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)

4
votes
2answers
221 views

If $A$ is a diagonalizable $n\times n$ matrix for which the eigenvalues are $0$ and $1$, then $A^2=A$.

If $A$ is a diagonalizable $n\times n$ matrix for which the eigenvalues are $0$ and $1$, then $A^2=A$. I know how to prove this in the opposite direction, however I can't seem to find a way prove ...
0
votes
1answer
79 views

Hermitian matrices [duplicate]

Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such that $\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H. How to prove ...
1
vote
1answer
48 views

solving linear matrix equation problem

Given two matrices $X$ and $Y$, with $Y$ invertible. Suppose that $$X=YZY^{-1}.$$ so $$Z=Y^{-1}XY.$$ In what order should I do the corresponding matrix multiplications to compute $Z$ ? Thanks.
4
votes
3answers
135 views

number of 8 x 8 matrices with specific conditions

How can we find the number of 8 by 8 matrices in which each entry is 0 or 1. In addition each row and each column contains odd number of 1's. Thanks for help.
1
vote
2answers
209 views

Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line

Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line $x_2=2x_1$ followed by reflection through the line $x_1=3x_2$ I am ...
0
votes
1answer
38 views

how to multiply particular type of matrix

Suppose you have $B =\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}$ , and $A =\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$ . I want a resulting matrix $C =\begin{bmatrix} 1 & 2 ...
0
votes
1answer
330 views

Counterexample for linear algebraic equations AX=Y

Say you have $k$ linear algebraic equations in $n$ variables; in matrix form we write $AX=Y$. Give a proof or a counterexample for each of the following: a) If $n=k$ there is always at most one ...
2
votes
1answer
64 views

Matrix of an invertible operator is also invertible -proof

Let $V$ be a finite dimensional vector space over an arbitrary field $F$ and let $S$ be an ordered basis of $V$. Let $A$ be an operator from $V$ to $V$. How can we prove that the matrix of $A$ with ...
5
votes
1answer
239 views

Is $AB+BA$ positive definite too if $A$ and $B$ are positive definite?

I have a question: Is $AB+BA$ positive definite too if $A$ and $B$ are positive definite matrices?
1
vote
0answers
38 views

Transformation to swap entries in a matrix

I have a 4x4 transformation matrix $$\begin{bmatrix} i_x & j_x & k_x & t_x \\ i_y & j_y & k_y & t_y \\ i_z & j_z & k_z & t_z \\ 0 & 0 & 0 & 1 ...
1
vote
1answer
202 views

Number of 4 × 4 Matrices Having Odd Determinants

How many 4 × 4 matrices with entries from {0, 1} have odd determinant? I was trying to partition the matrix as four block matrices with size 2 × 2, and consider all combinations of block matrices with ...
2
votes
2answers
220 views

Determine all $2\times2$ matrices $A$ such that $A^2=0$.

This is from Lang's introduction to Linear Algebra page no 61. Determine all $2\times 2$ matrices $A$ such that $A^2 = 0$. Let $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ ...
1
vote
1answer
61 views

Further explanation needed for this first order system of linear equations which is as follows:

I was trying the following problem which was as follows: Consider the first order system of linear equations: $\frac{dX}{dt}=AX; \space A=\begin{pmatrix} 3 &2 \\ -2&-1 ...
0
votes
1answer
119 views

Explanation on step $\rho$ of the SHA-3 algorithm

I'm working on implementing SHA-3 in a PIC microcontroller. In the block permutation, I don't quite understand step $\rho$: Bitwise rotate each of the 25 words by a different triangular number 0, ...
5
votes
3answers
3k views

Prove the following using induction on n (matrices)

Prove the following using induction on n: $$\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}^{n} = \begin{pmatrix} n+1 & n \\ -n & -n+1 \end{pmatrix}$$ I know that multiplication of ...
1
vote
1answer
53 views

Rearragning matrix equation to find a matrix with multiple occurences

Given the equation: $T+TD+TR=Y$ I need to solve this for $T.$ I have dealt with more basic matrix equations and using the inverse to re-arrange formulas, but since there is an addition of the ...
1
vote
1answer
77 views

If $f(X) = a_0 + a_1 X + a_2 X^2 \in \mathbb{F}[X]$ then show $f$ is uniquely determined by $f(x)$, $f(y)$, $f(z)$?

