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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2answers
197 views

Given a matrix $A$, is there a matrix $C$ with $AC = CA + A$?

Given this matrix A \begin{pmatrix}7+a&2&3&3+a\\2&7&7&11\\3&7&7&2\\3+a&11&2&11\end{pmatrix} where $a \in \mathbb{R}$ Is there a matrix $C \in ...
1
vote
1answer
121 views

Intuitive interpretation of the 3D to 2D mapping

Suppose a 3D configuration of points is given, $X\in\mathbb{R}^{n\times 3}$, and a non-zero matrix $Q\in\mathbb{3\times 2}$, with orthonormal columns. Now, suppose a mapping to 2D is obtained as ...
4
votes
2answers
274 views

$\det(A \otimes B - B \otimes A) = 0$ why? Why $rk(M) = n^2-n$ ? Why x and -x in Spec(M) ?

Let $A$, $B$ be $n\times n$ matrices. It seems $\det(A \otimes B - B \otimes A) = 0$. Moreover it seems that the kernel of $A \otimes B - B \otimes A$ contains $n$ vectors. Here is MatLab code to ...
1
vote
3answers
83 views

Show that the dimension of a particular linear space is $2$

Question: A Linear transformation $T: \mathbb R^4 \to \mathbb R^4$ is represented by the matrix $$\mathbf A=\begin{pmatrix} \\1&-1&2&3 \\ 2 & -3 & 4 & 5\\ 5 & -6 & ...
1
vote
1answer
120 views

V is a vector space such that $V = A\oplus A^\perp$ also $V = A \oplus C$ then can we say that $A^\perp = C$?

I have a vector space $V$ such that $V = A\oplus A^\perp$ i.e. $V$ is a direct sum of its subspace $A$ and orthogonal complement of $A$. Suppose we also have $V = A \oplus C$ Then can we say that ...
2
votes
1answer
299 views

How to show that $N(A) = R{(A^*)}^\perp$ and $N(A^*)=R({A})^\perp$?

How to show that for a given square matrices $N(A) = R{(A^*)}^\perp$ and $N(A^*)=R{(A)}^\perp$ where $N(A) $ and $R(A) $ are the null and range spaces of matrix $A$, respectively? I am not able to ...
2
votes
1answer
151 views

Solving matrices for certain variables using Cramer's rule

I have the following matrix equality: $$\left( \begin{array}{ccc} u_{11} & u_{12} & -p_1 \\ u_{21} & u_{22} & -p_2 \\ p_1 & p_2 & 0 \end{array} \right).\left( ...
1
vote
1answer
370 views

Given a matrix, find a linear transformation that uses it

The matrix is: $$\begin{pmatrix} 3+l & 8 & 3 & 3+l \\ 8 & 9 & 3 & 7 \\ 3 & 3 & 7 & 8 \\ 3+l & 7 & 8 & 13 \end{pmatrix}$$ I'm given the above ...
0
votes
1answer
107 views

About $P_{{L},{M}}$, projection transformation onto subspace $L$ along subspace $M$ .

I need help to study following theorem: For every idempotent matrix $E\in\mathbb{C}^{n\times n}$, $R(E)$ and $N(E)$ are complementary subspaces with $E = P_{{R(E)},{N(E)}}$. Conversely, if $L$ and ...
1
vote
2answers
496 views

$R(AB)=R(A)$ iff rank$(AB)$=rank$(A)$, $N(AB)=N(B)$ iff rank$(AB)$=rank$(B)$

$A$ and $B$ are two square matrices then show that $R(AB) = R(A)$ iff $\mathrm{rank} (AB) = \mathrm{rank} (A)$, and $N(AB) = N(B)$ iff $\mathrm{rank} (AB) = \mathrm{rank} (B)$. Here is my ...
3
votes
2answers
76 views

