0
votes
0answers
27 views

How can I calculate matrix differentiation? [duplicate]

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
2
votes
2answers
27 views

Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
0
votes
0answers
34 views

Solve linear equations [on hold]

\begin{bmatrix} 0 & 0& 1& 1& 1&0 \\ 0 & 0& 0& 0& 2&1 \\ -3 & 0 & 0 & 0 & -2& 0\\ 4& 4& 0& 0& 0 & -1\\ ...
2
votes
1answer
71 views

Why we use $\mathbb{R}^{m \times n}$ notation instead of $\mathbb{R}^{n \times m}$?

I just realised, that I use all the time the notation $\mathbb{R}^{n \times m}$, and all books and papers use $\mathbb{R}^{m \times n}$. $\mathbb{R}^{n \times m}$ is more sympathetic for me, because I ...
-2
votes
3answers
52 views

Is the basis of null space of a matrix always a subset of the basis of its column space?

Given an $m\times n$ matrix $A$, is the basis of its null space (set of $x$ such that $Ax=0$) always a subset of the basis of the row space of $A$? In general, the basis of a subspace may not be a ...
3
votes
2answers
45 views

finding the closest matrix of a given form

let's say I have a vector $(a_1\dots a_n)$, where each component is between $-1$ and $1$. Now from this vector I define a $n\times n$ matrix $M$ such that $$M_{ij} = \begin{cases} 1&\,& i = ...
2
votes
1answer
65 views

Which two matrices will create the zero matrix multiplication

I was thinking, which property of matrices could help me determine if the multiplication of some $A$ and $B$ result the zero matrix?
1
vote
2answers
56 views

Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$

Let $J$ be a $k \times k$ jordan block, prove that any matrix which commutes with $J$ is a polynomial in $J$. I appreciate your hints, Thanks
0
votes
2answers
41 views

Adding a constant to a matrix

Find p(A) if p(x) = $2x^2 - x + 1$ where A is the below matrix: $$ \begin{bmatrix} 3 & 1 \\ 2 & 1 \\ \end{bmatrix} $$ Attempt at a solution p(A) = $2 \cdot ...
3
votes
1answer
45 views

Is $\mathbf {B^TAB}$ non-singular for a non-singular $\mathbf A$, and $\mathbf {B}$ with full column-rank?

If $\mathbf A$ is any square non-singular matrix of dimension $n \times n$. And $\mathbf B$ is a $n \times m$ matrix with $\mathrm{rank(\mathbf B)} = m$. Is the full rank condition of matrix $\mathbf ...
0
votes
1answer
15 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 ...
0
votes
0answers
31 views

Transpose/multiplication of 3D matrices

I have $A(p)=\begin{bmatrix}p_1 &p_2 & p_3\\ 2p_1 &2p_2^2 & 4p_3^3\\ 3p_1 &3p_1 & 10\\ \end{bmatrix}\tag 1$ $ p= {\left(\begin{array}{c}p_1\\p_2\\p_3\\p_4 ...
0
votes
1answer
42 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
3answers
34 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 ...
4
votes
0answers
97 views
+50

Eigenvalue test faster than $O\left(n^3\right)$?

Given a real $n\times n$ matrix $A$, one can find the eigenvalues in $O\left(n^3\right)$ by using say, the $QR$ algorithm. Now, what if we guess an eigenvalue $\lambda_0$, and we want to know if it's ...
0
votes
0answers
12 views

Definite integral - Finding an equivalent form

I have the following definite integral $ \int_{0}^{L} {\psi(t) }_{1 \times 5}{A(s)}_{5 \times 5}(\psi(t) _{1 \times 5})^{T} {B(s)}_{5 \times 5} ds \tag 1 $ Given data All dimensions are ...
2
votes
1answer
42 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
27 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 ...
0
votes
1answer
17 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: ...
1
vote
1answer
25 views

Each element of a real orthogonal matrix is equal to its cofactor

If $A =(a_{ij})$ be a real orthogonal matrix with $\det A = 1$, prove that each element $a_{rs}$ of $A$ is equal to its cofactor $A_{rs}$ in $\det A$. I got this basic problem from my text book and ...
11
votes
3answers
152 views

Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its ...
0
votes
1answer
26 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
37 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
0answers
36 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\\ ...
0
votes
0answers
26 views

Looking for reference for the criterion of inveribility of a difference of two invertible matrices

It is pretty easy to show that $A-B$ is invertible if either $AB^{-1}$ or $BA^{-1}$ have all eigenvalues of absolute value less than $1$. But I am specifically looking for a handy reference of this ...
-4
votes
1answer
30 views

What is the reduced row echelon form of A?

let $A = \left( \begin{array}{cccc} 7 & 7 & 9 & -17\\ 6 & 6 & 1 & -2 \\ -12 & -12 & -27 & 1 \\ 7& 7 & 17 & -15\end{array} \right)$ What is the reduced ...
0
votes
0answers
19 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
24 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
22 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
60 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 ...
0
votes
2answers
52 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
0answers
13 views

Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
12
votes
0answers
94 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
4
votes
1answer
41 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 ...
2
votes
2answers
40 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 ...
0
votes
0answers
66 views

What does $\langle x,X\rangle$ mean?

I encountered lots of times but can't find the meaning of $\langle x,X\rangle$. I know in general $\langle a,b\rangle$ is the inner product of two vectors, but this is obscure. I'm talking about real ...
5
votes
3answers
118 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 ...
1
vote
0answers
14 views

Is there any restriction to the sum of eigenvalues for non-negative, irreduceble and square matrices?

I'm trying to find if there is a restriction in tr(A) or eigenvalues sum for a non-negative, irreducible square matrix A. As an additional information, the row sums and the order of the matrix is ...
5
votes
1answer
65 views

Mutually commuting matrices

Let $A_{1},..., A_{m}$ be $n \times n$ matrices with entries in a field $K$ such that $A_{i}A_{j} = A_{j}A_{i}$ for all $ 1 \leq i, j \leq n$ and the product $A_{1}A_{2} ... A_{m} = 0$ is the zero ...
1
vote
1answer
16 views

Finding variables, basis and dimension given linear transformation with representing matrice

Let $T:R^4 \to R^4$ be linear transformation such that $T(x)=Ax$, when $A = ...
6
votes
4answers
368 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 ...
1
vote
1answer
23 views

Lower bounds on eigenvalues of a symmetric matrix based on the diagonals

A symmetric matrix $A$ always has real eigenvalues. If I know the elements on the diagonals, is it possible to have a lower bound on the smallest eigenvalue? How sharp would this bound be? For now I ...
0
votes
1answer
27 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
3
votes
1answer
60 views

how to find $(I + uv^T)^{-1}$

Let $u, v \in \mathbb{R}^N, v^Tu \neq -1$. Then I know that $I +uv^T \in \mathbb{R}^{N \times N}$ is invertible and I can verify that $$(I + uv^T)^{-1} = I - \frac{uv^T}{1+v^Tu}.$$ But I am not able ...
0
votes
1answer
23 views

How do get eigenvalues of a matrix B if add a row/column pair of a matrix A?

I have a matrix of size N×N of the form: where and A is N-1 x N-1 matrix, a=0. I known the eigenvalues of A. Any possible for getting eigenvalues of B from eigenvalues of A?
1
vote
1answer
39 views

Jordan-Chevalley decomposition of $T$ acting on $k[T]/(\pi(T)^e)$

Given an algebraically closed field $K$, a f.d. vector space $V$ over $K$ and $A\in{\rm GL}(V)$, we can view the space $V$ as a $K[T]$-module, where $T$ acts by $A$. Using the fundamental theorem of ...
0
votes
0answers
13 views

Non-Unitarily Diagonalizable Matrices

When searching for matrices that are similar to a diagonal matrix but not in a unitary way then a first hint would be to exclude the normal ones. But apart from that is there a general form for such ...
0
votes
1answer
46 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 ...
0
votes
0answers
19 views

How to get the transformation matrix for Linear Discriminant Analysis?

I am trying to implement Linear Discriminant Analysis. Is the eigen vectors of the product of within scatter matrix and between scatter matrix inverse (Sw*Sbinverse), the transformation matrix? Could ...
1
vote
3answers
30 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?