This is the exact question: It's part(ii) that I don't understand - what does it mean and what is it asking me to do? How would I go about constructing a proof? Any help would be much appreciated.
1
vote
1answer
71 views

Characteristic polynomial question with Rank

Q: Suppose that $A$ is a $5x5$ matrix with characteristic polynomial $x^3(x-8)(x-3)$ a) Explain why Rank$(A)$ must be either $2, 3,$ or $ 4$? b) Suppose that Rank$(A)$ = $3$, is $A$ diagonalizable? ...
3
votes
3answers
264 views

Prove that if $\mathrm{rank}(A) < n$ then $\det(A) = 0$?

If $A$ is an $n \times n$ matrix with $\DeclareMathOperator{\rank}{rank}$ $\rank(A) < n$, then I need to show that $\det(A) = 0$. Now I understand why this is - if $\rank(A) < n$ then when ...
1
vote
1answer
21 views

Prove that $(I-ix)^{-1}(I+ix)(I-ix)(I+ix)^{-1} = I$

I have a question Prove $$(I-ix)^{-1}(I+ix)(I-ix)(I+ix)^{-1} = I$$ with $x$ being a $n \times n$ matrix.
7
votes
3answers
1k views

$I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible [duplicate]

assume $A,B\in M_n(F)$ if $I-AB$ be invertible then how to prove $I-BA$ is invertible and how find inverse of $I-BA$ Thanks in advance
4
votes
2answers
818 views

how to prove following matrix is invertible? [duplicate]

how to prove A is invertible or $\ detA\neq 0$ $$A=\begin{pmatrix} \frac11 & \frac12 & \frac13 & \cdots & \frac1n \\ \frac12 & \frac13 & \frac14 & \cdots & ...
3
votes
2answers
114 views

One interesting question from linear algebra

I got one interesting question from matrix theory. I tried but I am not finding any clue to solve this question. I need help and suggestions. Thanks in advance. Let $A$ be a real $n\times n$ matrix. ...
3
votes
2answers
99 views

How to solve this determinant?

I have to solve determinant of the following form: $$a_{ij}=|i-j|+1$$ It looks like this: $$ \begin{pmatrix} 1 & 2 & 3 & 4 & \cdots & n \\ 2 & 1 & 2 & 3 & ...
2
votes
0answers
29 views

how the number of steps needed depends on the number of nodes and depends on the transmission range?

I run the consensus algorithm, and for each round k, we record the norm of the disagreement vector(|(|δ(k)|)|>〖10〗^(-6)). We stop, at a predefined value|(|δ(k)|)|>〖10〗^(-6) and we call this ...
1
vote
1answer
58 views

Finding Pi variables from matrix. From PageRank Algorithm.

$$\pmatrix{\pi_1 & \pi_2 & \pi_3} = \pmatrix{\pi_1 & \pi_2 & \pi_3}\pmatrix{\frac{1}{6} & \frac{4}{6} & \frac{1}{6} \\ \frac{5}{12} & \frac{2}{12} & \frac{5}{12} \\ ...
1
vote
1answer
47 views

How to do QR Factorisation of a matrix

given A = $\begin{bmatrix}1 & 0 & 3\\2 & -6 & 3\\ -2 & 3 & -3\end{bmatrix}$ How would i find the QR factorisation? Well i have a guide on how to do this and have attempted ...
0
votes
1answer
101 views

real spectrum of an almost symmetric stochastic matrix

Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each ...
1
vote
1answer
31 views

Rearranging Matricies

I am little confused about re-arranging matrix equations... So I know you cannot rearrange multiplication of matricies like you would normally with algebra as you cannot divide, but can you still do ...
1
vote
2answers
130 views

Adjacency matrix

Let $G$ be any simple and undirected graph. Let $A$ be the adjacency matrix of $G$. 1) Let $B$ be the number $\tfrac16\mathrm{tr}(A^3)$. What does $B$ count? That is $B$ counts the number of....? 2) ...
0
votes
0answers
121 views

Relation between the block inverse and the inverseof the matrix itself?

I have been trying to solve the relation between the block inverse and the inverse of the matrix itself. Hopefully I can get some insights here. Consider the following vector x consists of the two ...
1
vote
1answer
141 views

What happens to this infinite sum?

