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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-2
votes
0answers
10 views

E as an expectation of a quadratic form

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
0
votes
2answers
27 views

Technical question in Vandermonde determinat proof

I can follow the proof given in (2nd proof, or the induction proof), until the sentence: "From the Expansion Theorem for Determinants‎, we can see that the coefficient of $x_k$ is:". I don't ...
2
votes
1answer
22 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
4
votes
2answers
67 views

Is this matrix invertible?

I have been working on a proof and am stuck with showing that the below matrix is invertible. I am not interested in the explicit inverse, only showing it has a nonzero determinant as the existence of ...
1
vote
2answers
24 views

spectral radius

Does the spectral radius of a matrix defines a norm? I mean does it satisfy the properties of norm, ie. $$||x|| \ge 0$$ $$||x|| = 0 \implies x=0$$ $$||kx|| = |k|\;||x||$$ $$||x+y||\le ||x||+||y||$$
-2
votes
3answers
29 views

Find the matrix product when possible

$\begin{bmatrix}-1& 3\\ 3 & 4 \end{bmatrix} * \begin{bmatrix}0& -2 & 4\\ 1 & -3 & 2\end{bmatrix} $ I realized that there is no third matrix column so does that mean I assume ...
0
votes
4answers
30 views

Find the indicated matrix

$$ A=\begin{bmatrix} 2& 3\\ 2 & 4 \end{bmatrix} $$ $$ B=\begin{bmatrix} 0& 4\\ -1 & 6 \end{bmatrix} $$ Find 2A+B Would I go 2*2+2*3+2*2+ *4+ 0+4+-1+6?
0
votes
0answers
17 views

Maximization of sum of convex functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) is concave ...
0
votes
0answers
20 views

Can I do Gaussian Elimination on this? (mod 2)

I have this matrix in GF(2): [0, 0, 1, 0] [1, 1, 0, 0] [0, 0, 0, 1] It's not a square matrix but I tried to do Gaussian elimination on it anyway after adding a ...
1
vote
1answer
52 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
1
vote
1answer
19 views

What is the characteristic polynomial of power of a matrix

If the c.p. of A is $(\lambda-2)^3(\lambda+2)^2(\lambda+3)$, how can I find the c.p. of $A^2$? Would it be $(\lambda-4)^3(\lambda+4)^2(\lambda+9)$? Thanks!
0
votes
1answer
33 views

Perform the operation or operations when possible.

$\begin{bmatrix}-5& -9\\ 9 & 3 \end{bmatrix} + \begin{bmatrix}8& 5\\ -4 & -1\end{bmatrix} -\begin{bmatrix} 4& -7\\ -9 & -6 \end{bmatrix} $ Also, I was trying to add ...
3
votes
4answers
56 views

Does an $n\times n$ matrix $A$ only have an inverse if $rank(A)=n$? If so, why?

I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example ...
6
votes
2answers
37 views

minimum eigenvalue for difference of two matrices

Let $A$ a symmetric positive definite matrix, and $B$ a matrix constructed from $A$ by setting all its off-diagonal elements to zero. Then is there a way to see for which values of positive scalars ...
0
votes
0answers
14 views

Asymptotical stability of a discrete dynamical system

There is a linear time invariant discrete system, \begin{align} x_{k+1}&=\tilde{A}x_k, \end{align} where $\tilde{A}$ is a block matrix represented by \begin{align} \tilde{A}= ...
1
vote
0answers
13 views

Determine mutual location of two coordinate systems, given two sets of points

My problem is: we've got tracking device and a robot. Tracking device provides set of $n$ points in cartesian coordinates(taken from marker on robot arm) and robot driver returns position of TCP(tool ...
1
vote
0answers
19 views

Negative definiteness of a block matrix

There is a block matrix, \begin{align} M=\left(\begin{array}{cc} A & B\\ C & I \end{array}\right)\quad\text{where}\quad A<0 \end{align} I am curious whether the matrix $M$ is negative ...
0
votes
0answers
19 views

