ How do you find $M^{2006}$? My thinking was that you ...
2
votes
1answer
33 views

Find projection matrix using partitioned matrices

If X is a ($n$, $p+1$) design matrix, partition $X$ to be $X$=[$J$ $X$*] where $J$ is a ($n$,$1$) vector of all $1$'s, and $X$* is a ($n$,$p$) matrix. Let $H_X$ be a projection matrix, where $H_X$ =...
1
vote
1answer
14 views

Linear Functional - Basis change

Having a trouble understanding what you have to do in case you have a "simple" linear functional. The basis change and stuff for Quadratic and Bilinear forms are everywhere in every book, but there is ...
1
vote
1answer
25 views

3D Matrix Transformation

This is a question that stems from a programming problem I am having, but I do not understand the math behind it. So forgive me if there is some Stack Overflow bleed in my question. I have an array. ...
2
votes
1answer
23 views

A matrix being symmetric/orthogonal/projection matrix/stochastic matrix

I am trying to do some practice questions and wanted to check the following properties and confirm my definition of projection matrix: Let $$A = \left[\begin{matrix} 1/2 & 0 & 1/2 \\ 0 &...
4
votes
1answer
111 views

Can I factor a rational expression of the form…

Given two equations $\displaystyle P_1 = \frac{1-X^2}{1-X^2}$ and $\displaystyle P_2 = \frac{1-aX^2}{1-bX^2}$ I am told that there is a relationship between P1 ...
1
vote
1answer
40 views

Can $A^2\preceq \gamma^2 B^2$ lead to that $A\preceq \gamma B$?

In the question, $A$ and $B$ are positive semi-definite matrices, $\gamma\geq 0$ is a constant, and $A\preceq \gamma B$ means that $\gamma B-A$ is positive semi-definite. We have known another fact ...
1
vote
1answer
23 views

The inner product matrix with zero determinant implies that all the vectors are linearly dependent.

Let $(V,\langle \cdot,\cdot\rangle)$ be an inner product space over the real field $\mathbb{R}$ and $v_1,\dots,v_n\in V$. Suppose that $A=(a_{ij})\in M_n(\mathbb{R)}$ with $a_{ij}=\langle v_i,v_j\...
0
votes
1answer
36 views

Prove or disprove: if $A$ is an $n \times n$ complex matrix, and $\ker(A) \cap \operatorname{range}(A) \neq \{0\}$, is $A^n = 0_{n \times n}$?

Question: Prove or disprove: if $A$ is an $n \times n$ complex matrix, and $\ker(A) \cap \operatorname{range}(A) \neq \{0\}$, is $A^n = 0_{n \times n}$? My attempt: I've shown that the converse of ...
0
votes
0answers
30 views

Minimizing trace with equality constraints

I would like to solve the following trace-minimization under equality constraints optimization problem: $$W^* =\arg\min \operatorname{Tr}[WCW^T] \text{ s.t. } A=B^TW^TWB$$ where $W,C\in\mathbb{R}^{...
0
votes
1answer
34 views

Given the parallel and perpendicular component of a vector in terms of another vector, how do you determine the tensor that connects both?

Sorry for the awkwardly phrased title, I wasn't sure how to properly word it. I want to do the following: I have a vector $\vec J$ and a vector $\vec E$ with the following relation (with the ...
1
vote
3answers
82 views

Is function invertible?

Reflection on the unit circle: Let $E=\mathbb R ^{2} - \left\{0,0\right\} $ be perforated plane and $f: E \mapsto E$ defined by $f\left(x,y\right)=\left(\frac{x}{x^{2}+y^{2} } , \frac{y}{x^{2}+y^{2}...
0
votes
1answer
29 views

Determining if system is consistent, and if it is determine if the solution is unique

In the following matrices [] is a nonzero entry and ∗ is a entry that may or may not be zero. For each of these (augmented) matrices determine if the associated system is consistent, and if it is ...
-1
votes
3answers
52 views

Choose $h$ and $k$ so that the system has no solution, one solution andinfinitely many

Choose $h$ and $k$ so that the system has no solution, one solution and infinitely many $x_1 + 3x_2 =2$ $3x_1 + hx_2 = k$ So I put it into a augmented matrix and row-reduced to get it in ...
0
votes
0answers
15 views

What is the Euclidean Metric of the Right-Most Column of a Matrix called?

