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
15 views

Proof for Semidefinite symmetric matrix product

If A is symmetric positive semidefinite, show that: (a) For any matrix B,$ BAB^T $is also positive semidefinite. (b) All the diagonal elements of A are nonnegative.
1
vote
1answer
55 views

How to determine the diagonalizability of a matrix

I am wondering there are any simple rules to determine whether a matrix $X$ is diagonalizable. I find from Wiki that tell me: $X$ is diagonalizable over the field $F$ if it has $n$ distinct ...
0
votes
4answers
62 views

How do you show that A has a eigenvalue equal to $0$ $\iff$ A is not invertible?

The only thing I can think of I that suppose A is \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} After subtracted from ...
0
votes
0answers
9 views

dependence between cholesky decomposition and positivity of matrix

Could you tell me: What is dependence between cholesky decomposition and positivity of matrix ? I know that when matrix is positive definited and symmetrical then there exists Cholesky decomposition. ...
0
votes
0answers
14 views

campbell-baker-hausdorff with one small matrix

Let $A$ and $B$ be non-commuting matrices. (Probably for the purposes of this question it is fine to assume that they are Hermitian.) I am interested in computing $\log(e^{A} e^{t B})$ in a formal ...
1
vote
2answers
78 views

Find the value of $s$ for which the matrix is positive definite

I am given the following Matrix $$ A= \begin{pmatrix} s & -4 & -4 \\ -4 & s & -4 \\ -4 & -4 & s \\ \end{pmatrix} $$ and asked to find ...
2
votes
2answers
20 views

How do you prove basis associated with col/row space?

$A$ is $m\times n$ matrix with $c_1, ...,c_n$. If $\mathrm{rank}\, A = n$, how would you show that {$A^Tc_1,...,A^Tc_n$} is a basis of $ℝ^n$? I think that in this case, since there are $n$ vectors ...
0
votes
0answers
18 views

Payoff Matrix/ How to fill positions in the matrix — Game Theory.

The Problem state: P1 and P2 each have three cards: a king, a queen, and a jack. They play their cards, one at a time, with the high card winning the trick (K>Q>J) and the playing of equal cards ...
0
votes
0answers
40 views

If $A$ and $B$ have independent columns/rows then $AB$ has them as well.

To prove this on column bases, I wrote out $r_1 c_{a_1} + \dots + r_n c_{a_n} = 0 \implies r_1 = \dots = r_n = 0$ and $r_1 c_{b_1} + \dots + r_n c_{b_n} = 0 \implies r_1 = \dots = r_n =0$. But I am ...
0
votes
0answers
18 views

Confusion on columna and row spaces.

If A is a 5$\times$4 matrix and dim(imA) is 2, how do you know that m=dim(row A)=rank A=dim(col A)=n. Is there a specific theorem for this?
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votes
1answer
39 views

How do you prove null $A$={0} $\iff$ {$c_1,…,c_n$} is independent?

So $A$ is a $m \times n$ matrix with columns $c_1, c_2...c_n$. How do you prove null $A$={0} $\iff$ {$c_1,...,c_n$} is independent? Since null $A$ ={$x$ in $ℝ^n$ |$Ax=0$}, how is it possible to say ...
0
votes
1answer
22 views

Problems on span of col and row space

Let $A=cR$ where $c\ne 0$ is a column in $ℝ^m$ and $r\ne 0$ is a row in $ℝ^n$, how do you show that row $A=$span {$r$} and col $A=$ span {$c$}. (I assume the $R$ is $r$?) Before answering the ...
0
votes
2answers
74 views

is determinant of A times A transposed bigger than or equal to zero?

