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
3answers
272 views

Basic question: what does the notation $[A,B]$ mean?

If $A$ and $B$ are both matrices, what is $[A,B]$? I understand that it is a commutator and that $[A,B]=AB-BA$, but since I don't know what a commutator is, none of this information is telling me ...
4
votes
1answer
118 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
1
vote
1answer
56 views

Calculating an integral with a matrix

I want to calculate the following integral: Let A be a symmetric, invertible matrix. $\int_{K}<A^2x,x>dx$ where $K:=\{x\in \mathbb R^n : \|Ax\|_2\leq1\}$ A is symmetric, hence there is an ...
7
votes
4answers
136 views

How to find x so that $\|A x\| = \|A\| \|x\|$ holds

The subbordinance property of matrix-vector multiplication states that $\|A x\| \le \|A\| \|x\|$ where $\|x\|$ is the norm of vector $x$ and $\|A\|$ is the induced norm of matrix $A$. Many textbooks ...
5
votes
3answers
176 views

Let $A$ be an $n\times n$ invertible complex matrix such that $A^7 = A^*$. Show that $A^8 = I$.

Let $A$ be an $n\times n$ invertible complex matrix such that $A^7 = A^*$ (where $*$ denotes conjugate transpose). Show that $A^8 = I$. Here are my thoughts so far: I was able to show that all ...
2
votes
2answers
54 views

How to show the identity relating to Matrix

Suppose that $$ A=\begin{bmatrix}a_{11}&a_{21}\\a_{21}&a_{22}\end{bmatrix}, \ \ B=\begin{bmatrix}d&-1\\1&0\end{bmatrix}. $$ and $$A=B^N$$ Show that $$a_{11}=\sum_{i=0}^{[N/2]}(-1)^i ...
0
votes
1answer
18 views

Basic Matrix Properties

I know its basic but I am not quite getting it. I have two matrices W and U. W has 3M rows and M columns while U is M into M diagonal matrix. I want to ask if R1 and R2 are equivalent. If yes then ...
5
votes
1answer
500 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
17
votes
2answers
274 views

Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
0
votes
1answer
249 views

Unique least square solutions

There is a theorem in my book that states: If $A$ is $m\times n$, then the equation $Ax = b$ has a unique least square solution for each $b$ in $\mathbb{R}^m$. But can we find a counter-example to ...
0
votes
1answer
42 views

Transformation of inverse to a system of linear equations

I have $X = (U'WU)^{-1}U'$ to be solved. Suppose $U'$ is $3 \times 7, W$ is $7 \times 7$ positive definite matrix, $U'$ is of rank 3. So, I transformed $(U'WU)^{-1}U'$ as $(U'WU)^{-1}U'WU = I\\ XWU ...
1
vote
1answer
49 views

Why are points from this matrix geometric sequence co-planar?

Let $ M= \left[ {\begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ \end{array} } \right] $, such ...
0
votes
3answers
40 views

Is This A Image Of A linear Transformation?

Let there be $T:R^3 \rightarrow R^3$ $T(0,-1,1)=(3,3,3)$ $T(1,0,-1)=(0,1,1)$ $T(1,1,0)=(1,2,-1)$ Is (1,2,3) is the only image of the vector $(1, \frac{-7}{9}, \frac{-8}{9})$? I have thought to ...
1
vote
1answer
60 views

Complex matrix and diagonalizablity

Let $A\in\mathcal{M}_4(\mathbb C)$ such that $\operatorname{rank}(A)=2$ and $A^{3}=A^2$ $\neq0$. Suppose that $A$ is not diagonalizable. Then 1. One of the Jordan blocks of the Jordan cannonical form ...
0
votes
1answer
69 views

Characteristic polynomial and characteristic equation

What is the major difference between the characteristic polynomial and the characteristic equation?
4
votes
4answers
47k views

Online tool for diagonalizing matrices?