Eigenvalues of integrals over similar matrices

Let $\rho = \rho(x)$ be a $2\times2$ matrix (don't know if it is necessary, but $\rho$ is a density operator) and $I$ be the (two-dimensional) identity matrix. I have two matrices $A$ and $B$, where ...
0
votes
1answer
15 views

Derivative of vector and vector transpose product

I saw this answer here : Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$. I am finding difficult to understand the part in red. What rule is that ? If I apply multiplication rule, ...
0
votes
0answers
15 views

Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? [duplicate]

Question: Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? Someone comes along and gives us the partition $P=\{2,2,3,3,4\}$ of $14$. How can we ...
0
votes
1answer
16 views

Characterising Adjugate(adjoint) of a matrix

If $A$ is an $n\times n$ matrix over a field, then adj$(A)$ is an $n\times n$ matrix (obtained from $A$) such that $$\mathrm{adj}(A)\,A=A\,\mathrm{adj}(A)=\mathrm{det}(A)I_n.$$ Question: If $B$ is ...
0
votes
1answer
49 views

Linear Algebra: Question about determinants

The following matrices are $4 \times 4$ matrices. $$A=\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\\ 1 & 1& 1 &0\\ 1 &1 &0 &0 \end{bmatrix}\\ B= ...
0
votes
1answer
14 views

Sylvester's law of inertia for generic matrices.

By Sylvester's law of inertia, the positive and negative indices of a symmetric matrix $A$ are also the number of positive and negative eigenvalues of $A$. I was wondering if a similar result is known ...
0
votes
2answers
33 views

Eigenvalues of $\mathbb E\pmatrix{2X&X\\ 1-X&2X}$. [on hold]

Let $X$ be a random variable between $0$ and $1$, such that: $\mathbb{E}[X]=\frac{1}{2}$. We have a matrix: $$A=\left( \begin{array}{cc} 2X & X \\ 1-X & 2X \\ \end{array} \right)$$ ...
0
votes
3answers
39 views

Inverse of partitioned matrices [on hold]

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
2
votes
0answers
57 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
1
vote
1answer
33 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
2
votes
0answers
15 views

show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
2
votes
2answers
43 views

Is this a circulant matrix?

It's symmetric, but I'm not sure whether it is circulant. In a question that I had asked on MSE a couple of weeks ago, several commenters had said that this is a circulant matrix, and to study the ...
2
votes
3answers
54 views

Producing lower bounds for $\text{trace}(A^2)$ for a positive semidefinite, symmetric matrix $A$

Are there any lower bounds on $\DeclareMathOperator{trace}{trace}$ \begin{align*} \trace(A^2), \end{align*} where $A$ is positive semi-definite and symmetric? I am aware of the inequality $$ ...
-1
votes
0answers
31 views

Find the solution to the following LPP by solving its dual. [on hold]

Minimize : $ Z = 300X_1 + 110X_2$ Subject to : \begin{align*} 30X_1 + 5X_2 &\geq 6 \\ 20X_1 + 10X_2 &\geq 8 \\ X_1, X_2 &\geq 0 \end{align*}
0
votes
0answers
17 views

How to multiply the elements within a vector using matrix operations (e.g., dot product)?

Suppose a vector $\vec{v}^T=(v_1, v_2, \ldots, v_n)^T$. To sum the elements within the vector, I can use the dot product with a column vector of ones, $\sum_i v_i = \vec{v}^T \cdot \vec{1}$. My ...
1
vote
0answers
22 views

how to understand a matrix with order $O(n^{-1})$

I am reading a paper in which an assumption is that a matrix (for example $A_{n\times n}$) is $O(n^{-1})$. I have difficulty to understand that assumption. Does that mean the norm of the matrix is ...
0
votes
1answer
40 views

Mean value for $\tiny\left( \begin{array}{cc} X & X \\ -X & 1-X \\ \end{array} \right)$

