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
0answers
201 views

Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
1
vote
1answer
84 views

confusion about rank and nullity of matrix and rank-nullity theorem

I can see that rank + nullity = number of columns of the matrix. Does that mean the dimension of matrix is the number of columns? Isn't dimension of a $m\times n$ matrix $mn$? Thanks.
4
votes
2answers
217 views

Injectivity of $A-\lambda I$

I'm reading a paper on determinants and on one point the author states that: A complex number $\lambda$ is called an eigenvalue of matrix $A$ if $A-\lambda I$ is not injective. Why is this? Could ...
0
votes
2answers
69 views

Rank of a 2 x 2 matrix

Prove that the rotation matrix is invertible. Let $$ \begin{pmatrix} \cos 2\pi t & \cos \frac{\pi}{6}t\\ \sin 2\pi t & \sin \frac{\pi}{6}t\\ \end{pmatrix} $$ What is the rank of the ...
1
vote
0answers
653 views

What do you call the product of a matrix's diagonal elements?

The trace of $A$, an $N\times N$ matrix, is $\sum_{i=1}^N A_{ii}$. What do you call $\prod_{i=1}^N A_{ii}$?
1
vote
1answer
46 views

Solving a system of nonlinear equations

Consider a machine that operates using the following equation: O $_3\ _×\ _1 = $ X $_3\ _×\ _3 ×$ [I $_3\ _×\ _1$ − Y $_3\ _×\ _1 $] + Y $_3\ _×\ _1,$ where I $_3\ _×\ _1$ is the input and O $_3\ ...
0
votes
0answers
64 views

Sum of two matrices

Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define ...
0
votes
2answers
43 views

Product of invertible matrix and basis

I am confused about this problem. Let $S$ be a subspace of $\mathbb{R}^k$ of dimension $m\leq k$ and $\{b_1,...,b_m\}$ is a basis of $S$. Now, given an invertible matrix $A$. I have a feeling the set ...
2
votes
2answers
57 views

Calculate the eigenvectors

We calculate the eigenvectors for the matrix $$ \begin{equation*} \mathbf{A} = \left( \begin{array}{ccc} 2 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & -1 & 3 \\ ...
4
votes
1answer
96 views

Name of a special matrix

I have a matrix which is kind of symmetrical with the other diagonal, i.e., something like $$A = \left[ \begin{array}{c c c c} a & b & c & d \\ e & f & g & c \\ h ...
1
vote
1answer
33 views

$A$ is singular and has nonzero row sums that are the same for every row. then $A+\lambda 11^{\prime}$ is singular

$A$ is singular and has nonzero row sums that are the same for every row. then $A+\lambda 11^{\prime}$ is singular, where $1$ is a vector of one's. Let $A=\{a_{i1} a_{i2}\ldots a_{i(n-1)}\space ...
0
votes
1answer
58 views

relations between two linear operators

Let $\alpha,\beta$ be linear operators on a finite dimensional vector space $V$ over field $F$. Let $\gamma=\alpha\circ\beta$ and $\delta=\beta\circ\alpha$. Prove that: (1). $m_\delta(x)$ divides ...
0
votes
0answers
135 views

Wikipedia's Cayley Table and Pictures for 3 by 3 Permutation Matrices

Are there any explanations or clarifications of the pictures at https://en.wikipedia.org/wiki/Permutation_matrix#Permutation_of_rows_and_columns? ...
1
vote
1answer
42 views

Prove two covariance matrices have same eigen values?

There are two covariance matrices $U$ and $V$ such that $U= X^\top X$ and $V = XX^\top$ where $X$ is a $d\times N$ matrix. How can I prove that $U$ and $V$ have the same non-zero eigenvalues?
2
votes
1answer
95 views

about symmetric and hermitian matrices

Using the frobenius norm $\langle A,B\rangle=\textrm{Tr}(B^{\star}A)$, where $B^{\star}=\overline{(B^T)}$, How can I prove that $\mathbb{S}(\mathbb{C})^{\perp}=A\mathbb{S}(\mathbb{C})$, that is that ...
0
votes
1answer
287 views

Invariant subspaces linear algebra problem

So I have this matrix of a linear map $ \varphi$: $ \begin{pmatrix} 8 &-6 &-1 \\ 5 &-3 &-1 \\ 4 & -2 & -2 \end{pmatrix} $, where $ \varphi \in (\mathbb{Q}^3)$. I have to ...
8
votes
5answers
670 views

What is the motivation defining Matrix Similarity?

