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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0
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1answer
91 views

Gaussian elimination with partial pivoting with dominant diagonal

Why is it that there is no need for row replacement when performing Gaussian elimination with partial pivoting on a matrics with a dominant diagonal?
2
votes
3answers
2k views

find all values of k for which A is invertible

$\begin{bmatrix} k &k &0 \\ k^2 &2 &k \\ 0& k & k \end{bmatrix}$ what I did is find the det first: $$\det= k(2k-k^2)-k(k^3-0)-0(k^3 -0)=2k^2-k^3-k^4$$ when $det = 0$ ...
1
vote
0answers
77 views

Measure of closeness of a matrix to triangular form

Given a square $n\times n$ matrix $A$, I want to develop a measure of how close the matrix $A$ is to a triangular form.
4
votes
3answers
96 views

Matrix inequality for square matrices

Does the following hold for any square matrix $A$, $(AA^*)^{1/2}\geq (A+A^*)/2$, where superscript $*$ denotes the Hermitian transpose. Proof/any comment would be appreciated.
1
vote
1answer
38 views

Is this matrix-vector equation with given properties always solvable?

Consider a standard n-dimensional matrix-vector equation with a square matrix, $A\textbf{x} = \textbf{b}$ The matrix $A$ is not precisely known in advance, but it is known to have the following ...
6
votes
1answer
2k views

How to prove that the inverse of a matrix is unique?

The ring of matrix is not an integral domain. How to prove that the inverse is unique?
0
votes
1answer
45 views

Prove a statement for the infinite matrix

We are given infinite two dimensional matrix $\{a_{i,j}\}_{i,j=1}^\infty$. And we know that matrix contain only natural values and each number appears in the matrix exactly 8 times. Task is to prove ...
1
vote
1answer
127 views

Proof about weakly elementary matrices;

A square matrix is said to be $weakly \ elementary$ if it is either elementary or it is obtained from an identity matrix by replacing one diagonal entry by zero. Prove that every square matrix is ...
1
vote
0answers
26 views

Algorithm for finding only the $k$-th singular vectors

I know that we have truncated SVD that can compute the first say $k$ largest singular values (and corresponding singular vectors). However, I'd like to know if there is an algorithm that can find only ...
1
vote
0answers
403 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
0
votes
1answer
609 views

Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues - Proof Strategy [Lay P397 Thm 3]

Herein, I denote the Hermitian conjugate by * (ie: $A* = \bar{A}^T) $. Let $v_i$ and $v_j$ be two eigenvectors of an Hermitian matrix H. First of all suppose that their respective eigenvalues i and j ...
5
votes
3answers
1k views

Reason for reversing the order when transpose and inverse of a group of matrices

Whenever there is a transpose or inverse of a group of matrices, I just reverse their order. For eg: $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$ and $(ABC)^{T} = C^{T}B^{T}A^{T}$ But usually, I am taking this ...
7
votes
1answer
328 views

Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$ a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right). $$ The first ten terms are: $0.75$ ...
2
votes
0answers
63 views

Best way to solve specific block-tridiagonal linear system (10000x10000 and larger)

To provide more context, this system came from energy balance equation on a mesh with (n,m) nodes in each direction. It's a linear system that looks like this (size of system in blocks n = 4, size of ...
7
votes
2answers
127 views

How to find determinant of this matrix?

Is there a manual method to find $\det\left(XY^{-1}\right)$ ? Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 1 & 3 & 3^2 & \cdots & 3^{2012} \\ ...
1
vote
1answer
52 views

Prove that $ A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$

I'd like to get some help So I need to prove that when $A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$ Linear Algebra, of course. Thanks
2
votes
1answer
149 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
0
votes
1answer
63 views

Induced Matrix Norm

I have trouble following a proof of the induced Norm $||\cdot||_1$ The proof can be found here: ...
3
votes
1answer
141 views

Rank one plus diagonal matrix approximation

Given $A \in R^{n \times n}$, $A$ symmetric. I'm trying to solve the following minimization problem: $\underset{u \in R^n, d \in R^n} \min \, \frac{1}{2} \|X - A\|_F^2$ subject to $X = u u^T + ...
1
vote
0answers
44 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
3
votes
3answers
48 views

