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
24 views

Number of matrices satisfying given conditions

Let A be the set of all $3*3$ symmetric matrices all of whose entries are either $0$ or $1$.Five of these entries are $1$ and four of them are $0$. $1)$ The number of matrices in $A$ is ? ...
0
votes
1answer
70 views

Easy way to get Determinant of 4 by 4 matrix

I have learned one way to get $4\times 4$ determinant. That is, divide a matrix $A$ by 4 part where each part is $2\times 2$ matrix: $$A = \left(\begin{array}{cc} B & C \\ D & E ...
2
votes
2answers
883 views

formula for calculating determinant of the block matrix

I saw a formula on the wikipedia page about determinant that $\det\begin{bmatrix}A & B\\ C & D \end{bmatrix}$ = $\det(AD-BC)$, if $C$ and $D$ commute. Is this always true? Or is there a good ...
1
vote
1answer
54 views

How can we determine if the hyper-plane pass through the origin?

Let $A$ is an $n \times n$ matrix. Consider each row of $A$ as a point in $\mathbb{R}^n$; and assume these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point. The ...
12
votes
7answers
7k views

Relation between Cholesky and SVD

When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed). Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
4
votes
0answers
57 views

I have a answer to a question about trace. Is there an easier answer to this question?

Let $A\in M_n(\mathbb{C})$. Show that $$tr\left(\frac{A+A^*}{2}\right)\leq tr((A^*A)^{1/2}).$$ My answer: It is easy to see that $$tr\left(\frac{A+A^*}{2}\right)=\text{Re}(tr(A))\qquad and\qquad ...
2
votes
1answer
97 views

The form of the states on an algebra of $n\times n$ matrices with complex entries

I have a question concerning $n \times n$ matrices. Denote by $M_n(\mathbb{C})$ the algebra of all $n \times n$ matrices with complex entries. Let $\phi$ be a state on $M_n(\mathbb{C})$ i.e. a linear ...
0
votes
0answers
15 views

If a matrix is of rank one, and let $v$ be the eigen vector corresponding the eigen value $d$. Then, $A=dvv^T$

If a symmetric matrix is of rank one, and let $v$ be the eigen vector corresponding the eigen value $d$. Then, $$A=dvv^T$$ How to prove this??
2
votes
1answer
30 views

$LDL^t$ Factorization Algorithm to find a factorization of the form A

For $$ \begin{pmatrix} 4 & 2 & 2 \\ 2 & 6 & 2 \\ 2 & 2 & 5 \\ \end{pmatrix} $$ I found that $$ L=\begin{pmatrix} 1 & ...
0
votes
2answers
31 views

Can we say scaling matrix is necessarily diagonal?

Can we say scaling matrix is necessarily diagonal? According to wikipedia, yes According to this video, no $S$ is scaling along orthogonal directions according to this So, how to put them both ...
1
vote
1answer
57 views

Decode the message $(1,1,1,0,1,1,1)$ using the Hamming $ (7,4)$ code

The question is asking me to decode $(1,1,1,0,1,1,1)$ using Hamming $(7,4)$ code. I know that I am suppose to set a $3 \times 7$ matrix ${\bf H}$ and multiply it by ${\bf r}$ and set it equal to zero, ...
0
votes
1answer
30 views

Find the bases for the eigenspaces of the matrix. [on hold]

The question I have is to find the bases for the eigenspaces. I have already found the characteristic equation which is $(λ-1)^2=0$. I also found that λ=1 The matrix I'm using is {(1,0),(0,1)} ...
1
vote
1answer
25 views

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A$ invertible? [duplicate]

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A=\begin{pmatrix}1 & 1 & 1 &1 \\ x_1 & x_2 & x_3 &x_4 \\ x_1^2& x_2^2 & x_3^2 & x_4^2\\ x_1^3& x_2^3 ...
0
votes
1answer
15 views

Rotations about the origin

Let R(θ) denote a rotation matrix which rotates a point $x$ in $S^2$ anticlockwise about the origin through a given angle θ. (Where $S$ is the set of real numbers) How do you illustrate that this ...
3
votes
2answers
98 views

linearly independence of $e^{a_1x},… e^{a_nx}$

$a_1,\ldots,a_n$ are real different numbers. Prove that the functions $e^{a_1x},...,e^{a_nx}$ are linearly independent in $Fun(R,R)$. My way to try to prove it: I assumed: $b_1e^{a_1x} + \cdots ...
0
votes
0answers
11 views

Inverse kinematics - How do i compute the du?