I need some online tool for diagonalizing 2x2 matrices or at least finding the eigenvectors and eigenvalues of it. I don't like to download any stuf because I'm not able to, some online tool will do ...
2
votes
1answer
122 views

How to solve this $2\times2$ linear system of equations?

at the moment I am a little bit confused. Here is the matrix I am trying to solve $$ \left( \begin{array}{cc|c} 5 & -1& 12 \\ -1 & 2& 12 \end{array} \right) $$ I tried ...
1
vote
2answers
61 views

Finding the amount of solutions in a 3 equation solution

So, I'm not really sure how to calculate the amount of solutions for a system with 3 equations. All I know is that it has something to do with matrices and the discriminant, but I'm not sure where to ...
2
votes
1answer
74 views

Are these two permutation matrices equivalent?

Definition: The permutation matrix $P_{ij}$ is the identity matrix with rows $i$ and $j$ reversed. When left-multiplied with another matrix, it exchanges rows $i$ and $j$. Am I right in thinking that ...
2
votes
5answers
315 views

Calculation of $\lambda$ in determinant multiplication.

$$\begin{vmatrix} a^2+\lambda^2 & ab+c\lambda & ca-b\lambda \\ ab-c\lambda & b^2+\lambda^2& bc+a\lambda\\ ca+b\lambda & bc-a\lambda & c^2+\lambda^2 ...
2
votes
1answer
45 views

rref matrix equations k2 - 5

This question is about reduced row echelon form, Gauss-Jordan, inverting matrices, and solving systems of equations. I try to solve a system of equations with matrices. I know what operations are ...
1
vote
1answer
3k views

Combine transformation matrices

Question: Find the transformation matrix that combines the following transformation matrices, in order: $$\begin{bmatrix} &3 &0 &0 &0 \\ &0 &-1 &0 &0 \\ ...
0
votes
1answer
64 views

Open subset of the space of matrices

This question comes from the process of my learning about Grassmann manifolds. Suppose that $M(m,n)$ is the set of real $n \times m$ matrices, where $n>m$. Let $F(m,n)$ be a subset of $M(m,n)$ ...
0
votes
1answer
61 views

Similar vs congruent matrices

Suppose that some symmetric matrix $S$ (everything here is over the field of real numbers) is similar to a diagonal matrix $D$ via the invertible matrix $P$. We have: $P^{-1}DP=S.$ My question: ...
0
votes
2answers
45 views

Proving that dimension of two vector spaces are the same.

Let $V=\{A\in \mathbb{F^{n \times n}} \ \ | \ \mathrm{Trace}(A)=0\}$ and $W=\{B\in \mathbb{F^{n \times n}} \ \ | \ \ B=CD-DC, C \ \text{and} \ D \in \mathbb{F^{n \times n}} \}$. Prove that $V=W$. ...
0
votes
1answer
176 views

How do you calculate the dimensions of the null space and column space of the following matrix?

I understand you are supposed to get the reduced row echelon form, which I did, and this is what I came up with: 1 -2 0 19 -6 0 -37 0 0 1 -6 2 0 6 0 0 0 0 0 1 3 0 ...
0
votes
2answers
29 views

Do all positive definite matrices satisfy this?

For any positive definite matrix, $A$, is the following true? Let $A\succ 0$, then there exists some $m>0$, such that $A-mI\succeq 0$ ($A-mI$ is postive semi-definite)?
1
vote
0answers
271 views

Consistency of linear system of equations

Let $Ax = B$ be a system of linear equations, where $A$ is an $n \times n$ matrix and $x, B$ are $n \times 1$ vectors. Yesterday in class our teacher said that if det$A = 0$ and Adj$A.B = O$ (where ...
0
votes
0answers
165 views

Transformation matrix from one basis to another

Say the vector space $V$ has two bases, $B$ and $B'$. There exists a matrix $P$ such that $BP=B'$. $B$ is of the form $\begin{pmatrix} v_1&v_2&\dots&v_n\\ ...
2
votes
5answers
2k views

Finding matrix $B$ is not zero matrix where $AB= 0$

$A$ is defined as an $m\times m$ matrix which is not invertible. How can i show that there is an $m\times m$ matrix $B$ where $AB = 0$ but $B$ is not equal to $0$? For the solution of this question ...
3
votes
2answers
2k views

Best Books to learn Proof-Based Linear Algebra and Matrices

So I'm in a really serious problem. It's my first year at university and I'm doing a CS major. The math is already getting serious and I'm lost, really lost. It's all about matrices so far and the ...
2
votes
1answer
52 views

How do you prove this linear algebra matrix equality?