Performing matrix chain multiplication by hand

I'm trying to gain intuition for writing a matrix chain multiplication algorithm by working through a few problems by hand. I see plenty of worked-through solutions on sets of three or four solutions, ...
4
votes
2answers
124 views

Product of positive matrices

I have two positive-definite matrices $A$ and $B$ (not necessarily symmetric), and we have $AB=BA$, is there any theorem that ensures that the product of $A$ and $B$, $AB$ is positive definite? Or ...
2
votes
1answer
99 views

Least squares problem with orthonormality constraints

Given $y_1,\ldots,y_n\in \mathbb{R}$,$w\in \mathbb{R}^d$, and $x_1,\ldots x_n\in \mathbb{R}^D$, how do we solve the following optimization problem \begin{align} \min_A \sum_{i=1}^n (y_i-w^TA^Tx_i)^2\\ ...
1
vote
2answers
51 views

Prove that if $B=P^{-1}AP$, then $q(B)=P^{-1}q(A)P$

Is it possible to prove this using a similarity invariant? For example showing that $$\det(q(B))=\det(q(A))$$
0
votes
1answer
62 views

properties of left-invertible matrix

When reading the notes on Left-invertible matrix, where $A$ is a matrix of dimension of $m\times n$, and $X A=I$. It is claimed that it $m$ must be larger than $n$, and $rank(A)=n$.How to get these ...
1
vote
1answer
303 views

how to find matrix A from complete solution to Ax=b

I am trying to solve a problem. I was stuck.Any help is appreciated. The complete solution to $Ax=\left[\begin{array}[c]{rr}1 \\3 \end{array}\right]$ is $ x= ...
1
vote
2answers
336 views

Inverse of a sum of positive definite matrices

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ ...
1
vote
1answer
44 views

get an element by finitely generated set

I want to know the method to get a element in a finitely generated group by its generated set, is there a general way to calculate? For example, $SL(2,\mathbb{Z})=<a,b|a=\begin{pmatrix}0 &1\\-1 ...
1
vote
1answer
30 views

the differences and relationship between linear independent and affinely independent

When learning optimization, I heard the two related concepts on linear algebra: linearly independent and affinely independent. ...
0
votes
2answers
412 views

Gauss Elimination - Diagonal dominant matrices don't need row changes

I was asked to prove the following statement: let $A$ be an $n$ by $n$ matrix with real entries such that $\forall k \in \mathbb N, k\leq n$: $$\sum_{i \neq k} |A_{i,k}| < |A_{kk}|$$ Show that if ...
2
votes
1answer
486 views

Sum of principal minors

Is there any formula for the sum of principal minors? (note: $i^{th}$ principal minor which results from omitting the $i^{th}$ row and $i^{th}$ column)
1
vote
0answers
93 views

Calculating MSE for two different size matrixes

I have two $2$-column matrixes, one of the has $467$ rows while the other one has $61468$ rows. Both them are trajectory paths of same robot, the big matrix is kind of raw data and the smaller one is ...
3
votes
1answer
88 views

Find rank of the matrix $a_{ij}=(i-j)^2$, $i,j=1,\dots, n$

Let is $A$ $n\times n$ matrix defined in following way $a_{ij} = (i-j)^2$. For example when $n=4$ $$ A= \begin{pmatrix} 0&1&4&9\\ 1&0&1&4\\ 4&1&0&1\\ ...
3
votes
1answer
51 views

Diagonalisability…without the characteristic polynomial

Let us consider an $n\times n$ matrix $A$ defined as follows $$ A=\begin{pmatrix} 1+a&1&\cdots &1\\ 1&1+a&\ddots&\vdots\\ \vdots&\ddots&\ddots&1\\ ...
2
votes
1answer
42 views

Showing the existence of an eigenvalue whose real part is positive

$$M = \left(\begin{array}{cc|cc|cc|cc|cc} -b_1 &0 &b_2 &0 &0 &0 &\ldots &\ldots &0 &0\\ 0 &-a_1 &0 &a_2 &0 &0 &\ldots &\ldots &0 ...
0
votes
1answer
38 views

