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
31 views

Prove that $M$ is similar to a $n\times n$ real matrix

Let $M$ be a diagonalizable complex $n\times n$ matrix such that $M$ is similar to its complex conjugate $\overline{M}$. Prove that $M$ is similar to a $n\times n$ real matrix. My work: ...
1
vote
1answer
27 views

Why would a system of equations be solved this way?

If my instructions are to solve a system of equations using Gauss-Jordan elimination, and the matrix below is one of my final steps, why would I not be expected to fully row-reduce it to the second ...
1
vote
1answer
13 views

echelon form of matrices

I was wondering why the augmented matrix $M_1$ is in echelon form, while the augmented matrix $M_2$ is not in echelon form. $M_1 = \left[ {\begin{array}{cc} 1 & 2 & -3 & 2 & -4 & ...
2
votes
1answer
32 views

The orthogonal group $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$

I tried to do the following exercise: Prove that $\mathcal O_n (\mathbb K)$ is isomorphic to a subgroup of $\mathcal O_{n+1} (\mathbb K)$ The definition of $\mathcal O_n (\mathbb K)$ is ...
0
votes
5answers
115 views

Prove/disprove that $A^3+A=0$

Let $A$ be a square invertible matrix which its members are real numbers. Prove/disprove: There cannot be a matrix $A$ that satisfies: $A^3+A=0$ I did that: $$A^3=-A$$ $$A^{-1}A^3=-AA^{-1}$$ ...
0
votes
0answers
17 views

Solving the least squares problem for tridiagonal matrix

I'm looking for solution to solve the least squares problem for tridiagonal matrix in O(n), by using QR decomposition. Do you know any easy and readable solution or algorithm for this problem?
1
vote
1answer
20 views

Find unknown matrix in matrix equation

Given a matrix $A$ and a symmetric positive definite matrix $Y$, find a symmetric positive definite matrix $X$ which solves $$ X + AXA'+A^2X\left(A'\right)^2=Y $$ (This differs from the algebraic ...
1
vote
0answers
26 views

The relationship between singular values of two matrices

If $A,B\in \mathbb{R}^{m\times n}$ with $m\geq n$, assume singular values of $A$ are $\sigma_1\ge \sigma_2\ge \cdots\sigma_n;$ the singular values of $A+B$ are $\hat{\sigma}_1\ge \hat{\sigma}_2\ge ...
3
votes
2answers
22 views

Upper bound on trace of product of unitary and arbitrary matrix

Let the field be complex, $U$ be an $n\times n$ unitary matrix, $M$ be any $n\times n$ matrix, and $|M|$ denote the matrix formed by taking the absolute value of every entry of $M$. Edited Question: ...
1
vote
0answers
14 views

Eigenvectors of linear combinations of symmetric matrices

What can be said about the eigenvectors of the linear combination of several real symmetric independent matrices? I did find some papers about eigenvalues of sum of symmetric matrices, which are ...
1
vote
1answer
27 views

Define a matrix power by some scalars

Suppose I have an nxn matrix, for example: $$A=\begin{pmatrix}6&-2\\8&-2\end{pmatrix}$$ How is it possible to define the matrix $A^9$ using two scalars $b,c$ in R s.t.: $A^9 = bA + cI$ I ...
1
vote
0answers
37 views

Prove two matrices are similar (Frobenius normal form related)

Let $x^2+a_1x+a_0\in\mathbb{R}[x]$, irreducible polynomial. It's roots are $\alpha \pm \beta i$. Show that $A=\left( {\matrix{ 0 & { - {a_0}} \cr 1 & { - {a_1}} \cr } } \right)$ ...
2
votes
1answer
27 views

Prove $\sigma(\tau(I))=(\sigma\tau)(I)$

$I=(i_1,...,i_k)$ denotes an ordered $k$-tuple of indices. Given $\sigma\in S_k$, define $\sigma(I)=(i_{\sigma^{-1}(1)},...,i_{\sigma^{-1}(k)})$ Let $\sigma,\tau\in S_k$. Then ...
0
votes
0answers
18 views

Random Matrix Theory Noise

Hello and Merry (past) Christmas! I am new to random matrix theory, was reading an article about how to improve a correlation matrix (for portfolio optimization). And everywhere i see this "noise ...
0
votes
2answers
28 views

Non-singular matrices forms a group

Do the set of all non-singular matrices forms a group under multiplication. ? I don't think it does as the multiplication operator is not even defined for every two element in the set . Just need a ...
0
votes
1answer
16 views

Show $P(\sigma\tau)=P(\sigma)P(\tau)$

$\forall\sigma,\tau\in S_n$, $P(\sigma\tau)=P(\sigma)P(\tau)$ Definition: To each $\sigma\in S_n$ we may associate an $n\times n$ permutation matrix $P(\sigma)$ given by ...
4
votes
3answers
51 views

Assume that $A $ is an $n \times n$ symmetric positive-definite matrix.