Assume $W$ is $n$ by $n$ matrix and $r<1$ is a real number. Let $$Q = \sum_{i=0}^{\infty} (rW)^i=[I_n-rW]^{-1}$$ Now assume that the $i$'th row of matrix $W$ is multiplied by a constant real ...
1
vote
1answer
47 views

Decomposition of a single 4D rotation

I have a $4\times 4$ matrix $M$ which represents a general 4-dimensional rotation. $$ M = \pmatrix{a_{11} &a_{12} &a_{13} &a_{14}\\a_{21} &a_{22} &a_{23} &a_{24}\\a_{31} ...
0
votes
1answer
36 views

Calculate total number of matrices of all orders which contain $2013$ elements

Calculate total number of matrices of all orders which contain $2013$ elements My Try:: By Simple Guessing wecan say that there are two matrices of order $(1\times 2013)$ and $(2013 \times 1)$ ...
1
vote
2answers
853 views

Show that the identity matrix $I$ must have norm $1$.

I am trying to understand why the identity matrix $I$ must have a norm $1$, for any choice of matrix-norm $|\cdot|$? How would i show this?
0
votes
2answers
830 views

Inverse of an matrix exponential

Consider the matrix exponential $$ e^{At} = \frac{1}{4} \begin{bmatrix} -e^{-t} + 5e^{3t} & e^{-t} - e^{3t} \\ -5e^{-t} + 5e^{3t} & 5e^{-t} - e^{3t} \end{bmatrix} $$ And $$ ...
3
votes
1answer
159 views

Norm inequality involving matrices

Let $A$ and $B$ be two definite positive symmetric $n \times n$ matrices. Prove or disprove that $$ \Vert AB - B^{-1} A^{-1} \Vert \geq \Vert AB - I \Vert $$ where $\Vert . \Vert$ is the Frobenius ...
1
vote
1answer
128 views

What's this matrix called?

In an inner product space, $v_1,\dotsc,v_n$ are linear independent iff the matrix $A_{ij} := \langle x_i | x_j \rangle$ is invertible. What's the name of this matrix??
1
vote
1answer
107 views

Adjugate matrix product

I have some problems understanding the proof of the Caley-Hamilton theorem (saying that a matrix the root of ith characteristic polynomial), namely: Why $A \cdot A^D = A^D \cdot A = \det A \cdot I$ ? ...
0
votes
0answers
140 views

hermitian matrices, pauli matrices

These matrices are the Pauli matrices \begin{align} A_1 & = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right] \\ A_2 & = \left[\begin{array}{cc} 0 & -i \\ i & 0 ...
4
votes
4answers
819 views

Can you raise a Matrix to a non integer number? [duplicate]

So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
0
votes
1answer
52 views

About the spectral radius of a kind of matrices

Consider a square matrix $A=DS$ where $S$ is symmetric with diagonal entries being $0$ and $D$ is a diagonal matrix for normalizing $S$'s row sums so that $Ae=e$ where $e$ is a vector with all entries ...
2
votes
1answer
76 views

Condition of an eigenvector problem #2

This one of the problem, where the only thing I can do is ask for help. Let $A$ be a diagonalisable matrix, $\lambda_i\in\mathbb{R}$ a simple eigenvalue of $A$, and $B$ any matrix. Show that, for ...
5
votes
2answers
317 views

Determinant inequality about positive definite matrices.

Assume $A \in M_n(\Bbb{R})$ (not necessarily symmetric), and for $\forall \alpha\not=0$, $\alpha^TA\alpha>0$. Show that $$\det\left(\frac{A+A^T}{2}\right)\leq \det A.$$
2
votes
1answer
35 views

Delocalization of eigenvectors in Expanding Graphs

Given an adjacency matrix A, can we say something about whether the eigenvectors corresponding to its highest(or second-highest) eigenvalues are delocalized ? By delocalization I mean that every ...
1
vote
2answers
98 views

Prove that $A^*A + I$ is invertible

Let $A$ be an $m\times n$ matrix. Prove that $A^*A + I$ is invertible. I'm not sure what to do because everything I try ends up at $A^*A$ again and not $AA^*$.
0
votes
0answers
87 views

Matrix norm applications?

There'are many different ways to calculate Matrix norm. But once calculated, what is the practical use/application of it (e.g. in computer programming)? Or does it let define something that can be ...
2
votes
1answer
128 views

How to Solve Bilinear Matrix Equation?

How should I solve this matrix equation? What is the solution for $X$? \begin{equation} BXC +B^TXC^T=D \end{equation}
1
vote
1answer
70 views

Prove or disprove: the spectral radius of a matrix with negative entries and row sums as 1 is larger than 1

We all know that the spectral radius of a stochastic matrix is $1$. But how's the "negative" proposition: For a matrix $M=(m_{ij})_{n\times n}$, if there exists $m_{ij}<0$ for some $1\le i,j\le n$ ...
1
vote
1answer
78 views

Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$

SEE AUTHOR'S ANSWER BELOW So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...