We have an m by n matrix A of real numbers where n is bigger than m. Prove that determinant of A times A transposed is bigger than or equal 0.
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votes
0answers
17 views

Expected number of row operations of progressively populating a band matrix

Consider populate an $M\times M$ matrix $\mathbf{A}$ with randomly incoming length-$M$ row vectors: each row vector may have non-zero elements only at indices $i,i+1,\ldots, i+W-1$, where ...
1
vote
2answers
46 views

How would you show that $AB$=$0$ $\iff$ col $B \subseteq$ null $A$

Assuming $A$ is $m\times n$ and $B$ is $n\times m$, and to show that $AB$=$0$ $\iff$ col $B \subseteq$ null $A$, I set biconditional statement: if $AB=0$, then $R_a\cdot C_b =0$ for all A rows and ...
0
votes
1answer
37 views

What does ℝ$x$ stand for?

The question says suppose that $A$ is a $5\times4$ matrix and null($A$)=$ℝx$ for some column $x\ne0$. Can dim(im$A$)=2? First, I am not sure what $ℝx$ looks like and second, is the question equivalent ...
0
votes
2answers
24 views

Multiplication of an orthogonal matrix by its first column

I am given a real orthogonal matrix Q (nxn), where the first column of Q is the vector x (nx1) where the 2-norm of x equals 1. I am asked to prove that QTx has first entry 1 and all the others zero: ...
11
votes
3answers
566 views

Why is the matrix multiplication defined as it is? [duplicate]

Matrix multiplication is defined as: Let $A$ be a $n \times m$ matrix and $B$ a $m\times p$ matrix, the product $AB$ is defined as a matrix of size $n\times p$ such that $(AB)_i,_j = ...
0
votes
2answers
49 views

The matrix equation for solving a straight line

So, I'm looking at this paper and trying to understand where equation 5 comes from. Looking at wikipedia, I see that they would use $\mathbf{X} = (\mathbf{A^T}\mathbf{A})^{-1}\mathbf{A^T}\mathbf{y}$ ...
4
votes
1answer
70 views

Are there matrices $A$ and $B$ such that $AB = BA \neq I$

I've been learning about matrices and the identity matrix $I$. It says when $AB = BA = I$, then $A$ and $B$ are inverses of one another. Is it possible for $AB$ to equal $BA$ but not equal $I$?
0
votes
1answer
26 views

Explanation of the proof $Cov(X, Y) = E[X \cdot Y^T] - E[X] \cdot (E[Y])^T$

I was reading the proof of $$Cov(X, Y) = E[X \cdot Y^T] - E[X] \cdot (E[Y])^T$$ but there's a step (at least) that I don't understand. The proof is the following: The $(i, j)$ entry of $E[X \cdot ...
0
votes
0answers
27 views

Is there an eigen-decomposition for $X$

Let $A$ and $B$ be two rectangular $n \times m$ matrices $(n<m)$, whose elements can be 0 or 1. Also, we know that $(A+B)$ has no zero rows. Let $\bar A$ and $\bar B$ be two row-normalized matrices ...
1
vote
0answers
56 views

What is this matrix operation/symbol?

What is this matrix operation/symbol, the $\langle 0 \rangle$ in this expression? $$\Large {\left(\mathrm{s}_{a1}^{\;\;\mathrm{T}}\right)^{\langle 0 \rangle}}^{\mathrm{T}} $$ I've looked around on ...
0
votes
0answers
27 views

Finding Leading Principal Submatrices to Determine Positive Definite Matrices

I know how to compute the determinant, but I'm having trouble figuring out how to find the Leading Principal Submatrices. The example section of the book I'm reading doesn't have anything relating to ...
1
vote
0answers
33 views

Is normalized RBF always better than RBF

The question is as the title. Mathematically, I want to know does the following inequation always hold for any vector $\mathbf b$? $\mathbf b^T \mathbf B \mathbf B^+ \mathbf b \, \ge \, \mathbf b^T ...
2
votes
1answer
61 views

How to calculate a determinant of a 2x2 symmetry block matrix?