Assume that $A$ is an $n\times n$ symmetric positive-definite matrix. Prove that: the element of $A$ with maximum magnitude must lie on the diagonal.
1
vote
1answer
31 views

Algebra of matrix coefficient over a compact group is isomorphic to its dual.

Let $K$ be a compact group. Then we have the following definition of matrix coefficient: Definition: $f: K \rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite dimensional ...
2
votes
4answers
123 views

If $A^4=I$ then $A$ must be diagonalizable?

Suppose we have a real matrix $A$ which satisfies $A^4=I$, can we determine if $A$ is diagonalizable? I believe the answer is that we can't because all we know about the matrix $A$ is that it is ...
1
vote
0answers
53 views

a unitary relation between a matrix and its transpose

Let $F$ be a field, $E/F$ is a quadratic extension. The notions "Hermitian matrix, unitary matrix" make sense for $E/F$. In fact, a matrix $A$ is called Hermitian if $A=\bar A^T$. Given a Hermitian ...
0
votes
1answer
49 views

Complex square matrices, proving there exists x,y in C^n such that A=xy*

Suppose A $\in$ $M_{nxn}$ the set of complex square matrices. Show the following statements are equivalent a) A has rank 1 b) $\exists$x,y $\in$ $C^n$ such that $A=xy^*$ What are the right and ...
0
votes
2answers
26 views

Determining the dimension of the set of solutions of the system $Ax=b$ without solving for $x$ or applying row reduction

Consider the matrix $A=\begin{bmatrix}2&2&2&4\\1&2&0&-1\\1&3&-1&-4\end{bmatrix}$ part d) What is the dimension of the solution space of the homogeneous ...
2
votes
1answer
51 views

Is it possible to have an $a \times b \times c$ matrix?

The book Artificial Intelligence: A Modern Approach states that a certain variable is a $2 \times 2 \times 2$ matrix", but I thought that matrices could only be rectangular (i.e. $a \times b$). Is it ...
2
votes
3answers
47 views

Whether the set of functions $(1,e^{x},e^{-x})$ linearly independent

Are the set of functions $(1,e^{x},e^{-x})$ linearly independent? I wrote it as an augmented matrix but it brought me to nowhere. Can somebody help me?
-1
votes
0answers
31 views

Standard matrix for reflection about the plane $x=z$ followed by shifting [on hold]

Determine the 3x3 standard matrix of a linear transformation that permutes and translates the components of a 3-dimensional vector as $v = (v_{x}, v_{y}, v_{z})$ $T(v) = (v_{z}+1, ...
1
vote
3answers
97 views

Prove/disprove that the matrix $AB+BA=0$

Let $A,B$ be matrices of an order $5\times 5$ such that: $$AB+BA=0$$ Prove/disprove: at least one of the two matrices is not invertible. I wanna say I tried various ways before I posted here but I ...
0
votes
2answers
31 views

Express the polynomial $ax^2+2hxy+2gx+2fy+by^2+c$ in matrix notation

I'm given $$\begin{bmatrix}x & y & 1\end{bmatrix}*M*\begin{bmatrix}x \\ y \\ 1\end{bmatrix}$$ where $M$ is the polynomial $ax^2+by^2+2hxy+2gx+2fy+c$ in matrix notation. Im totally stumped ...
0
votes
1answer
7 views

Number of bit operations in nxn zero-one matrix boolean product

I was reading transitivity closure from the book Discrete Mathematics and Its Application by Kenneth Rosen It says that in the boolean product of nxn zero-one matrix, there are $n^2(2n-1)$ bit ...
3
votes
1answer
56 views

Determining $\det(\mathbf{A})$ using the characteristic polynomial

Let the 3x3 matrix be $ \mathbf{A} = \begin {bmatrix} 3&1&0\\1&3&0\\0&0&1 \end {bmatrix}$. a) Determine its eigenvalues and eigenvectors. b) Do the eigenvectors ...
0
votes
0answers
24 views

A simple matrix proof [duplicate]

Let $A$ be a matrix of an order: $(n\times n)$, Prove/disprove: $AC=CA$ for every invertible matrix $C$ of an order $n\times n$ if and only if: $$ a_{ij} =\begin{cases} c \in R, & i=j \\ 0 ...
5
votes
1answer
34 views

what can be derived from similar matrix

If $A=\begin{pmatrix} 0&\star&\star \\ \star&x&\star \\ \star & \star & 5 \end{pmatrix}$ is similar to $B=\begin{pmatrix} 1&0&0 \\ 0&y&0 \\ 0 & 0 & 10 ...
0
votes
0answers
26 views

update cholesky factorization

I need to compute cholesky(H'*H) where H is a big sparse rectangular matrix. After that H is modified by adding several lines. That is Hn = [H ; line_1 ; ... ; line_n] in Matlab. How can I recompute ...
4
votes
0answers
33 views

An inequality concerning restricted isometry property

Let $A\in \mathbb{R}^{m\times n}$ be a matrix and let us denote by $A_S$ the submatrix of $A$ with the columns restricted to a set $S\subset [n]:=\{1,2,\cdots, \ n\}$. Then one says that the matrix ...
2
votes
1answer
42 views

$n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Does a conjugated matrix with only $1$'s and $0$'s exist?