Determinant of symmetrical factorized matrix

Given $A, B \in \mathbb{R}^{n\times n}, t \in \mathbb{R}\setminus \{0\}$ with $b_{ij} = t^{i-j}\cdot a_{ij}$. Prove $\det(A) = \det(B)$. I first thought of induction. I can easily prove this for $n ...
0
votes
1answer
198 views

endomorphism as sum of two endomorphisms (nilpotent and diagonalizable)

$V$ is a field over $\mathbb{C}$. Show that $\phi: V \to V$ can be written as $\phi = \psi + \sigma$ where $\psi$ is diagonalizable and $\sigma$ is nilpotent. I managed to show this first part (you ...
2
votes
1answer
748 views

Is it possible to determine if this matrix is ill-conditioned?

I want to better understand ill-conditioning for matrices. Say we're given any matrix $A$, where some elements are $10^6$ in magnitude and some are $10^{-7}$ in magnitude. Does this guarantee that ...
0
votes
1answer
45 views

A matricial process to assign different values to elements of a diagonal matrix

Consider having vector $$v = \begin{pmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{pmatrix}$$ Consider the final result: $$ V = \begin{pmatrix} v_1 & 0 & \dots & 0\\ 0 & v_2 & \dots ...
1
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2answers
676 views

Finding Transition Matrix

Problem: Find the transition matrix P such that $P^{-1}AP=B$ where: $$A=\begin{bmatrix} 3 & -1 & 0 \\ -1 & 0 & -1 \\ 0 & 1 & 1 \end{bmatrix} \quad\text{and}\quad ...
1
vote
2answers
918 views

Finding Matrix Representation

Problem: Let T: $\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear map given by $$T\left[ \begin{matrix} x\\y\\ z\end{matrix} \right]= \left[ \begin{matrix} 3x-y\\z-x\\z-y\\\end{matrix} \right]$$ ...
0
votes
1answer
67 views

I need help to understand meaning of certain terms in a theorem

There are certain terms in the following theorem where I am finding difficulty to figure out. I need help. Theorem. Let $\mathbb{C}_{r}^{m\times n}$ denote the set of all complex $m\times n$ ...
1
vote
2answers
1k views

Determinant of symmetric Matrix with non negative integer element

Let \begin{equation*} M=% \begin{bmatrix} 0 & 1 & \cdots & n-1 & n \\ 1 & 0 & \cdots & n-2 & n-1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 ...
6
votes
1answer
693 views

Largest eigenvalue of a positive semi-definite matrix is less than or equal to sum of eigenvalues of its diagonal blocks

This question is very similar to this one. Let $B$ be a positive semi-definite matrix and $B = \begin{bmatrix} B_{11} & B_{12} \\ B_{12}' & B_{22} \end{bmatrix}$ where $B_{11}$ is $p \times ...
4
votes
2answers
231 views

Interesting Determinant

Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$. Define \begin{equation*} A=% \begin{bmatrix} 0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & ...
3
votes
1answer
145 views

Matrices of Trace $0$

The set of all $n$-square matrices with trace $0$ is a subspace of the set of all $n$-square matrices. Is there a standard notation and/or name for this subspace?
0
votes
1answer
112 views

Systems of linear equations

I am I right on the following: A 2X2 system of equation with no solution will look like this: x+3y=3 2x+6y =-8 1/2 + 3/6 ≠ 3/8 A 2X2 system of equations with in ...
1
vote
2answers
85 views

Finding the dot product.

Finding the dot product of $(-2w) \cdot w$ where $w=(0,-2,-2)$ $$ \text{dot product} = \frac{v . u}{\| v \| . \| u \|} $$ So $$-2 (0,-2,-2)=(0,4,4) \\ (0+4+4) = 8 \\ (0,-2,-2)=-4 \\ 8 \times ...
2
votes
1answer
273 views

Show that the set of matrices such that $\det A \neq 0$ is open [duplicate]

Possible Duplicate: Why do the $n \times n$ non-singular matrices form an “open” set? Like the title says how would you show that the set of matrices such that $\det A \neq 0$ ...
1
vote
1answer
399 views

power series expansion of the square root of a Hermitian matrix

Is there a power series expansion of the square root of a Hermitian matrix, as a procedure to calculate the square root without taking the inverse or diagonalizing the matrix? I find for scalar number ...
0
votes
0answers
141 views

Dimension reduction for non-full rank matrices

Is there a transformation to decrease the dimension of a matrix so that it becomes full rank?
3
votes
2answers
208 views

Let $A$ be real symmetric $n\times n$ matrix whose only eigenvalues are 0 and 1. Pick out the true statements.

