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)

1
vote
1answer
47 views

If $\vec{v}$ is an eigenvector of $A$, then also $B\vec{v}$, when $AB = BA$ [duplicate]

I have the following problem: Let $V$ be a finite dimensional vector space. Let $A$, $B$ be linear maps of $V$ into itself. Assume that $AB = BA$. Show that if $\vec{v}$ is an eigenvector of $A$, ...
0
votes
1answer
34 views

Formula for transpose matrix multiplication

I ran by an exercise in my textbook that got me writing sheets of paper of matrix multiplications, and I got a lot of ($ij$ entry)$^2$, but that was not enough for me to construct a convincing ...
2
votes
4answers
10k views

Help demystify the Navy PFA equations.

I need help finding an equation that the Navy's Physical Readiness Program Office (PRIMS) keeps unpublished for some unexplainable reason and will not share after numorous requests. Anyways, luckily, ...
1
vote
1answer
53 views

What is the spectrum of this matrix?

$$A_n=\begin{bmatrix} 1 & 1 & 1 & \cdots & 1 & 1\\ 1 & 2 & 2 & \cdots & 2 & 2\\ 1 & 2 & 3 & \cdots & 3 & 3\\ \vdots & \vdots & ...
1
vote
3answers
109 views

Show that matrices are not similar

I have to show that the following matrices are not similar: $$A = \left[\begin{matrix} 1 & 3 & -3 \\ -3 & 7 & -3 \\ -6 & 6 & -2\end{matrix}\right]$$ and $$A' = ...
0
votes
1answer
43 views

Hessian Matrix and critical point

Consider $f(x,y,z)\in C^2$. Suppose that $(0,0,0)$ is a critical point of $f$ and the Hessian Matrix of $f$ in $(0,0,0)$ is given by $\left(\begin{array}{ccc} 1 & 0 & \pi\\ 0 & \omega ...
1
vote
2answers
55 views

question in linear algebra on Hermitian matrices

Hello this indeed a very short question from Algebra that I have no real idea on and figured it is simple but for some reason I cannot seem to find it. I am given $A$ and $B$ complex square matrices ...
6
votes
2answers
9k views

Column Vectors orthogonal implies Row Vectors also orthogonal?

If the column vectors of a matrix $A$ are all orthogonal and $A$ is a square matrix, can I say that the row vectors of matrix $A$ are also orthogonal to each other? From the equation $Q \cdot ...
0
votes
0answers
28 views

Matrix of node's ranking in a graph

In any graph of n nodes in any dimension, define matrix $M_r$ of ranking as $\forall r_{ij}\in M_r$, $r_{ij}$ is the ranking of j to i. That is, j is the $r_{ij}$th nearest node to i. Therefore, any ...
0
votes
0answers
23 views

matrix input convergence

I'm hesitant to ask this here as I'm relatively new to matrix mathematics, so please go easy on me. I have an input vector of size n to a matrix of size ...
3
votes
1answer
57 views

Difficulties understanding these statements about change of basis

I understood more or less what a change of basis matrix is and how I can use it to pass to one coordinate system to another. Basically, a change of basis matrix is a matrix whose columns are the ...
0
votes
2answers
775 views

Find the value of $k$ for which matrix is diagonalizable

Consider the matrix $$A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & k \\ 0 & 0 & 2 \\ \end{bmatrix}$$ where $k$ is a real number. The ...
0
votes
5answers
220 views

$AB-BA=A$, Then A is singular? [duplicate]

Title is the question, I tried taking trace both side and got trace of $A$ is zero, now to conclude $A$ is singular, suppose $A$ is non singular, then multiplying both side by Inverse of $A$ we get ...
1
vote
1answer
46 views

Determinants using elementary row operations

Let matrix $A$ be defined as \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ ...
0
votes
0answers
58 views

Which classes of matrices contain $A$ and which contain $B$?

14. In the list below, which classes of matrices contain $A$ and which contain $B$? $$ A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 ...
23
votes
3answers
1k views

Are $10\times 10$ matrices spanned by powers of a single matrix?

I don't know how to answer this question: Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the ...
0
votes
2answers
33 views

Orthogonal Transformations and Eigenvalues

True or false: There exists and orthogonal matrix $T$ that has 2 as an eigenvalue. I think this is false, but I do not know how to prove it
0
votes
3answers
56 views

Diagonizable matrix

Got this matrix: \begin{bmatrix} 1 & 2 \\ -2 & 5 \end{bmatrix} I should determine if the matrix is diagonalizable or not. I found the eigenvalues ( only one) = 3. My eigenvector is then ...
0
votes
1answer
53 views

Magnitude of product of symmetric matrix and unit vector

If $A$ is any symmetric 2 by 2 matrix with eigenvalues -3 and 3 and $\vec{u}$ is a unit vector in $\mathbb{R}^2$, what is $||A\vec{u}||$? Any help would be appreciated, I haven't the slightest idea ...
2
votes
0answers
142 views

Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ ...
1
vote
1answer
39 views

What does it mean to take the power of a transition matrix? Or multiply it by a vector?

