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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1
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4answers
133 views

Upper bound for the rank of a nilpotent matrix , if $A^2 \ne 0$

I came across the fact that the rank of a nxn-matrix A with $A^2=0$ is at most $\frac{n}{2}$. The easiest way to proof this is using the inequality $rank(A) + rank(B) -n \le rank(AB)$. With $A=B$ and ...
0
votes
0answers
20 views

Is it a Wishart matrix? [duplicate]

Now I am having some problems about the Wishart matrix. Please help me, thank you! We know that $m \times m$ random matrix $\boldsymbol{A} = \boldsymbol{H} \boldsymbol{H}^H$ is a (central) ...
2
votes
0answers
30 views

Is it a Wishart matrix or not?

Now I am having some problems about the Wishart matrix. Please help me, thank you! We know that $m \times m$ random matrix $\boldsymbol{A} = \boldsymbol{H} \boldsymbol{H}^H$ is a (central) ...
1
vote
1answer
25 views

Cross-product is a left singular vector?

Assume A is a 3x2 matrix with rank(A)=2. u1 and u2 are already left singular vectors... How would I go about proving that the cross-product of the two is also a left singular vector? Hints would be ...
0
votes
1answer
33 views

Cholesky factorization and non-positive definite matrices

When Cholesky factorization fails, is there an alternative method to obtain the $\mathbf{L}$ matrix in: $\mathbf{A}=\mathbf{L}\mathbf{L}^{*}$ I'm dealing with a matrix not guaranteed to be ...
1
vote
2answers
110 views

On the nilpotence of the matrix $AB-BA$ [closed]

Given $n\times n$ matrices $A,B$ satisfy: $rank(AB-BA)=1$ Prove that $(AB-BA)^{2}=0$ Generalize the problem if possible. Any solution not mention Jordan canonical form would be appreciated!
-1
votes
2answers
58 views

When does the system of equations have initly solution, no solutions, and only one solution [closed]

I am trying to solve this math problem. So far I am bad at it. I nedd to determine for what given x does the system of linear equations has no solutions, has one solution, or infinetly many solutions. ...
0
votes
1answer
59 views

Finding the exponential relation between two 4x4 transition matrices

Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given $4\times4$ transition matrices, representing transitions in three dimensions with the fourth ...
6
votes
3answers
228 views

$A+A^2B+B=0$ implies $A^2+I$ invertible?

Let $A$ and $B$ be two square matrices over a field such that $A+A^2B+B=0$. Is it true that $A^2+I$ is always invertible ?
3
votes
3answers
55 views

Homotopy of Involutory Matrices?

I want to construct a homotopy from the matrix $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ ...
1
vote
1answer
45 views

Some “Product” of Positive Definite Matrices

I could remember that if $A,B$ are two positive definite matrices, then $(a_{ij}b_{ij})$ is positive definite also. But I could not see how to prove it then.
0
votes
3answers
76 views

About semipositive definite matrix

Suppose $A$ and $B$ are positive semidefinite matrices $A \ge B\ge 0$ Is the statement $A^2\geq B^2$ true or false? Why? $\geq$ means nonnegative pointwise
0
votes
1answer
35 views

Study endomorphism diagonalization

Given an endomorphism whose matrix is: $\begin{pmatrix} 1+a & -a & a \\ 2+a & -a & a-1 \\ 2 & -1 & 0 \end{pmatrix}$ How can I study if it's diagonalizable or not depending ...
1
vote
0answers
24 views

Fast way to find exponential of a matrix dot product where one of them is diagonal

Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ ...
0
votes
0answers
31 views

Explain why a matrix is orthogonally diagonalisable.

If people could tell me if I'm on the right track on and give me a push in the right direction for the ones I'm unsure of that would be much appreciated. Let A$\epsilon$M$_{3}$($\mathbb{R})$ and ...
2
votes
1answer
45 views

Matrix time derivative

Given a complex, square matrix $A$ that is diagonalizable, is it possible to write a simple formula for $\frac{d}{dt} A^t$ for a real, positive power $t$ and for $A$ a smooth function of $t$?
0
votes
0answers
48 views

Finding true bearings?