I am at the moment trying to implement at jacobian based inverse kinematics solver, which is given a current homogeneous Transformation matrix r(q) and a desired homogenous tranformation matrix ...
2
votes
0answers
28 views

Matrix Calculus and Linear Transformations

I'm working on making the jump from differentiating real valued functions ($f: \mathbb{R}^n \rightarrow \mathbb{R}$) and vector valued functions ($g: \mathbb{R}^n \rightarrow \mathbb{R}^m$) to matrix ...
1
vote
2answers
37 views

Prove the positive definiteness of Hilbert matrix

This is so called Hilbert matrix which is known as a poorly conditioned matrix. $$ A = \left(\begin{matrix} 1 & \frac{1}{2} & \frac{1}{3} & ... & \frac{1}{n} \\ ...
0
votes
1answer
18 views

how do you solve the simultaneous equations 2x + y = 9 and x - 2y = -8 using the matrix method?

i first convert the bases into a 2x2 matrix and then i multiplied the inverse matrix by the 2x1 e/f matrix of 9 (on top) and -8 (on the bottom) which gave me x = 5.4 and y = 5 however the answer ...
-3
votes
0answers
34 views

If I have matrix A, what is difference between $det(A), det(A_n), det(A_{n+1})$? [closed]

If I have matrix $A_n$, what is difference between $det(A_n), det(A_{n+1})$ and $det(A_{n+2})$? If somone wants to help and answer on question can that be with example?
1
vote
2answers
30 views

Basis that contains a basis for a subspace

I have this exercise and I want to know if my answer is correct. The exercise is: Consider the linear space $\mathbb{R}^{2\times2}$ of $2\times2$ matrices with real entries. Consider $W$ contained ...
1
vote
1answer
100 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
1
vote
2answers
71 views

How do I prove that $\det A_{n+2} = a \det A_{n+1} + b \det A_n$ for matrix $A$?

I have calculated: $\det A_1=2$, $\det A_2=3$, $\det A_3=4$, so I was putting some numbers in $\det A_{n+2} = a \det A_{n+1} + b\det A_n$ like $n=1$, $n=2$ ($\det$ $n\times n$ matrix) and get that ...
0
votes
1answer
1k views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 ...
3
votes
3answers
72 views

$A^{2014}=0$ for a matrix A

Let A be a 3*3 matrix and $A^{2014}=0$. Must $A^3$ be the zero matrix? I can work out that I-A is invertible, but I don't know how to proceed further.
1
vote
1answer
194 views

Determinant of a matrix minus its transpose

Let $A$ be an $n\times n$ matrix and $A^t$ be its transpose. I know how to show $\mathrm{det}(A-A^t)=(-1)^n\mathrm{det}(A^t-A)$. I would like to know how to show ...
1
vote
0answers
22 views

Eigenvalues of a tridiagonal block matrix

When a tridiagonal matrix is also Toeplitz, there is a simple closed-form solution for its eigenvalues, namely $ a + 2 \sqrt{bc} \, \cos(k \pi / {(n+1)})$ , for $ k=1,...,n$. Now my question is that ...
3
votes
1answer
50 views

Determinant of $P_n$

I am preparing for an exam on linear algebra within few days, so I am in desperate need for a solution for the following question: Question: Let $P_n$, $n\ge2$, be the $n\times n$ matrix whose ...
0
votes
1answer
25 views

Symmetric block matrix related

How to find eigenvalues of following symmetric matrix $\begin{bmatrix} kI-A & -A & -A & \cdots & -A\\ -A & kI-A & -A & \cdots & -A\\ -A & -A & kI-A & ...
0
votes
0answers
17 views

Neural Net Matrix Multiplication

I'm trying to figure out the matrix multiplications for the implementation of a single hidden layer neural net for MNIST digit recognition. Like the following: ...
0
votes
0answers
18 views

making sense of this polar decomposition [on hold]

I saw that the polar decomposition of $\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ is $\sqrt{\rho}\sqrt{\sigma}U$. $U$ is an unitary matrix. However, I cannot see how the $\rho^{1/2}$ can merge together. ...
1
vote
1answer
19 views

Transition matrix of polynomial.

Good night, i need help with this. Find the transition matrix that goes from the basis W to the basis $\left\{ 2,1-2x,x^{2}-1,x^{3}-x^{2}+x\right\} $ I found a basis for W, $\left\{ ...
0
votes
1answer
17 views

Can a Hermitian Matrix be Decomposed into a Sum of Unitary Matricies?