I understand the general idea that a matrix $M$ has some cell $M_{ij}$ that denotes the number of ways we can go from state $i$ to state $j$, but what does $(M^t)_{ij}$ represent? The number of ways ...
6
votes
2answers
193 views

Efficient way to compute $(A+D)^{-1}$ when $A^{-1}$ is known

I need to compute the inverse of a matrix sum $A+D$, where the inverse of $A\in\mathbb{R}^{n\times n}$ is known. The matrix $D\in\mathbb{R}^{n\times n}$ is a diagonal matrix which can be thought of as ...
2
votes
2answers
88 views

Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places

Let $a>0$. Let $A$ be the $n\times n$ matrix with $a+1$ on the diagonal and $a$ in all other entries. How can one compute $A^{-1}$ as a function of $n$?
2
votes
1answer
92 views

Is there a way to directly solve this matrix equation: $XAX^{T} = B$

$X^{T}$ is the transpose of $X$. $A$ is a $n$ x $n$ matrix and $B$ is a $m$ x $m$ matrix, $m$ > $n$, both of them are known, $A$ is positive definitive and $B$ is symmetric. I would like to find $X$. ...
0
votes
1answer
45 views

Properties of Positive Semidefinite Matrices.

I am looking to do some modeling involving matrices and a requirement is the matrices be positive semidefinite and complex. However, the modeling tool I'm using does not handle complex values well, so ...
1
vote
3answers
73 views

Book on linear algebra containing interesting problems

Could anyone suggest me a problem book on linear algebra that contains interesting problems on rank, nullity, nullspace, linear transformations, eigenvalues, eigenvectors and characteristic ...
3
votes
2answers
629 views

Determinant of matrix with trigonometric functions

Find the determinant of the following matrix: $$\begin{pmatrix}\cos\left(a_{1}-b_{1}\right) & \cos\left(a_{1}-b_{2}\right) & \cos\left(a_{1}-b_{3}\right)\\ \cos\left(a_{2}-b_{1}\right) ...
5
votes
3answers
160 views

Matrix P to the power of 4, i.e $P^4$, is this the same as $P^2 \cdot P^2$?

Basically, what it says in the title. I have a $5 \times 5$ matrix and I need to work out $P^4$, is it possible to just do $P^2$ and multiply this with itself?
3
votes
3answers
80 views

Matrix multiplication: What is $\mathbf A^3$ and $\mathbf A^n$?

Suppose there is matrix A. I know that A2 = A $\cdot $A But what if it is A3? Is it A $\cdot $A $\cdot$A OR A2 $\cdot$ A OR A $\cdot$ A2? So basically my question is what is An?
1
vote
0answers
68 views

Eigenvalues and eigenvectors of certain diagonal constant matrices

Suppose I have an infinite complex diagonal constant (Toeplitz) matrix, which is also Hermitian. This is given by finite number of complex parameters $z_1, z_2, \cdots, z_k$. If, $z_1$ is the ...
0
votes
1answer
208 views

Symmetric Permutation Matrix

I am trying to prove that an nxn permutation matrix P that is formed by switching two rows of an nxn identity matrix will always be symmetric. This is what I am trying to use thus far but I can't ...
0
votes
1answer
222 views

Show without expanding that the two determinants are equal

Let $$ A= \begin{bmatrix} 0 & a^2 & b^2 & c^2\\ a^2 & 0 & z^2 & y^2\\ b^2 & z^2 & 0 & x^2\\ c^2 & y^2 & x^2 & ...
0
votes
1answer
61 views

Diagonally dominant matrix

Assume $A$ is a positive definite matrix, and $B$ is a matrix with zero row sum. Does matrix $A$ exist such that $AB$ is strictly diagonally dominant?
3
votes
1answer
182 views

DE solution's uniqueness and convexity

I am lost and don't know how to prove the following: If $M$ is a positive definite symmetric square matrix and if $\overrightarrow {v}(t)$ is a solution of: $$\overrightarrow {v'}(t) = ...
5
votes
6answers
486 views

Is $A^2=I \implies A=\pm I$ necessarily true?