Symmetric Matrix Algebra

This should be a straightforward answer but my matrix algebra skills are weak. If I have a symmetric matrix $X$, is $A^TX^{-1}A$ symmetric for any matrix $A$? I know the inverse of a symmetric ...
1
vote
2answers
2k views

How to test if a graph is fully connected and finding isolated graphs from an adjacency matrix

I have large sparse adjacency matrices that may or maybe not be fully connected. I would like to find out if a matrix is fully connected or not and if it is, which groups of nodes belong to a ...
13
votes
1answer
224 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
1
vote
1answer
68 views

How do I rewrite vectors in other basis' given change of coordinate matrices?

$\displaystyle β= \begin{bmatrix}2\\2\\\end{bmatrix}$,$\displaystyle \begin{bmatrix}4\\-1\\\end{bmatrix}$ $\displaystyle C= \begin{bmatrix}1\\3\\\end{bmatrix}$,$\displaystyle ...
0
votes
0answers
63 views

Matrix Induction proofs using $\rm mod$.

Please could someone advise me on where to even start with this question. Thanks in advance Let $A$ be an $n\times n$ matrix, for some $n\geqslant 1$. Given any prime number $p\gt1$, write $A_p$ for ...
1
vote
1answer
69 views

matrix representation of linear transformation

For a set $N$ let $id_N:N \rightarrow N$ be the identical transformation. Be $V:=\mathbb{R}[t]_{\le d}$. Determine the matrix representation $A:=M_B^A(id_V)$ of $id_V$ regarding to the basis ...
0
votes
0answers
58 views

Notation in Linear Algebra

What does $(A\mid b)$ denote in Linear Algebra? Specifically in the context of the following question: "If $(A\mid b)$ is in reduced row echelon form, prove that A is also in reduced row echelon ...
9
votes
2answers
136 views

How many Matrices exist with increasing row and increasing column condition?

Given $N$, I would like to know the number of matrix constructed from $1$ to $N$ which satisfies the following condition: 1. The each row entries should be in increasing order. 2. The each column ...
0
votes
1answer
417 views

Eigenvector when all terms in that column are zero?

so I have this matrix: $$ \begin{matrix} 0.7 & 0 & 0 \\ 0.1 & 0.6 & 0 \\ 0 & 0.2 & 0.8 \\ \end{matrix} $$ I managed to solve ...
0
votes
1answer
22 views

Multiplying matrices, need some clarification simple dot product

I understand that when you want to multiply two matrices that the number of rows in the left matrix have to be equal to the number of columns in the right, otherwise the result of the multiplication ...
3
votes
2answers
163 views

Eigenvectors of the Zero Matrix

Given the following matrix: $ \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix} $. I have to calculate the eigenvalues and eigenvectors for this matrix, and I have calculated that ...
1
vote
2answers
146 views

Find a matrix such that $Ax=0$

Let $$W = span\left\{ {\left( {\matrix{ 1 \cr 0 \cr 0 \cr 1 \cr } } \right),\left( {\matrix{ 0 \cr 2 \cr 1 \cr { - 1} \cr } } \right)} \right\}$$ I was asked ...
0
votes
3answers
110 views

What is the dot product of two or three vectors graphically or visually?

I don't understand what the dot product actually is. I understand when and where to use it, but when it comes to proving things with it, I don't really grasp what it actually is making it difficult ...
1
vote
0answers
39 views

Product of two matrices with simple spectrum

We are given two square matrices $A$ and $B$ of the same size over the field of complex numbers and $\epsilon > 0$. Then it can be shown that there exist non-singular (even diagonalizable) ...
0
votes
1answer
31 views

For how many values of $a,b,c\in(1,2\ldots,p-1)$ does $p$ | $({a^2}-bc)$ where $p$ is an odd prime number

In a mock test for an entrance exam I am preparing for came the following question: Let $p$ be an odd prime number and $T_p$ be the following set of matrices $$ T_p= \left( ...