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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3
votes
1answer
44 views
+100

How to prove this result about the interlacing of eigenvalues.

Let $A$ be a real symmetric matrix of order $n$ with eigenvalues $\mu_1\ge \mu_2,\ldots, \mu_{n}$. $B=\begin{bmatrix} x&x&x&x&x \end{bmatrix}$, where $x$ is non zero column vector in ...
0
votes
3answers
47 views

Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
0
votes
1answer
23 views

Determinant of the matrix representation of an isomorphic linear transformation

Are there any theorems or special properties about the determinant of a matrix representation of an isomorphic linear transformation?
0
votes
0answers
23 views

Linear Algebra Base Change Matrix

I'm currently learning eigenvectors and Linear Transformations. And this basis change part is particularly confusing for me.. I feel like there are two types of questions, First type, you have a ...
1
vote
1answer
31 views

Form of pure states on $M_n(\mathbb{C})$

Related question: The form of the states on an algebra of $n\times n$ matrices with complex entries I have tried to show that pure states on $M_n(\Bbb{C})$ are of the form $\phi(A)=Tr(\rho A)$, just ...
0
votes
2answers
26 views

Show Matrix relationship

Say $A$ and $B$ are 2x2 matrices with integer entries in the group of matrices with determinant 1, under matrix multiplication. Let $A$ and $B$ have the same first column, show there is exists an ...
1
vote
1answer
30 views
+50

Block matrix of order $m$ with three block matrices

How to find eigenvalues of following block matrices? $M=\begin{bmatrix} A & B & O & O & O & O & O & \cdots & O & O\\ B & A & B & O & O & O ...
2
votes
1answer
34 views

Find $A^{20}x$ using eigenvectors and eigenvalues.

Find $A^{20}x$ A is a 3X3 matrix with the following eigenvectors and eigenvalues: $V_1 = [1, 0, 0]... V_2 = [1, 1, 0]... V_3 = [1, 1, 1]$ corresponding to Eigenvalues.. $\lambda1 = -1/3, \lambda2 = ...
0
votes
1answer
11 views

Find invariant factors of a power of a matrix, given a matrix in Jordan canonical form

Suppose \begin{align} A&=(I_4+(N_1 \oplus N_3))\oplus(N_2\oplus N_4 \oplus N_5)\oplus(-I_3+N_3) \\ &= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ...
-1
votes
0answers
29 views

Gradient of a vector [on hold]

Matrix $V$ is 200 by 785. Matrix $X$ is 785 by 1. Matrix $W$ is 10 by 201. Matrix $y$ is 10 by 1. First, I do: $ V * X$ Then, I apply $tanh()$ to every element of that resulting matrix. The result ...
2
votes
0answers
32 views

3D shaped matrices - how would multiplication work? [duplicate]

I've been thinking about vectors and matrices lately, and I got a little curious. Why don't we have cubic shaped matrices? After all, vectors are 1-dimensional matrices, so it follows that there ought ...
0
votes
3answers
43 views

finding an orthogonal basis for $\mathbb{R}^3$

I have a vector $[-1/3, 1/3, 4/3]$ and another vector $[1, 1, 0]$ and I need to find an another orthogonal vector that are orthogonal to both vectors but could not find a method. Any help would be ...
0
votes
1answer
22 views

on Matrix Inequality

Let $A=(a_{ij})$, and $B=(b_{ij})$ be two $n$ by $n$ real symmetric matrices such that $$ a_{ij}\leq b_{ij}+\alpha, \quad \alpha>0. $$ Can we conclude that $A\leq B +\textbf{1}\alpha$? Note ...
0
votes
1answer
25 views

Relationship between the ranks of matrices A and B , where B is obtained by changing one element of A.