We have a matrix $$\left( \begin{array}{cc} X & X \\ -X & 1-X \\ \end{array} \right)$$ where $X$ is a random variable between $0$ and $1$. I heard about "random matrices". Is it an ...
2
votes
1answer
20 views

properties of the solution to a non-homogeneous matrix equation with a non-singular M-matrix

I have a matrix equation $Ax=b$, where $A$ is a $4\times4$ non-singular M-matrix ($A$ has negative off-diagonal and positive diagonal entries) and $b$ is a strictly positive vector. Let $x=(x_1, x_2, ...
1
vote
1answer
37 views

Finding the Jordan form and basis failing

Let $$A = \left(\begin{array}{cccc} 3&4&-1\\0&-2&0\\1&-4&1 \end{array}\right)$$ Find the Jordan form $J$ and $P$ such that $P^{-1}AP = J$. So here's what I did: $f_A(x) = ...
4
votes
3answers
54 views

$A$ is a symmetric postivie definite matrix. Prove that $A^k$ is also a positive deinite

Let $A\in M_n(\mathbb{R})$, a symmetric positive-definite matrix. Prove that for every $k\in\mathbb{N}$, $A^k$ is also positive definite. So since $A\in M_n(\mathbb{R})$ is symmetric and positive ...
2
votes
1answer
34 views

$\left\| A \right\| \le \varepsilon \Rightarrow \left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon$

Suppose $A \in {C^{n \times n}}$ $\left\| A \right\| \le \varepsilon$ such that $\left\| . \right\|$ is matrix norm subordinate to the euclidean vector norm. Is this true that $\left\| {\mathop ...
2
votes
1answer
14 views

Commutative Monoid - matrix set

Let $M$={$\begin{bmatrix} a & b & c \\ c & a & b \\ b & c & a \end{bmatrix}|a,b,c\in \mathbb{R}, a+b+c=0$}. The matrices in $M$ are a special kind of Toeplitz matrices ...
2
votes
1answer
38 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
0
votes
1answer
23 views

Analytical result for element-wise vector division?

I have two vectors $$a=[a_1,a_2,...,a_n], b=[b_1,b_2,...,b_n]$$ Is it possible to express the result $$c=[a_1/b_1,a_2/b_2,...,a_n/b_n]$$ by some standard matrix operations such as matrix ...
0
votes
1answer
53 views

Which matrices diagonalizes a diagonal matrix? [on hold]

I think the answer is the set of all diagonal matrices but I am not sure. Can anyone give the answer with a proof?
2
votes
4answers
145 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
1
vote
2answers
36 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
0
votes
1answer
13 views

Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...
3
votes
2answers
36 views

For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
0
votes
1answer
25 views

how to calculate the projection of a vector onto a closed convex set? [duplicate]

suppose we have a vector $x \in \mathbb{R}^{n}$, and a closed convex set $C \in \mathbb{R}^{n}$. $C =\{x|Ax=b\}$ how to calculate the vector $y \in \mathbb{R}^{n}$, which is the projection of $x$ ...
1
vote
1answer
27 views

Proving space of skew-symmetric matrices is orthogonal complement of symmetric matrices

Problem: Prove that $\left\{ A \in \mathbb{R}^{n \times n} \mid A \text{ is symmetric}\right\}^{\bot} = \left\{ A \in \mathbb{R}^{n \times n} \mid A \ \text{is skew-symmetric}\right\}$ with $\langle ...
1
vote
1answer
28 views

Matrix Properties Problem

If $A\in M(n\times n;R)$ and $K= \dfrac {A+A^T}{2} $ and $L= \dfrac{A-A^T}{2}$. Prove: i) that $K$ and $-L$ are symmetric ii) that $K+L=A$ iii) that $K$ and $L$ are unique matrices with the properties ...
0
votes
0answers
18 views

Notation sumation confusion

I am reading paper about additive schwarz preconditioner, where following notation is used in order to obtain matrix C $$C_i = \sum_k (I^k B^k (P^k u_i)R^k)$$ . My question is, what's dimension of ...