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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3
votes
0answers
67 views

How to check whether it is possible to solve this problem?

We have a matrix with numbers. We can add $1$ to any selected element and this action adds $1$ to elements according to some function, which I'll call the $X$-function. For instance, $1$ could be ...
0
votes
2answers
60 views

What is $[T]^{\scr{C}}_{\scr{B}}$?

What does it mean for $[T]^{\scr{C}}_{\scr{B}}\in M_{m\times n}(F)$ to be a matrix of $T$ in basis $\scr{B}$ in $\scr{C}$?
1
vote
1answer
78 views

Show that $P = Q^2$

Suppose $P$ is a positive semi-definite $n\times n$ matrix. Show that there exists a unique positive semi-definite matrix $Q$ such that $P = Q^2$. In class we've been going over singular value ...
1
vote
0answers
611 views

Does a Symmetric Matrix with main diagonal zero is classified into a separate type of its own? And does it have a particular name?

For example, I have a Matrix as shown below. Does this Matrix belong to a particular type. I am CS student and not familiar with types of Matrices. I am researching to know the particular Matrix type ...
4
votes
2answers
475 views

Matrix Calculus in Least-Square method

In the proof of matrix solution of Least Square Method, I see some matrix calculus, which I have no clue. Can anyone explain to me or recommend me a good link to study this sort of matrix calculus? ...
2
votes
1answer
1k views

What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?

Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this ...
0
votes
2answers
41 views

Finding vector of co-ordinates

How do we find a co-ordinate vector in Algebra? For example, given: $$ \begin{align*} \left(\begin{matrix} 2 & -3 \\ 0 & -4 \end{matrix}\right) & \left(\begin{matrix} v_1 \\ v_2 ...
4
votes
2answers
69 views

Does there exist an $n\times n$ real matrix $A$ exist such that $e^{e^{A}} - I_n$ is singular?

I have one doubt whether an $n\times n$ real matrix $A$ exist such that $e^{e^{A}} - I_n$ is singular? I think I have to show that 1 is the eigenvalue of $e^{e^{A}}$ in case answer is yes. But I am ...
2
votes
1answer
39 views

Find the basis for $\text{Im} \, ψ$ of a matrix transformation

Let $\psi\colon\mathrm{Mat}_{ 2\times 2 }(\mathbb R) \to \mathrm{Mat}_{ 2\times 2 }(\mathbb R)$ be defined by $$\psi\colon \pmatrix{a&b\\c&d}\to\pmatrix{a+b&a-c\\a+c&b-c}.$$ Find ...
2
votes
1answer
1k views

Rotation Matrices - Rotating a point on a graph

I'm trying to understand how rotation matrices work in Linear Algebra... I don't think I'm visualizing it correctly though... I'd like to rotate a point (-2, 1) around a graph... the point (-2, 1) ...
0
votes
1answer
76 views

proof about deteminant of a complex linear transformation

say I have a linear space $V$ over $\Bbb C$ and a linear transformation $T:V \to V$ such that $T=A+iB$ where $A,B \in \Bbb R^{n \times n}$ I proved already that $T_\Bbb R = \begin{pmatrix} A & -B ...
0
votes
1answer
52 views

Question on matrix stability and definiteness

Is it true that $AX + XA^*$ is necessarily negative definite if $A$ is stable and $X$ is positive definite. If so, give a proof. If not produce a counter example. I am unable to solve it.
0
votes
1answer
49 views

Question related to non-negative matrix

I am facing a problem in this question. Let $y$ be fixed. Prove that $x \geq y$ implies $Ax \geq y$ for all $x$ if and only if $A \geq 0$ and $Ay \geq y$. A≥0 implies each entry in the matrix A ...
2
votes
2answers
323 views

Find the basis for kernel of a matrix transformation

$Let \ ψ\ :{Mat }_{ 2x2 }(ℝ)\ →\ { Mat }_{ 2x2 }(ℝ)\ be\ defined\ by$ $ ψ : \pmatrix{a&b\\c&d}→\pmatrix{a+b&a-c\\a+c&b-c}$ . Find basis for ker ψ I'm not sure how to do it for a ...
1
vote
0answers
36 views

Demonstrate basic property of Hermitian

I want to demonstrate that $(A|v\rangle)^* = \langle v|A^*$ Given only this $(AB)^*=A^*B^*$ and $(|v\rangle, A|w\rangle) = (A^*|v\rangle, |w\rangle)$ Is it possible? The difficulty is that i don't ...
0
votes
5answers
127 views

Having Difficulty Finding Eigenvectors

I'm having a lot of problems with the following problem in Steven J. Leon's "Linear Algebra with Applications" 8th edition. Problem 6.1.1.I asks the reader to "find the eigenvalues and the ...
1
vote
0answers
50 views