Let $A$ be a $5 \times 5$ matrix and let $B$ be obtained by changing one element of $A$. Let $r$ and $s$ be the ranks of $A$ and $B$ respectively. Which of the following statement is/are correct? $s ...
3
votes
1answer
56 views

Validity of a formula for the $n$-th power of a general $2 \times 2$ matrix

I am taking an optics course and at one point$^1$ we need to find the $n$-th power of the $2 \times 2$ matrix $$M := \begin{bmatrix} a & b \\ c & d\end{bmatrix}$$ where $a, b, c, d$ are real ...
0
votes
1answer
74 views

Find determinant of given matrix

Let $A$ be an $n × n$ matrix of the following form. What is the value of the determinant of $A$? My attempt: I've used brute force to identity correct option. When I put $n=1$, then ...
0
votes
0answers
15 views

What are the eigenvalues of the following Hermitian matrix?

Let $\mathtt{i}=\sqrt{-1}$ and $p=1+\mathtt{i},q=1-\mathtt{i}$. Let $A$ be an $n\times n$ matrix such that $$A=\begin{bmatrix} 0 & p & p & \cdots & p & \color{blue}{q}\\ q & 0 ...
1
vote
2answers
27 views

How to find a matrix $B$ and an invertible matrix $P$ such that this matrix $A$ is in Jordan Canonical Form?

I am working on the following exercise: Find a matrix B and an invertible matrix P such that $$A = \begin{bmatrix} 1 & -2 & 1 & 0 \\ 1 & -2 & 1 & 0 \\ 1 & -2 & 1 ...
1
vote
2answers
25 views

If matrix $A$ is similar to matrix $D$ and $B$ is similar to $E$, than: $AB$ is similar to $DE$?

More specifically: if $A$ & $B$ are diagonalizeable, than is it correct to say that $AB$ is diagonalizeable? (Hints would be more appreciated)
1
vote
1answer
38 views

Inverse of a “Vandermonde-like” matrix composed of power series

Does it exist analytical inverse of a complex matrix whose elements are sets of "power series" except the last term is scaled? Let $0<x_1<x_2<...<x_n$ are monotonically increasing, the ...
-1
votes
2answers
72 views

Intrinsic proof for $(I + AB)^{-1}A = A(I + BA)^{-1}$ by using Schur complements on matrix block elimination

Given $(I + B(I - AB)^{-1}A)$ to be inverse of $(I + BA)$, how could we derive that the following alternative form holds $(I + AB)^{-1}A = A(I + BA)^{-1}$. This is easy to verify(direct proof). ...
0
votes
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 ...
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 ...
3
votes
1answer
65 views
+50

Eigenvalues of block matrix of order $m+1$

How to find eigenvalues of following matrix? $\begin{bmatrix} mkI-A & -A & -A & \cdots & -A\\ -A & kI-A & O & \cdots & O\\ -A & O & kI-A & \cdots & O\\ ...
7
votes
2answers
117 views

Existence of an $n\times n$ real matrix $A$ such that $A^2=-I$.

Let $A$ be a $n\times n$ real matrix $A$ such that $A^2=-I$. Such an $A$ cannot be, Orthogonal. Invertible. Skew-symmetric. Symmetric. Diagonalizable. I tried to figure out the answer by looking ...
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 ...
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 ...
1
vote
3answers
48 views

Linear algebra: Show that there exists vectors $c_i \in \mathbb R^n$ such that $A\cdot c_i=b_i$

Let $A\in \operatorname{Mat}_{m,n} (\mathbb R)$ be a real matrix of rank $r=n$. Let $(b_1,b_2,..,b_n)$ be an orthonormal basis for the column space $R(A)$ (terms. the scalar product) Show that for ...
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 ...
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 ...
-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?
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 ...
1
vote
2answers
46 views

Nullspace and column space of invertible matrix

I want to show that the matrix $A$ $n\times n$ is invertible if and only if $N(A) = {0}$ and $C(A) = R^n$. So far, this is what I've got: Theorem: A is invertible $\implies N(A) = 0$ and $C(A) = 0$. ...
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
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 ...
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.
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
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 [closed]

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. ...
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 ...
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\{ ...
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 ...
-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 ...
1
vote
2answers
24 views

Expectation and variance of matrix valued random variable

Suppose I have a discrete matrix-valued random variable $X$, that is, I have defined a set of fixed matrices $\{Y_i\}_{i=1}^n$, and the random variable $X = Y_i$ with probability $\frac{1}{n}$. Is ...