What is the true north bearing of NNE on 16 point cardinac compass? I just wanna know that is there any exact bearing or do we have to only give an approximate bearing?
0
votes
1answer
40 views

A and B are nxn matrices. A = $B^{T}B$ Prove that if rank(B)=n, A is pos def, and if rank(B)<b, A is pos semi-def.

A and B are nxn matrices. A = $B^{T}B$ Prove that if rank(B)=n, A is positive definite, and if rank(B) My current understanding is that if rank(B)=n, then rank($B^{T}B$)=n then rank(A)=n, making A ...
0
votes
3answers
34 views

Prove that $b^2 pr =q^2 ac$ using matrices

Let $i_1,i_2$ and $j_1,j_2$ be non-zero real roots of $ax^2+bx+c$ and $px^2+qx+r$ respectively, where a,p $\neq$0. If the system of equations $ i_1y+i_2z=0$ and $j_1y+j_2z=0$ has a non-trivial ...
1
vote
1answer
183 views

Find a 2x2 matrix with positive eigenvalues, but a negative quadratic form for some x in $R^{2}$

Find a 2x2 matrix with real and positive eigenvalues, but a negative definite quadratic form. Also, find a 2x2 matrix with real and positive eigenvalues, but an indefinite quadratic form. Isn't this ...
0
votes
1answer
65 views

Connection between Eigenvectors and linear equations

I'm trying to understand the connection between Eigenvectors/Eigenvalues and linear equations: $Ax=b$ If you are given the eigenvectors and eigenvalues of $A$, can you construct the solution for the ...
0
votes
0answers
45 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
83
votes
0answers
2k views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
0
votes
1answer
21 views

invert lower triangular matrix

I am sorry if the question is simple,am trying to find the quicker method to invert a triangular matrix. Could you please provide some references where i could refer? Moreover,is there any known way ...
1
vote
1answer
48 views

Solving an augmented coefficient matrix so there are infinitely solutions

I am trying to figure out this math problem. For what values $a,b$ does the linear system have infinitely many solutions? This is the matrix $$ \left[ \begin{array}{ccc|c} ...
0
votes
1answer
32 views

what is the name of this matrix? does it have any special characteristics?

does anyone know the name of this matrix or if it has any special characteristics or how to calculate its inverse efficiently e.g. in a closed-form? [ \begin{array}{llllll} ...
0
votes
0answers
28 views

Correlating random numbers seems to skew the data

I am trying to generate a series of correlated random numbers that represent currency exchange rates for a Monte-Carlo simulation. I am attempting to do this via a Cholesky decomposition of the ...
2
votes
1answer
58 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
1
vote
2answers
65 views

For which values of $a, b$ does the system of equations NOT have any solutions?

I am trying to solve this math problem: For which values of $a$ and $b$ does the linear system represented by the augmented matrix not have any solution? $$ \left[ \begin{array}{ccc|c} ...
2
votes
2answers
115 views

Square root of a $3\times3$ matrix

Here is $3\times3$ matrix$$\begin{pmatrix} 0& 0& 1\\ 0 & -1 & 0\\ 1& 0 & 0\end{pmatrix}$$ How can I solve this by using Cayley-Hamilton? I know how to ...
2
votes
1answer
37 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
0
votes
1answer
19 views

Difference between matrices with altered eigenvalues

Given two p.s.d. matrices $X_1$ and $X_2$ with eigen decomposition $X_1 = U_1V_1U_1^T$ and $X_2 = U_2V_2U_2^T$ and a constant $\lambda > 0$ Now consider an altered version of the eigenvalue ...
4
votes
3answers
94 views

Cross product: matrix transformation identity

How can one prove the following identity of the cross product? $$(M a)\times (M b)=\det(M) (M^{\rm T})^{-1}(a\times b)$$ $a$ and $b$ are 3-vectors, and $M$ is an invertible real 3x3 matrix.
4
votes
2answers
121 views