Basically My professor wrote down simplified this expression $Av_1 =$ $Av_2 =$ to $\begin{bmatrix}A\end{bmatrix} \begin{bmatrix} v_1 & v_2 \end{bmatrix}$ Where $A$ is a matrix, $v_1$ and ...
3
votes
0answers
93 views

Matrix partwise multiplication

I am working on an artificial intelligence application that (among other things) combines "opinions" of several "experts" who each have access to different aspects of a "situation". I can build this ...
0
votes
0answers
35 views

Matrix Algebra - Linear dependency

We have a given equation $ \frac{\mathrm{d}R(t) }{\mathrm{d} t}=R(t) \{(1-t)U_0+t U_1\}\tag 1$, all variables except scalar variable 't' has dimension $3 \times 3$. Given data $R(t)$ is ...
0
votes
1answer
94 views

How do you find the standard matrix for a transformation

How do you find the standard matrix for a transformation from $\mathbb{R}^2$ to $\mathbb{R}^4$ where $T(e_1) = (3, 1, 3, 1)$ and $T(e_2) = (-5, 2, 0, 0)$? I do not know how to approach this ...
2
votes
1answer
66 views

Role of metric in the matrix representation of Hermitian adjoint

I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ ...
2
votes
0answers
131 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
1
vote
1answer
110 views

Matrix - Commutative property

I have a rotation matrix represented as $R(t)=e^{B(t)},\tag 1$ where $B(t)$ is a skew symmetric matrix (since any rotation matrix can be expressed as a matrix exponent of a skew symmetric matrix), ...
0
votes
2answers
98 views

Eigenvalues of a matrix that is a product of a vector and transpose vector

Find eigenvalues, eigenvectors and rank of matrix $A$. $$\textbf{a}=\begin{bmatrix}a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{bmatrix}, \quad \textbf{b} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ ...
0
votes
1answer
29 views

Find a basis $B$ such that the matrix has the desired shape.

Let $p=(x-(a+bi))(x-(a-bi))$ be the characteristic polynomial of linear operator $T$ ($T$ in $\Bbb C$) and its basis of eigenvector is $A=\{u+iv,u-iv\}$. Find a basis $B$ in $\Bbb R^2$ such that ...
4
votes
1answer
90 views

Any article expounding the difference between matrix analysis and functional analysis?

I do theoretical physics. For quantum mechanics, the mathematical foundation is rigorously functional analysis. However, people generally take matrix analysis (for finite dimensional vector spaces) to ...
1
vote
2answers
113 views

Estimate eigenvectors of symmetric matrix with almost vanishing diagonal

Is there a way to approximate the eigenvectors of a symmetric matrix with almost vanishing diagonal elements, i.e. with the block matrix form, \begin{equation} M=\left( \begin{array}{cc} ...
2
votes
0answers
62 views

Counting symmetric binary matrices with constant line-sum

I'm trying to count, as the title suggests, symmetric matrices with entries $0$ or $1$ and constant line-sum $k$. ($0 \leq k \leq n$). If you start listing the number of these on a table you'll get a ...
2
votes
2answers
59 views

Families of Square Roots of Identity Matrices

I just analysed this equation for real matrices $$ A^2=\begin{pmatrix}a&b\\c&d\end{pmatrix}^2=I $$ From the main diagonal of $A^2$ we must have $a^2+bc=bc+d^2=1$ showing that $d=\pm a$. CASE ...
4
votes
2answers
2k views

Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
6
votes
4answers
638 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
0
votes
1answer
124 views

Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
0
votes
1answer
119 views

Null space of a matrix mcq

Let $M$ be the set of all $m\times n$ matrices with real entries. Which of the following statement is correct? There exists $A$ of order $2\times 5$ belonging to $M$ such that the dimension of the ...
2
votes
0answers
73 views

What do you call a matrix where the rows sum to zero and the columns sum to zero?

What do you call a matrix where the rows sum to zero and the columns sum to zero? Or is there no standard name for this type of matrix?
2
votes
3answers
187 views

non-symmetric matrix with orthogonal eigenvectors

Given that a symmetric matrix with real entries has orthogonal eigenvectors, is the converse true? That is, if a matrix has orthogonal eigenvectors, does it have to be symmetrical and real?