I'd like to calculate the determinant of the matrix: $$ \begin{pmatrix} -A & B^\star \\ -B & A^\star \\ \end{pmatrix} $$ $A$, $B$ are $L\times L$ complex ...
2
votes
2answers
76 views

Write $1, 2, \dots, n^2$ into a $n \times n$ square grid such that sum of each row and column is a power of 2

Let $n$ be a positive integer. Show that one cannot fill a $n \times n$ square grid with numbers $1, 2, \dots, n^2$ such that sum of numbers on each row and column is a power of 2. My attempt: Assume ...
-2
votes
1answer
47 views

Solve for x in matrix equation [closed]

Solve for $X$ $$ A^{-1} \, X \, B-B=-A^{-1} \, X $$ $$ A = \begin{bmatrix}1&-1\\1&2\end{bmatrix} \quad B = \begin{bmatrix}-1&1\\2&1\end{bmatrix} $$
-1
votes
1answer
20 views

Convenient representation for matrix equation through rvec/devec operator and kronecker

I have matrix equation in the form $y = Ax$, where $y=\begin{pmatrix} y'_1 \\ y'_2 \\ \vdots \\ y'_M \\ y''_1 \\ y''_2 \\ \vdots \\ y''_M \end{pmatrix}; $$ and, $$ x = \begin{pmatrix} x'_1 \\ ...
-1
votes
1answer
31 views

Relationship of two Linear Transform Matrices in two different equations

Assume that vector $\alpha \in \Bbb{F}^n$ and matrix $A_{n\times n}$, we define a linear transform $\mathscr{A}: \Bbb{F}^n \rightarrow \Bbb{F}^n$. $$\mathscr{A}(\alpha)=A\alpha$$ Now assume that ...
1
vote
0answers
31 views

symmetric matrix rank and leading minor

Let $A$ be a symmetric matrix with rank $r$. Is it true that there exists at least one non-zero $r\times r$ leading minor . Here, a $r\times r$ leading minor is the determinate of a submatrix of $A$ ...
1
vote
1answer
52 views

Find change of basis matrix

I'm asked to find the change of basis matrix from basis $\underline{e}$ to $\underline{f}$ given the following information: The coordinate relationship is given by: $$3y_1 = -x_1 + 4x_2 + x_3$$ ...
7
votes
2answers
103 views

Why does $A \circ {B^{ - 1}} + {A^{ - 1}} \circ B \ge 2{I_{n \times n}}$?

Let $A, B \in M_n$ be positive definite and $A \circ B = \left[ {{a_{ij}}{b_{ij}}} \right]$. Why does $A \circ {B^{ - 1}} + {A^{ - 1}} \circ B \ge 2{I_{n \times n}}$ ?
1
vote
1answer
31 views

lower bound on the minimum singular value of $\underline{\sigma} (A+B)$

what can we say about the lower bound on $\underline{\sigma}(A+B)$? Can we say the following? $\underline{\sigma}(A+B)>\underline{\sigma}(A)+\bar{\sigma}(B)$, where $\underline{\sigma}$ denotes ...
1
vote
2answers
41 views

Relationship between Nullspaces

Let $A$ be an $m*n$ matrix, and let $A'$ be an $m'*n$ matrix. Is there a relationship between:
4
votes
2answers
53 views

prove that if A is M-matrix then A is also a P-matrix

$A \in \mathbb{R}^{n \times n}$ is a $P$-$matrix$ if all its principal minors are positive. Let $I$ be the identity matrix of rank $n$. $A \in \mathbb{R}^{nxn}$ is a non-singular $M$-$matrix$ if ...
2
votes
1answer
109 views

Matrix diagonalization takes infinitely many operations?

I was reading chapter 7 (Tridiagonal Form) of the book The Symmetric Eigenvalue Problem by Parlett, and I came across a curious passage. When explaining the advantages/usefulness of the tridiagonal ...
0
votes
1answer
11 views

Finding the matrix of a relation $R$ on $X$

Given the relation: $$ R = \{(x, y)|x < y\} $$ and the ordering of $x$ is $\{ 1,2,3 \}.$ How to find the matrix of this relation? I honestly cannot understand the question. Any ...
0
votes
1answer
14 views

Fisher distance in regards to classification

I'm trying to understand what Fisher Distance actually is: Ruiz et al As well I am unsure how the writer of the paper has gotten from equation 4 to 5 by substituting in the activation function as ...
1
vote
0answers
20 views

How do I decompose the 3-D idenity matrix into lower dimensions?