I'm taking the course Linear Algebra 1, and recently we've learned about matrix similarity. What is the motivation defining it? or, What are the uses/applications for this definition? Thanks
4
votes
3answers
168 views

Diagonalization and eigenvalues

Let $A$ $\in M_3(\mathbb R)$ which is not a diagonal matrix. Pick out the cases when $A$ is diagonalizable over $\mathbb R$: a. when $A^2 = A$; b. when $(A - 3I)^2 = 0$; c. when $A^2 + I = 0$. I ...
1
vote
1answer
45 views

Is there a smarter way to solve? Matrix multiplication

So i have this Problem: I know how to solve it basically finding the inverse matrices and so on, but i was wondering if there isn't a quicker and smarted way, because the matrix on the right is ...
0
votes
2answers
79 views

Rank of triangular matrices of special form

Am I right about the following matrices: $$A= \left[\begin{matrix} x&x&x&x\\ 0&x&x&x\\ 0&0&0&0\\ \end{matrix}\right] $$ $$B= \left[\begin{matrix} x&x&x\\ ...
2
votes
0answers
201 views

Diagonalization of Vandermonde matrix

Is there a method to diagonalize (at least some) $ n \times n $ Vandermonde matrices? For example invertible matrices which has method to invert them with Cramer method for example, but there is some ...
0
votes
1answer
51 views

Calculating a linear map using a transformation matrix

Let: $T:\mathcal M_{2\times2 }(\mathbb R)\to \mathbb R^3$ be a linear map that has the following transformation matrix in relation to the following bases. Calculate $T\begin{pmatrix} 1 ...
0
votes
2answers
157 views

How to find out if it is possible to contruct a binary matrix with given row and column sums.

How to find out if it is possible to contruct a binary matrix with given row and column sums. Input : The first row of input contains two numbers 1≤m,n≤1000, the number of rows and columns of the ...
4
votes
0answers
91 views

Non Existence of matrices $A,B\in M_n(\mathbb{R})$ such that $(I-(AB-BA))^n=0$

Question is to Prove: Non Existence of matrices $A,B\in M_n(\mathbb{R})$ such that $(I-(AB-BA))^n=0$. This question has already been asked already but then i am asking for clarification of another ...
1
vote
2answers
145 views

Matrix diagonalizable or not [duplicate]

Let $A$ is in $M_3(\mathbb R^3)$ which is not a diagonal matrix. Pick out the cases when $A $ is diagonalizable over $\mathbb R$: a. when $A^2=A$; b. when $(A-3I)^2=0$; c. when $A^2+I=0$. My ...
1
vote
1answer
30 views

How to name a matrix with restricted input values?

How should I refer to a matrix with a restricted domain of possible values that can be stored inside?
3
votes
0answers
63 views

Partial order on the orbits of the variety of commuting nilpotent matrices

The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are ...
5
votes
2answers
78 views

Necessary condition for have same rank

Let $P,Q$ real $n\times n$ matrices such that $P^2=P$ , $Q^2=Q$ and $I-P-Q$ is an invertible matrix. Prove that $P$ and $Q$ have the same rank. Some help with this please , happy year and thanks.
5
votes
4answers
167 views

What's a matrix?

What is a matrix exactly? What are matrices used for? I have read some of the Wikipedia article, but since my math knowledge is pretty basic, I didn't understand much. Could you explain to me in ...
5
votes
5answers
683 views

What is the intuition for using definiteness to compare matrices?

If $a$ and $b$ are two numbers on the real line, we compare $a$ and $b$ by knowing which of them comes first as we move from $-\infty$ to $\infty$ on the real line. However when $A$ and $B$ are ...
0
votes
1answer
61 views

Hermitian Matrix n x n

Sea M an Hermitian matrix that satisfies the condition : $$M^5 + M^3 + M = I $$ with I the identity matrix n x n. How can i prove that $M = I$. Please help...
3
votes
0answers
91 views

Minimal polynomial of eigenvector entries

Suppose that $M$ is a matrix with integer entries and that $\lambda$, $v = (v_1, \ldots, v_n)^{T}$ are an eigenvalue and eigenvector of $M$. Then $\lambda$ is an algebraic number and we can see this ...
5
votes
0answers
154 views

How to prove the positive-definiteness of a symmetric Toeplitz matrix like this?

Define a symmetric Toeplitz matrix by $$\begin{pmatrix}c_1 & c_2 & c_3 & \cdots & c_n\\c_2 & c_1 & c_2 & \cdots & c_{n-1}\\c_3 & c_2 & c_1 & \cdots ...
2
votes
1answer
49 views

matrix norm derivative with respect a parameter

What is the result of the following expression $\frac{d}{dt}\left( \|A(t)-B(t) \|\right) $, where $\|\cdot \|$ can be for instance the Frobenius norm?
2
votes
1answer
97 views

How to prove this inequation of matrix norm?