Linear Algebra: Linear transformation and eigenvalues [duplicate]

Hi could some one please help. I am having problems proving this. Let $A$ be an $n \times n$ matrix with complex entries and let $f (t) =\det(A - tI)$ be its characteristic polynomial. Prove ...
0
votes
1answer
152 views

Vectors Subspace Basis / not all linearly independent

If I have a matrix $$A = \begin{vmatrix} a & 1 & 1 & 1 \\ 1 & b & 1 & 1 \\ 1 & 1 & c & 1 \\ 1 & 1 & 1 & d \end{vmatrix} = 0$$ Edit Note: I want ...
2
votes
1answer
766 views

Cholesky factorization for a non-positive semidefinite matrix

I was trying to build a random positive definite matrix and I came across this case. $a=\begin{bmatrix}5 & 5\\ 16 & 7\end{bmatrix} $. The eigenvalues for this matrix are $eig=(-3, 15)$. So ...
1
vote
2answers
220 views

deteminant of a block skew-symmetric matrix

If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix} how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix} so the determinant is ...
0
votes
3answers
264 views

Eigenvalue of all matrix with all duplicate rows

I've been asked to determine an eigenvalue given the following matrix without any calculations: $\begin{bmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{bmatrix}$ My ...
0
votes
1answer
508 views

Finding the Matrix Power of a matrix and limit

Find the matrix power, $A^k$, of $$A=\begin{pmatrix}a & 1-a \\ b & 1-b\end{pmatrix}$$ $$D=P^{-1}AP$$ $$A^k=PD^kP^{-1}$$ I think that $$P=\begin{pmatrix}1 & \frac{a-1}{b} \\ 1 & ...
1
vote
1answer
31 views

Express Matrix 1-norm into other matrix operation

Consider a square matrix $A$, The Forbenius norm $||A||_F$ can be expressed into $\sqrt{Tr(AA^T)}$ Is there are similar expression of matrix one norm $||A||_1$?
4
votes
1answer
95 views

What graph Laplacians commute

I know that the graph Laplacian of a fully connected graph commutes with the Laplacian of any other graph. Is there any theorem stating something similar about some more general family of graphs? ...
0
votes
1answer
76 views

Local expansion of a polynomial

I've been staring at this local expansion, from a paper, of a multivariate polynomial but I can't quite understand it. It goes as follows: $\mathcal{C}$: Field of Complex numbers Let $x=(x_1, ...
3
votes
1answer
64 views

Construct matrix

Let $B$ any square matrix. Is possible to construct an invertible matrix $Q_B$ such that $$\|Q_BBQ_B^{-1}\|_2\ \leq\ \rho(B)?$$ Thanks in advance for the help. Edit: $Q_B$ only need to be ...
0
votes
1answer
70 views

Show that $\langle x,y\rangle_T$ is an inner product

Let $T$ be an $n\times n$ matrix. Define a form on $\mathbb{R^n}$ by $\langle x,y\rangle_T=\langle Tx,Ty\rangle$ for $x,y \in \mathbb{R^n}$. Show that this is an inner product iff T is invertable. My ...
2
votes
3answers
228 views

An $r\times r$ submatrix of independent rows and independent columns is invertible (Michael Artin's book).

Let $A$ be an $m \times n$ matrix of rank $r$, let $I$ be a set of row indices such that the corresponding rows of $A$ are independent and let $J$ be a set of $r$ column indices such that the ...
0
votes
1answer
99 views

Jordan Canonical form of the matrix

Using Jordan Canonical theorem, prove that the matrix sequence $\lbrace A^k\rbrace\rightarrow 0$ (i.e matrix sequence tending to zero matrix) if and only if $|\lambda_i|<1$ i.e the absolute value ...
1
vote
3answers
78 views

Rank of matrix of order $2 \times 2$ and $3 \times 3$

How Can I calculate Rank of matrix Using echlon Method:: $(a)\;\; \begin{pmatrix} 1 & -1\\ 2 & 3 \end{pmatrix}$ $(b)\;\; \begin{pmatrix} 2 & 1\\ 7 & 4 \end{pmatrix}$ $(c)\;\; ...
1
vote
0answers
71 views

Differences between a norm in $\mathbb{C}^n$ and a norm in $\mathbb{R}^n$

Consider the definitions of matrix norm and subordinate matrix norm from Matrix Norm set #2 and let $A$ a real matrix and $\|\cdot\|$ a vector norm over $\mathbb{C}^n$. Define \begin{eqnarray*} ...
2
votes
1answer
518 views