Prove that $\det(M-I)=0$

$M$ is a $3 \times 3$ matrix such that $\det(M)=1$ and $MM^T $= I, where $I$ is the identity matrix. Prove that $\det(M-I)=0$ I tried to take $M$ $=$ $$ \begin{pmatrix} a &b & c \\ ...
1
vote
3answers
90 views

How find the matrix $K$ such $AKB=C$

Question: Find a matrix $K$ such that $$AKB=C$$ given that $$A=\begin{bmatrix} 1&4\\ -2&3\\ 1&-2 \end{bmatrix},B=\begin{bmatrix} 2&0&0\\ 0&1&-1 \end{bmatrix} ...
1
vote
0answers
34 views

Characterize matrices complying to certain constraints.

Characterize those matrices $ X $ (real symmetric), $ Y$ (real positive definite), $ R$ (real diagonal) and $ F $ (real diagonal) such that $ XRY + YRX = 0$ , (1) $ YRY - XRX = F$ . (2) ...
0
votes
1answer
75 views

Cramer's rule and linear dependence/independence test

When you have the system of equations: $$ax + by = e\\cx + dy = f$$ And you do some row operations to eliminate $y$, we get: $$x = \frac{ed-bf}{ad-bc}\tag{1}$$ If we eliminate $x$ we get: $$y = ...
1
vote
3answers
65 views

Tridiagonal Symmetric Matrix

Could anyone help me to find the determinant of a $N\times N$ tri-diagonal symmetric matrix, named "$A[i,j]$" with $i,j \le N$, that has all the elements in the super-diagonal and sub-diagonal equal, ...
0
votes
1answer
47 views

Are circulant matrices open

Are the set of positive definite symmetric circulant matrices open in the set of positive definite symmetric matrices?
1
vote
0answers
37 views

Matrix with trace zero [duplicate]

Question is that : Suppose a matrix $A\in M_n(\mathbb{C})$ is a commutator by which i mean $A=BC-CB$ for some $B,C\in M_n(\mathbb{C})$ then we see that Trace of $A$ is $0$ But Suppose a matrix is of ...
1
vote
0answers
22 views

Matrix that shows how close indices are to each other

I have a vector of n words, not all distinct. [the, quick, brown, fox, jumps, over, the, lazy, dog] From this, I want to make an n×n matrix that shows how ...
1
vote
4answers
70 views

TT* + I is invertible

I've the following exercise which I can't solve: Prove that: $$ AA^* + I $$ is invertible for all Matrix $ A $ in finite-dimensional field $V$ with inner product. $ A^* $ is the adjoint operator. ...
9
votes
2answers
124 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
2
votes
2answers
197 views

Matrix Manipulation Question

Given that A $\begin{bmatrix}1\\-2\\1\end{bmatrix}=\begin{bmatrix}2\\-3\\-5\end{bmatrix}$ and A $\begin{bmatrix}1\\4\\-2\end{bmatrix}=\begin{bmatrix}0\\2\\-1\end{bmatrix}$ find A $\begin{bmatrix}1 ...
0
votes
2answers
22 views

Solving a system of equations using the inverse of the coefficient matrix

Let A be 2*2 matrix with A={{1 1}, {2 -2}} use A^-1 to solve the matrix equation A{{x_1 x_2}} ={{3 5}} I got A^-1 ={{1/2 1/4 },{1/2 -1/2}} just need to know how to solve equation .
1
vote
1answer
335 views

find an elementary matrix $E$ such that $EA=B$

Let matrix $$A = \begin{pmatrix} 1 & 2 & 0 \\ -3 & 1 & 1 \\ 0 & 4 & 2 \end{pmatrix} $$ and $$B = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 7 & 1 \\ 0 & 4 ...
0
votes
0answers
34 views

Can I solve this problem with matrices?

So I have some two dimensional data sets thats I want to analyse. They can be viewed in 2D form as below: $M1$: $$\begin{matrix}00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 ...
1
vote
1answer
29 views

$A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$

How can i prove that $A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$ i stuck in this problem i know that if $A$ ...
0
votes
2answers
41 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
14
votes
4answers
429 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...