Suppose a square matrix $A=(a_{ij})_{n\times n}$ is irreducible. It is given that there exits $i_0$ for $$\sum_{j=1}^{n}{|a_{i_0j}|}<\|A\|_{\infty}$$ Out goal is to prove: $$\rho(A)< ...
0
votes
1answer
131 views

Skew-symmetric matrix mapping vector to orthogonal vector

I am trying to prove that if $-a^2,-b^2$ are distinct non-zero eigenvalues of $M^2$ where $M$ is a real square skew-symmetric matrix, then there exist orthogonal vectors ...
0
votes
1answer
79 views

Contrary interpretations of Least Squares for Regression

According to the original thought, our goal is to minimize the quadratic error $$\min\{\frac{1}{2}(Ax-b)^2 \}$$ Then, we search the extremum by the derivation of $x$ $$A^T(Ax-b)=0$$ $$A^TAx=A^Tb$$ ...
3
votes
1answer
56 views

Matrix Exponent

I know that the exponential function for an $n\times n$ non-degenerate square matrix $A$ is: $$e^{At}=S \operatorname{diag}(e^{\lambda_1t},\dots,e^{\lambda_nt}) \space S^{-1}, $$ where $S$ is the ...
3
votes
3answers
122 views

Determinant of a special $n\times n$ matrix [duplicate]

Compute the determinant of the nun matrix: $$ \begin{pmatrix} 2 & 1 & \ldots & 1 \\ 1 & 2 & \ldots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 ...
0
votes
1answer
50 views

A problem about the sign of pitch angle in rotation.

The sign of pitch angle in rotation. In the Yaw-Pitch-Roll convention, for example, XYZ-Zup coordinates system. When I'm reading page 194-195 of 《Linear Algebra with Applications》 7ED, of STEVEN ...
1
vote
1answer
37 views

Equivalence involving diagonalization of block matrix

Given $T^\top L T = diag(A, B, C)$ being a block diagonalization with $L$ symmetric $A,B$ positive definite (p.d.) $T = \left( \begin{array}{ccc}I & 0 & T_2 \\ T_1 & I & T_3 \\ 0 ...
1
vote
1answer
40 views

Invertibility of a Matrix Given Some Conditions

Let $A$ and $B$ be different $n\times n$ matrices with real entries. Suppose that $A^3=B^3$ and $A^2B=B^2A$, can $A^2+B^2$ be invertible?
2
votes
0answers
29 views

$C$ Hermitian, show $\text{Tr}(C)=0 \iff \exists P,Q \text{ Hermitian s.t.} \,\, PQ-QP=iC$ [duplicate]

Let $C$ be an $n\times n$ Hermitian matrix. Show that $$\text{Tr}(C)=0 \iff \exists P,Q \text{ Hermitian s.t.} \,\, PQ-QP=iC.$$ Ideas: The right-to-left direction I have no problem with. For the ...
1
vote
1answer
49 views

Proving a determinant inequality

Let $A$ be a square matrix in $M_n(\mathbb R)$. Prove that: $$det(A^2+I_n) \ge 0$$ I wrote $A^2+I_n=A^2 I_n+I_n=I_n(A^2+1)$: $$det(I_n)\cdot det(A^2+1)=det(A^2+1)$$ How can I prove that is $\ge 0$ ...
4
votes
2answers
697 views

Cauchy-Schwarz matrix inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ and $Y$ are random vectors, is there a way to bound ...
0
votes
3answers
50 views

Can functionals be independent?

I was asked this question: Let $A$ be a 2x2 matrix with real entries. Let $\xi_1, \xi_2, \xi_3$ be functionals from $mat_{2x2}(\mathbb R)\to\mathbb R$ such that: $\xi_1(A)= tr(A)$, ...
1
vote
0answers
282 views

Area optimization: Packing rectangles inside rectangle

Background: I am scrambling to figure out the optimization algorithm to build sprite image, which is essentially a big container rectangular image, with multiple rectangular images. I have found an ...
0
votes
1answer
641 views

Norm of Matrix transpose

I have a problem below: Let $\|\cdot\|$ denotes the norm matrix \begin{equation} \|A\|=\max \frac {\|Ax\|}{\|x\|}, \end{equation} for every $A$. Now suppose that $H: \mathbb{R}^k \rightarrow ...
1
vote
3answers
185 views

Sum of the matrix series

Let $A\in\mathbb R^{n\times n}$ be a symmetric matrix which $0\preceq A\preceq I$ ($I$ is identity matrix), and $w_k\in\mathbb R^n$ are arbitrary certain vectors which $\|w_k\|\leq1,\,\,k=0,1,\ldots$ ...
2
votes
2answers
391 views

Given a perturbation of a symmetric matrix, find an expansion for the eigenvalues

Let $A$ be a real, symmetrix $n\times n$ matrix with $n$ distinct, non-zero eigenvalues, and let $V$ be a real, symmetric $n\times n$ matrix. Consider $A_{\varepsilon}=A+\varepsilon V$, a ...