If I have matrix like, $$M_2=\begin{bmatrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{bmatrix}$$ and I apply an operator such that, $$r_e(M_2)=m_{12}^2+m_{22}^2$$ What is that called? For ...
0
votes
1answer
56 views

Differentiation w.r.t. the $\mbox{vec}$ operator

I am stuck at solving the following derivative $$\frac{d \mbox{vec} (X^T X)}{d \mbox{vec} (X)}$$ where $X$ is an $m \times n$ matrix and $\mbox{vec}$ is the vector/stack operator. I have tried using ...
1
vote
0answers
23 views

What is the opposite of “sparsity” in a matrix?

If a sparse matrix has only 1% non-zero entries, I find it weird to speak of "1% sparsity". In particular, "increasing sparsity" goes along with a smaller percentage of non-zero entries, so this is ...
6
votes
4answers
189 views

What is the relation between $\det(A^TA)$ and $\det(AA^T)$?

In the question, $A \in \mathbb R^{m\times n}$ is a matrix, and $\det(\cdot)$ denotes the determinant.
1
vote
1answer
18 views

On some matrix inequality

Suppose we have as symmetric positive definite matrix $A$ such that \begin{align} 0 \preceq A \preceq ( C+ D)^{-1} \end{align} where $C$ and $D$ are both symetric positive definite. My question does ...
1
vote
1answer
10 views

Trace form of Frobenius Norm of Matrix approximation

I'm a CS Student and I've implemented the Convex Non-Negative Matrix Factorization (Convex-NMF) Algorithm for a project. Now, for "classic" NMF algorithms, you get an approximation: $$ \mathbf{A} \...
0
votes
1answer
49 views

Why $\frac{{\partial D}}{{\partial x}}$ and $\frac{{\partial D}}{{\partial y}}$ don't have any common factor?

Let ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$. ${\rm{P(}}\lambda {\rm{) = ...
1
vote
1answer
25 views

Name of this almost diagonal matrix

I am guessing this type of matrix has a name: $$M = \begin{pmatrix} d & 0 & 0 & 0 & 0 \\ a_2 & d & 0 & 0 & 0 \\ a_3 & 0 & d & 0 & 0 \\ a_4 & 0 &...
1
vote
0answers
52 views

explicit form of partitioned matrix

Consider a partitioned matrix: $D=\left( \begin{array}{cc} a & b' \\ b & C \end{array} \right)$ where $a$ is a scalar, $b$ is $m \times 1$ and $C$ is $m \times m$. I want to compute $a-b'C^...
2
votes
1answer
33 views

Proof from Matrix

I have tried upto where I could. please anyone help me to complete my proof.. Help much appreciated
4
votes
1answer
85 views

General solution of a vectorial ecuation

This is the first time that I ask a question here. When I was looking for the maximum of a multivariable vector function, I encountered the following problem: I cannot find the general solution of the ...
2
votes
1answer
70 views

Determinant of a tridiagonal matrix with a superdiagonal of ones and a subdiagonal of minus ones

$$ D_n = \begin{vmatrix} a_1 & 1 & 0 & \cdots& 0 & 0\\ -1& a_2 & 1 & \cdots & 0 & 0 \\ 0 & -1 & a_3 & \cdots & 0 & 0 \\ \vdots & \vdots &...
1
vote
1answer
35 views

Hamming's code is perfect

How does one prove that Hamming's code is perfect (i.e. it is the 1-error correcting code that has the smallest possible size). I haven't found a complete proof using Google.
1
vote
0answers
39 views

Determinant of a block rectangular matrix

Assume that $a \in \mathbb{R}^{n \times 1}, b \in \mathbb{R}, P \in \mathbb{R}^{n \times n}, u \in \mathbb{R}^{n \times 1}, x \in \mathbb{R}^{n \times 1}$ and $\lambda \in \mathbb{R}$. Now I want to ...
2
votes
3answers
69 views

Maximum Eigenvalue of the Discrete Laplace Operator (Image Processing)

Given the derivative operator in image processing $$ A = \begin{bmatrix} 1 & -1 & 0 & 0 & \ldots & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 & \ldots & 0 & 0 &...
1
vote
2answers
51 views

We know that D is a diagonal matrix and M has only ones in the entries. Can we say something about the eigenvalues of D+M?