I have been comfortable writing things like $\mathbb{I}_{x,y}\otimes\mathbb{I}_{z}=\mathbb{I}_{x,y,z}$ but now I realize I have no idea how the actual mechanics of it work. My first, naive attempt was ...
0
votes
1answer
25 views

Two questions on rotation matrices and eigenvalues

Let $A,B\in\mathbb{R}^{3\times 3}$ satisfy the relations: \begin{align*} (Id_{3}+A)(Id_{3}-A)^{-1} = B\ &\Longleftrightarrow(Id_{3}+A)=B(Id_{3}-A)\\ &\Longleftrightarrow\ A = ...
1
vote
0answers
18 views

Tracking Matrix Decomposition after row subset selection

Imagine we have $X$, a $n\times m$ non-negative matrix. We take the rank-r SVD of X $$ X = U\Sigma V^T$$ I'm now interested in knowing the decomposition of $X_2$, a $n_2 \times m$ matrix formed by a ...
1
vote
2answers
43 views

Question on eigenvalue properties

If $\lambda$ is eigenvalue of matrix $A$ and $B=S^{-1} A S$ show that $\lambda$ is also eigenvalue of matrix $B$. Also show that if matrix $A^{2015}$ is a zero matrix then its only eigenvalue is $0$. ...
0
votes
3answers
33 views

Test for linear dependence of 3 matrices

I have these 3 matrices and I have to check if they're linearly independent or dependent without the notion of rank or determinant. $ A=\begin{bmatrix}1&1\\0&0\end{bmatrix} ...
0
votes
0answers
17 views

Matrix obtained from derivatives

Consider the matrix $M$ of entries: $$ m_{ij} = \int_Y \dfrac{\partial f(x,y)}{\partial x^i}\,\dfrac{\partial f(x,y)}{\partial x^j}\,dy\;, $$ where $x$ is in $\Bbb{R}^n$, $f$ is a function ...
1
vote
1answer
48 views

If $A \ge B$ then $A[α] \ge B[α]$

Let $A, B \in M_n$ be Hermitian. If $A \ge B$ and $α ⊂ {\rm{\{ 1, }}{\rm{. }}{\rm{. }}{\rm{. , n\} }}$. Why does $A[α] \ge B[α]$ ? (Note: $A[α]$ is submatrix of $A$ )
0
votes
2answers
67 views

If $\left( {\begin{array}{*{20}{c}} A & B \\ B & A \\ \end{array}} \right) \ge 0 \Rightarrow A \ge B$

Let $A,B\in M_n$ and $\left( {\begin{array}{*{20}{c}} A & B \\ B & A \\ \end{array}} \right) \ge 0$ Why does $A \ge B$ ?
-1
votes
1answer
37 views

Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial, find the matrix

Let $M$ be an $n\times n$ diagonalizable matrix with characteristic polynomial $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$, Find the matrix $a_nM^n+a_{n-1}M^{n-1}+...+a_1M+a_oI$, where $I$ is the ...
0
votes
0answers
19 views

Connection between 2 different norms of normalized eigenvector

For a matrix $ A\in \mathbb{R}^{n \times n} $, we calculate its normalized eigenvector (in terms of $2$ norm) say $v_1,v_2,\cdots,v_n$. Therefore, $\|v_i\|_2 = 1$ for $i=\{1,2,\cdots,n\}$. In the ...
3
votes
2answers
41 views

Matrix Equations

I have worked it out and determined that both equations hold. I am wondering why this is the case. Is there a reason why the equation holds for these two types of matrices?