Let $A$ be real symmetric $n\times n$ matrix whose only eigenvalues are $0$ and $1$. Let the dimension of the null space of $A-I$ be $m$. Pick out the true statements. The characteristic ...
7
votes
3answers
7k views

Determinant of symmetric matrix with the main diagonal elements zero

How to prove that the determinant of a symmetric matrix with the main diagonal elements zero and all other elements positive is not zero (i.e., that the matrix is invertible)? EDIT: OP indicates in a ...
4
votes
2answers
346 views

Finding dimension of a vector space

Let $H_n$ be the space of all $n\times n$ matrices $A = (a_{i,j})$ with entries in $\mathbb{R}$ satisfying $a_{i,j} = a_{r,s}$ whenever $i+j = r+s$ $(i, j , r , s = 1, 2, \ldots, n)$. What would be ...
1
vote
3answers
420 views

Finding the rank of a matrix

Let $A$ be a $5\times 4$ matrix with real entries such that the space of all solutions of the linear system $AX^t = (1,2,3,4,5)^t$ is given by$\{(1+2s, 2+3s, 3+4s, 4+5s)^t :s\in \mathbb{R}\}$ where ...
3
votes
3answers
135 views

Let $\alpha$ and $\beta$ be two distinct eigenvalues of $A$ then $ A^3 = \frac{\alpha^3-\beta^3}{\alpha-\beta}A-\alpha\beta(\alpha+\beta)I$?

Let $\alpha$ and $\beta$ be two distinct eigenvalues of a $2\times2$ matrix $A$. Then which of the following statements must be true. 1 - $A^n$ is not a scalar multiple of identity matrix for any ...
4
votes
3answers
150 views

$T :\mathbb {R^7}\rightarrow \mathbb {R^7} $ is defined by $T(x_1,x_2,\ldots x_6,x_7) = (x_7,x_6,\ldots x_2,x_1)$ pick out the true statements.

Consider the linear transformations $T :\mathbb {R^7}\rightarrow \mathbb {R^7} $ defined by $T(x_1,x_2,\ldots x_6,x_7) = (x_7,x_6,\ldots x_2,x_1)$. Which of the following statements are true. 1- ...
1
vote
1answer
273 views

Strassen Multiplication?

How are the values of the 7 new matrices derived? I'm referring to the values that reduce matrix multiplication to 7 multiplications per level: $M_1 = \left(A_{1,1} + A_{2,2}\right)\left(B_{1,1} + ...
2
votes
1answer
175 views

laplace form and matrix exponential

Given the matrix $(I-A)^{-1}$ and $B$, can we compute $e^{A+B}$, where $e^X$ is defined to be $\sum_{i=0}^{\infty} \frac{X^i}{i!}$. (Note that $A$ and $B$ do not commute, and hence $e^A \cdot e^B ...
7
votes
4answers
585 views

Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$

Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be ...
1
vote
1answer
547 views

a matrix inverse laplace transform problem

Let $\mathcal {L}^{-1}[\cdot]$ be an inverse Laplace transform. Let $A$ be a square matrix, and $I$ an identity matrix. Based on the fact that $\mathcal {L} ^{-1} [{(sI-A)}^{-1}] = e ^{tA}$, how can ...
2
votes
1answer
284 views

Matrix exponential and rank

Let $A$ be a square matrix. Invertibility of $\exp(A)$ follows easily from properties of the matrix exponential. Is $\int_0^t \exp(A u)du$ also invertible? I believe it should be, and that the ...
1
vote
1answer
162 views