Both matrices are trivial and it is easy to find their eigenvalues, but I can't find the eigenvalues of the sum D+M.
0
votes
0answers
18 views

How to construct a unimodular polynomial matrix?

Suppose that we have the nonsquare polynomial matrix \begin{equation} M(s)=\left[\begin{array}{cc}A(s)\,\,\,B(s)\end{array}\right], \end{equation} which has more columns than rows. Under what ...
1
vote
1answer
41 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
5
votes
2answers
91 views

Show that the map is onto

Suppose $R=\{A\in GL(n,\mathbb C):A^{-1}=\bar A\}$, where $\bar A$ is the conjugate matrix of $A$. Show that $$b:GL(n,\mathbb C)\to R,\quad A\mapsto A\bar{A}^{-1}$$ is onto. I don't know where to ...
0
votes
1answer
23 views

Quadratic form of a symmetric indefinite matrix

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric indefinite matrix. Furthermore, we let $e_i$ be the $i^{th}$ vector for standard basis. Is it possible to have $(e_i-e_j)^{T}A(e_i-e_j)>0$ for all ...
2
votes
2answers
38 views

Sum of elements of the inverse matrix (without deriving the inverse matrix) using elementary methods.

I have the matrix $$\begin{pmatrix} 3&2&2&\\ 2&3&2\\ 2&2&3 \end{pmatrix}.$$ Find the sum of elements of the inverse matrix without computing the inverse. I have seen ...
3
votes
1answer
100 views

Is $\ln \det F$ a rank one convex function of $F$

Let $F=F_{ij}$ be a matrix where $i,j \in \{1,2,..,d\}$. $F_{ij} \in \mathbb{R}$. And $F$ is taken to be invertible. Is Is $\ln \det F$ a convex function of $F$? Or is it a rank-one convex function of ...
2
votes
1answer
46 views

Fixed points of linear transformation and commutativity

Is it true the statement that if two square matrices $A$ i $B$ have the same fixed points $≠0$ for their linear transformation (i.e. for some vectors $v_i ≠0 $ we have $Av_i=v_i$ and $ Bv_i=v_i$) ...
1
vote
0answers
28 views

Is $1\le 3\le 4\le 4$ a possibility for the sequence $k_1\le k_2\le k_3\le k_4$?

Let $T$ be a $4\times 4$ real matrix such that $T^4=0$ .Let $k_i=\dim \ker T^i;1\le i\le 4$. Is $1\le 3\le 4\le 4$ a possibility for the sequence $k_1\le k_2\le k_3\le k_4$? My try: Consider ...
0
votes
1answer
29 views

What values of 'a' and 'b' would create a unique, no, and infinite solution(s)?

The following problem has really troubled me: I have row reduced it, so now it looks more like this: How do I really figure out what values of a and b would create infinitely many solutions, no ...
0
votes
1answer
72 views
+50

Is this a matrix notation of standard error?

What is the matrix notation of the standard error? A friend is referencing the standard error as: $$SE^2=(XX^T)^{-1}\sigma^2$$ $$\sigma^2 = \frac{1}{n-p}\sum\limits_{i=1}^n |\hat{y_i}-y_i|$$ ...
3
votes
1answer
66 views

Understanding Ax = 0 in Linear Algebra

If we have 3 systems of equations, that intersect at the point (1,2,3) does it have trivial or nontrivial solutions? Let's assume it is this system of equations which intersects at (1,2,3) and row ...