Monotone matrix

A real matrix A is called monotone if Ax≥0 implies x≥0. If inverse of A exists and is real, then prove that A is monotone if and only if inverse of A≥0. (x≥0 means x is a column vector whose all ...
4
votes
2answers
188 views

Linear Algebra matrix $Ax=b$ true or false nullspace

$Ax=b$ $m$ number of Rows $n$ number of columns true or false A) If $n > m$, given any $b$ you can always solve $Ax=b$. The answer: False. Counterexample: A is the zero matrix. We have ...
0
votes
1answer
69 views

Question related to Hermitian Matrix

If A is a Hermitian matrix of order n-cross-n. Then show that the rank of A is equal to the number of nonzero eigen values of A,but this is not generally true for non- Hermitian matrices.
2
votes
1answer
1k views

If $A$ is an invertible skew-symmetric matrix, then prove $A^{-1}$ is also skew symmetric

Let $A$ be an invertible skew-symmetric $(2n \times 2n)$-matrix. Prove that $A^{-1}$ is also skew-symmetric. (You may assume that $(AB)^T = B^TA^T$). I did this with a $2 \times 2$ matrix and got ...
1
vote
1answer
50 views

My question is related to advanced course in matrix theory.

Suppose Ax=b is any underdetermined linear system. Prove that the minimum-norm solution to an underdetermined system can be obtained by projecting any solution of the system onto range space of ...
6
votes
1answer
105 views

My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible

Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial. Here's my ...
1
vote
1answer
224 views

How to deduce the psition mapping of entries of a matrix?

I would be thankful if any peer shed light on me. Assume that the mapping of a set is unknown. By knowing n number of E element sets and the transformed sets with positioned elements, How can I ...
0
votes
2answers
48 views

Polynomial on Matrices

Assume for any $t \in \mathbb{C}$, where $\mathbb{C}$ denotes set of complex numbers, we have $P(t) = 1$, where $P$ is a polynomial. Does it imply that for any matrix $A$, $P(A)=I$?
4
votes
1answer
199 views

Matrix Norm set #2

As a complement of the question Matrix Norm set and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions: (3) ...
0
votes
1answer
46 views

Matrix equation, how to

$A = \begin{bmatrix} 3 & 1 & -1 \\ 2 & 5 & a \\ 1 & a & 5 \end{bmatrix};$ $AX = 2X+4A$ How would one solve this matrix equation, for all $a$ for which it is solvable? I need ...
2
votes
3answers
141 views

Prove that $\operatorname{adj}A^t = \operatorname{adj} A$

Let $A$ be an anti-symmetric ($A^t = -A$), squared matrix ($n \times n$, while $n$ is uneven). Prove that ${\rm adj}\;A^t = {\rm adj}\;A$.
2
votes
2answers
171 views

If $A$ is a diagonalizable $n\times n$ matrix for which the eigenvalues are $0$ and $1$, then $A^2=A$.

If $A$ is a diagonalizable $n\times n$ matrix for which the eigenvalues are $0$ and $1$, then $A^2=A$. I know how to prove this in the opposite direction, however I can't seem to find a way prove ...
0
votes
1answer
70 views

Hermitian matrices [duplicate]

Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such that $\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H. How to prove ...
1
vote
1answer
47 views

solving linear matrix equation problem

Given two matrices $X$ and $Y$, with $Y$ invertible. Suppose that $$X=YZY^{-1}.$$ so $$Z=Y^{-1}XY.$$ In what order should I do the corresponding matrix multiplications to compute $Z$ ? Thanks.
4
votes
3answers
132 views

number of 8 x 8 matrices with specific conditions

How can we find the number of 8 by 8 matrices in which each entry is 0 or 1. In addition each row and each column contains odd number of 1's. Thanks for help.
1
vote
2answers
184 views

Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line

Find the standard matrix of the transformation $T:\mathbb{R}^2\to \mathbb{R}^2$ that corresponds to the reflection through the line $x_2=2x_1$ followed by reflection through the line $x_1=3x_2$ I am ...
0
votes
1answer
37 views

how to multiply particular type of matrix

Suppose you have $B =\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}$ , and $A =\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$ . I want a resulting matrix $C =\begin{bmatrix} 1 & 2 ...
0
votes
1answer
294 views

Counterexample for linear algebraic equations AX=Y

Say you have $k$ linear algebraic equations in $n$ variables; in matrix form we write $AX=Y$. Give a proof or a counterexample for each of the following: a) If $n=k$ there is always at most one ...
2
votes
1answer
60 views

Matrix of an invertible operator is also invertible -proof

Let $V$ be a finite dimensional vector space over an arbitrary field $F$ and let $S$ be an ordered basis of $V$. Let $A$ be an operator from $V$ to $V$. How can we prove that the matrix of $A$ with ...