Given a Hermitian matrix $A$, when is it possible to write $A$ as a sum of unitary matricies as in the following form? $$ A = \sum_{i} a_i U$$ Where $U$ is unitary. Intuitively, because you have a ...
-2
votes
1answer
32 views

Linear Algebra - Row echelon form [closed]

Find two different row echelon forms of: $$\left(\begin{matrix} 1&4\\ 3&11 \end{matrix}\right)$$ this exercise shows that a matrix can have multiple row echelon forms. I konw it is easy, ...
0
votes
0answers
17 views

Finite difference of radial Laplace operator doesn't give a symmetric (hermition in general) matrix

I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial^2 u}{\partial r^2}+$ $\frac{1}{r}$ $\frac{\partial u}{\partial r}$ which ...
0
votes
0answers
9 views

Determinants using Row Reduction replacement

I am aware replacement does not affect the value of determinant when doing a row reduction. However, I realised there isn't a good explanation on how to handle different forms of replacement when ...
0
votes
1answer
10 views

Derivative of a matrix times vector with respect to an entry of the matrix

I'm trying to find the derivative of $$\frac{\partial{A\vec{x}}}{\partial{A_{ij}}}$$ but I'm having some trouble figuring it out. $A\vec{x}$ will be a vector where the $i^{th}$ entry is $$A_{i1}x_1 + ...
1
vote
2answers
18 views

matrix operations with transpose like properties

Has any one studied operations on matrices with transpose like properties - for example (A+B)' = A' + B' where ' stands for transpose. Also ' is its own inverse. Is there a common name for such ...
1
vote
0answers
35 views

Characteristic polynomial of matrix

If I wanted to find the eigenvalues of a matrix $\mathbf{A}$, then I could use these two options. $$\lambda\mathbf{I}-\mathbf{A}=\mathbf{0}$$ $$\mathbf{A}-\lambda\mathbf{I}=\mathbf{0}$$ However, ...
0
votes
2answers
158 views

Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices [closed]

Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
0
votes
2answers
30 views

How do I get the Rational Canonical Form from the minimal and characteristic polynomials?

Let's say I have the minimal polynomial and characteristic polynomial of a matrix and all its invariant factor compositions. How do I get a rational canonical form matrix from this?
0
votes
0answers
12 views

the classes of similarity relation on matrices

we say two matrices $A,B∈M_n(F)$ similar if there exists a matrix $P$ that $$B=P^{-1}AP$$ Now suppose $F$ be an infinite field and assume the similarity relation and let $C$ be a finite class of ...
1
vote
1answer
44 views

LU factorisation

I am studying the LU factorisation. What I have learned is that with this technique we start with a matrix $A$ and result into two matrices $L$ and $U$ where $L$ is a Lower Triangular matrix and $U$ ...
2
votes
1answer
34 views
0
votes
1answer
30 views

Number of errors detected from a generator matrix

Consider the encoding function $\alpha : \mathbb{Z_2^2} \rightarrow \mathbb{Z_2^5} $ given by the Generator matrix $$ G = \begin{bmatrix}1&0 &1& 0& 0 \\0& 1 & 0 & 1 & ...
0
votes
1answer
31 views

eigenvalues and eigenvectors of 2x2 block matrix

My question is a really straightforward one: Is there an easier way to find the eigenvalues and/or eigenvectors of a 2x2 block diagonal matrix other than direct diagonalization of the whole matrix? $ ...
1
vote
1answer
38 views

Various matrix manipulations effect on determinant

Suppose that the matrix $A$ below has determinant $−3$. Find the determinants of $B$, $C$ and $D$. $$ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} ...
0
votes
3answers
34 views

Inverse matrix of sin and cos being multiplied.

find the inverse $$\begin{pmatrix} 5 e^{2 t} \sin(2 t) & 5 e^{3 t} \cos(2 t)\\ -6 e^{2 t} \cos(2 t)& 6 e^{3 t} \sin(2 t) \end{pmatrix}$$ I understand the inverse of $$\begin{pmatrix} ...
1
vote
1answer
43 views

Special skew-symmetric matrices

A skew symmetric matrix $J$ is "special" if for any matrix $X$ with determinant equal to one, it satisfies $$ XJX^{-1}=XX^TJ.$$ For $2 \times 2$ matrices one can easily verify that any multiple of the ...
-1
votes
0answers
39 views

What do i need to construct a 3x3 rotation matrix? [closed]

What do i need to construct a 3x3 rotations matrix, that tells how a point should oriented to face object.