Let $A$ be an $n\times n$ matrix with all eigenvalues equal to $1$ or $0$. Is there a conjugated matrix $B = XAX^{-1}$ for some $X$ such that all the elements equal either $1$ or $0$? My thoughts so ...
1
vote
3answers
30 views

How close apart are two message - “Document Distance” algorithm

I was looking at this algorithm that computes how close apart are two texts from one another and the formula seems a bit weird to me. The basic steps are: For each word encountered in a text you ...
0
votes
1answer
22 views

Shifting of the spectrum of a linear operator - in both the symmetric and non-symmetric cases,

a) I finished a problem that sort of highlighted the fact that if a real symmetric matrix $A_2$ = A + I, where A is also real and symmetric, then $A_2$ has the same eigenvectors as A, but its spectrum ...
0
votes
1answer
23 views

Align the cube's nearest face to the camera

I have a cube and 4x4 transformation matrix Cube is rotated randomly I need to find the nearest face of cube regarding to camera and rotate the cube by aligning that face to the camera. How can I do ...
0
votes
2answers
28 views

Square block matrix, with Hermitian, non-negative definite blocks, prove that the matrix is also non-negative definite,

Consider the square block matrix $$S= \begin{bmatrix} R & RQ^* \\ QR & QRQ^* \\ \end{bmatrix} $$ where $R$ is a Hermitian, non-negative definite square matrix ...
3
votes
1answer
82 views

I can't understand a step in the proof of the associativity of matrix multiplication

Matrix multiplication is proven by the following reasoning: Let there be matrices $A^{m \times n}$, $B^{n \times k}$ and $C^{k \times l}$. Then $$ ...
3
votes
1answer
26 views

Calculating a bound on the norm of a matrix exponential

The problem is this: Let A be a square $n \times n$ matrix, and define $$e^A=\sum_{k=0}^\infty \frac{1}{k!}A^k$$ Find a bound for $\lvert e^A \rvert$ in terms of $\lvert A \rvert$ and $n$. I was ...
0
votes
1answer
27 views

Over-specified linear system

Consider the matrix $A $ with RREF consisting of three of the 4, 4- dimensional standard vectors: $[\mathbb {e_1}, \mathbb {e_2}, \mathbb {e_3} ] $ Since the rank is 3 the matrix has one solution ...
1
vote
0answers
29 views

matrix with positive diagonal elements

I was wondering if a symmetric matrix with positive elements only in the diagonal (negative elsewhere) is any special beside the symmetry. Thanks in advance
0
votes
1answer
13 views

Generation of rank-$2$ matrices from a dictionary of rank-$1$ matrices.

I have a question about the construction of rank-$2$ matrices from a dictionary of rank-$1$ matrices. Consider the set $\mathcal{D} = \{\mathbf{A} \in \mathbb{C}^{2 \times 2} \mid rank(\mathbf{A}) = ...
2
votes
1answer
33 views

how to find the dimension of the image of $f$ in this case?

Let $A \in M_{m \times n}(\Bbb R)$ be fixed, and let $B \in M_{m \times l} (\Bbb R)$. Consider the map $f: M_{n \times l}(\Bbb R) \to M_{m \times l}(\Bbb R)$ defined by $f(X) = AX + B$ for all ...
1
vote
1answer
24 views

Determining the standard matrix from the images of the standard basis vectors

Let a linear transformation $T:$ $\mathbb{R}^3$ → $\mathbb{R}^3$ rotate a vector around the z-axis by $45^{o}$ followed by an orthogonal projection onto the x-axis. Determine the standard matrix ...
0
votes
2answers
31 views

Relationship between eigenvalues of A symmetric matrices

Let $$A=\begin{pmatrix}a & b\\b & c\end{pmatrix} \in M_2\mathbb{(R)}$$ i) Find the eigenvalues of $A$ ii) If $\begin{pmatrix}1\\2\end{pmatrix}$ is an eigenvector of $A$, prove that ...
0
votes
0answers
10 views

Efficient inverses of many related matrices [duplicate]

Say I have a $N$-by-$N$ positive definite real matrix $\Sigma$ and I wish to compute the inverses (or equivalently Cholesky decompositions) of $(\Sigma + a_k I)^{-1}$ for a set of $K$ positive $a_k$. ...
-1
votes
0answers
15 views

Finding bases for kernel N(T) and image R(T)?

Goodday, I need some help with the following problem: Find bases for both N(T) and R(T) in the following transformation: $T: M_{2 \times 3}(F) \rightarrow M_{2 \times 2}(F)$ defined by: ...
1
vote
2answers
23 views

Inverse of nonnegative Toeplitz matrice

Consider a right-hand circulant matrice of size $n$ (called also Toeplitz matrice) \begin{equation} T= \left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_n \\ a_n & a_1 & ...
1
vote
2answers
73 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.