$A$ is $n\times n$ matrix. How to prove whether it is true or false $$A^2=I \implies A=\pm I$$ I was trying on $2\times 2$ case...multiplying general entries and then equating them to the identity ...
2
votes
0answers
58 views

Finding an orthogonal matrix with given absolute value

$\DeclareMathOperator{\Abs}{Abs}$Define the absolute value of a matrix $A = (a_{ij})$ by $$ \Abs(A) = \pmatrix{|a_{11}| & \cdots & |a_{1n}|\\ \vdots & \ddots & \vdots\\ |a_{n1}| & ...
1
vote
1answer
40 views

Relationship between the square of the Frobenius norm and the Frobenius norm of the square

I am looking to understand the following relationship: I have a matrix $A$, whose entries are all bounded by $0 \leq a_{i,j} \leq 1$, and follows the constraint $\|A\|_2 = 1$. Is there anything ...
0
votes
0answers
218 views

interpreting negative square root of a matrix

I am working on stability analysis of systems with impacts and in my algorithm, I have reached a state where I have $xPx = 1$, where $P$ is a positive semi definite matrix. now with my code, at a ...
0
votes
1answer
30 views
0
votes
1answer
62 views

Find eigenvalues of huge matrix

I have a huge matrix (7,000,000 x 7,000,000). It is singular sparse Laplacian matrix. I am looking for algorithm to find eigenvalues. I need to find the second eigenvalue (the first one is zero) ...
1
vote
0answers
128 views

Row degeneracy in systems of linear equations

I am trying to understand the concept of row degeneracy in a system of linear equations, but having trouble understanding this problem. \begin{align} x+2y+z &= 2 \tag{1} \\ 2x+y+3z &=5 ...
1
vote
0answers
14 views

Show that $A^{2m+1} + B^{2m+1} = (A+B)\prod_{j=1}^{m}{(A-x_jB)}{(A-\bar{x_j}B)}$

Suppose $m$ is a positive integer. Define a polynomial $x^{2m}+1$ and denote $x_0,x_1,...,x_{2m}$ as zeros of $x^{2m}+1$, where $x_{j+m}=\bar{x_j}$, $1 \leq j \leq m$. If matrices $A$ and $B$ ...
0
votes
1answer
53 views

Show that this map is well defined.

Consider the following map: $$SU(2) \to O_{3}^{+}(\mathbb{R})$$ $$h \mapsto (\langle i,j,k\rangle_{\mathbb{R}}\;\; \to \;\;\langle i,j,k\rangle_{\mathbb{R}} )$$ $$a \to h^{-1}ah$$ where $h$ is a ...
4
votes
1answer
116 views

How to prove that given set is a connected subset of the space of matrices?

Let $M$ be the space of all $m\times n$ matrices. And $C=\{X\in M|\operatorname{rank}(X)\leq k\}$ where $k\leq \min\{m,n\}$. Check whether the set $C$ is: Closed Connected Compact Open What are ...
16
votes
4answers
751 views

Avoiding the Cayley–Hamilton theorem [duplicate]

Every $n\times n$ matrix satisfies a polynomial equation of degree at most $n^2$, simply because the space of $n\times n$ matrices has dimension $n^2$. By the Cayley–Hamilton theorem, every matrix ...
1
vote
1answer
28 views

Figuring out the variables rows and colums for matrices

Let $P$ be a $2 \times 3$ matrix, $Q$ an $m \times 5$ matrix, and $R$ a $p \times q$ matrix. Find the values of $m$, $p$, and $q$ such that the operation $Q - PR$ is possible. So I figured that $p = ...
0
votes
0answers
61 views

inner product of left and right eigenvector of a matrix

Let $\mathbf{A}\in\mathrm{R}^{n\times n}$ be a matrix whose left and right normalized eigenvectors corresponding to $i^{th}$ eigenvalue are $\mathbf{u}_i$ and $\mathbf{v}_i$, respectively. Does ...
1
vote
2answers
32 views

Determine the rank of the linear map given by a $n \times n$ matrix , dependend on n. Proof by induction

The task: Let $$ A:= \begin{pmatrix} 1 & a & a & ... & a\\ a & 1 & a & ... &a \\ a & a & ... & a & a\\ ... & ... &... & 1 & a \\ a ...
0
votes
2answers
31 views

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible.

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible. I'm having trouble proving this, I tried saying that $|AB|=|A||B|=0 \implies |A|=0 \text{ or } |B|=0$ but that got ...
2
votes
1answer
10k views

What's the relationship between singular, nontrivial and linear dependent?

I understand that if a matrix is singular, it has no inverse. If it has nontrivial solutions, it means at least one solutions exists. If it is linearly dependent, it means that for $a_1 ...