Necessary and sufficient condition for the matrix $A = I-2xx^t$ to be orthogonal

Let $x$ be a non zeo (column) vector in $\mathbb{R}^n$. What is the necessary and sufficient condition for the matrix $A = I-2xx^t$ to be orthogonal?
2
votes
2answers
266 views

Let $B$ be a nilpotent $n\times n$ matrix with complex entries let $A = B-I$ then find $\det(A)$

Let $B$ be a given nilpotent $n\times n$ matrix with complex entries. Let $A = B-I$ find out $\det(A)$. What if B is orthogonal or skew symmetric matrix? Then can we say anything about its trace and ...
6
votes
4answers
115 views

Find out trace of a given matrix $A$ with entries from $\mathbb{Z}_{227}$

Let $A$ be a $227\times 227$ matrix with entries in $\mathbb{Z}_{227}$ such that all of its eigen values are distinct. What would be its trace? I think it is zero by adding all 227 elements but i am ...
3
votes
3answers
279 views

Linear algebra exam questions

Pick out the true statements: There exist $n\times n$ matrices $A$ and $B$ with real entries such that $(I-(AB-BA)^n) = 0$. If $A$ is symmetric and positive definite matrix then $tr(A)^n\geq n^n ...
2
votes
1answer
278 views

How do I prove positive definiteness for a matrix difference?

Let $A$ be a symmetric positive semidefinite matrix. Let $W$ be a diagonal matrix with the entries $w_i \in (0,1)$. I think $$A - WAW$$ should be positive semidefinite, but I don't know how to prove ...
1
vote
0answers
194 views

2D Cartesian Matrix / coordinate transformation.

I has initially asked this question in the programming site but did not get an answer that worked. This is my first question on this site so please bear with me. Consider a page with three distinct ...
0
votes
1answer
485 views

Find out dimension of the eigenspace of a given linear transformation $T$

Let $T:\mathbb{R^4}\rightarrow \mathbb{R^{4}}$ be defined by $T(x,y,z,w)=(x+y+5w,x+2y+w,-z+2w,5x+y+2z)$ then what would be the dimension of the eigenspace of $T$? One approach may be to find out ...
6
votes
5answers
333 views

Computing $A^{50}$ for a given matrix $A$

$A =\left( \begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array} ...
0
votes
1answer
174 views

Determinantal criterion for positive semi-definite matrix

Consider an $n\times n$ real matrix $K$ which satisfies $$ \det[K_{ij}]_{i,j=1}^k\geq 0,\qquad 1\leq k \leq n. $$ I know that if one assumes moreover that $K$ is symmetric, then $K$ is positive ...
5
votes
1answer
195 views

What is $\frac{\det(A+tI)}{\det(B+tI)}$ as $t\to0$?

If $A$ and $B$ are two real $2\times 2$ matrices with $\det A = 0 $ and $\det B = 0 $ and $\mathrm{tr}(B)$ is non zero. then what will be limit of $$\lim_{t\to0}\frac{\det(A+tI)}{\det(B+tI)}$$ I used ...
1
vote
2answers
305 views

rank one update

Given a matrix $X$, we can compute its matrix exponential $e^X$. Now one entry of $X$ (say $x_{i,j}$) is changed to $b$, the updated matrix is denoted by $X'$. My problem is how to compute $e^{X'}$ ...
2
votes
1answer
164 views

matrix differential equation

Given a matrix $X(t)=e^{tA}$, we know that $X(t)$ is the solution of the following matrix differential equation: $$ \frac{dX(t)}{dt} =X(t) \cdot A .$$ Now could anyone help to construct a matrix ...
4
votes
1answer
158 views

Can we say that there exist an integer n such $A+nB$ invertible?

If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exist an integer $n$ such that $A+nB$ invertible? I was trying by choosing n such that eigne values of $A+nB$ ...