Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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4
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1answer
130 views

Is there a name for the space of vectors orthogonal to a given vector in $\mathbb{R}^n$?

Given a vector $\mathbf{v} \in \mathbb{R}^n$, the set of vectors in $\mathbb{R}^n$ orthogonal to $\mathbf{v}$, namely $$\{\mathbf{u} \in \mathbb{R}^n: \mathbf{u} \cdot \mathbf{v}=0\},$$ forms a ...
1
vote
2answers
182 views

Quadratic forms of two matrices are equal then the matrices are equal

$A,B\in M_n$, then prove that if $x^HAx=x^HBx$ for all $x\in C^n$, then $A=B$
2
votes
4answers
2k views

Could a set of $3$ vectors in $\mathbb{R}^4$ span all of $\mathbb{R}^4$?

Could a set of $3$ vectors in $\mathbb{R}^4$ span all of $\mathbb{R}^4$; is this the same as asking if a 4 x 3 matrix could span $\mathbb{R}^4$ or if a ...
0
votes
1answer
80 views

$\sqrt{T^\ast T}$ is positive

I read that $\sqrt{T^\ast T}$ is positive operator. I tried to proof it but fail. Is it not true that $\sqrt{T^\ast T}$ is positive? If it is true anyone can show me how to proof it please? Positive ...
7
votes
2answers
210 views

Determinant of the product equal to the product of determinants?

Let $X$ be an $n\times p$ matrix and $A$ be a $n\times n$ matrix. When is it true that $$\det (X^{\top}AX) = \det(A)\det(X^{\top}X)?$$
13
votes
8answers
2k views

I need to calculate $x^{50}$ [duplicate]

$x=\begin{pmatrix}1&0&0\\1&0&1\\0&1&0\end{pmatrix}$, I need to calculate $x^{50}$ Could anyone tell me how to proceed? Thank you.
2
votes
0answers
86 views

The dimension of centralizer $\gamma=\{B\in M_n(\mathbb{R}):AB=BA\}$

Let $A$ be a $6\times 6$ matrix with charpoly $x(x+1)^2(x-1)^3$. We need to find the dimension of $$\gamma=\{B\in M_n(\mathbb{R}):AB=BA\}.$$ What is the relation of charpoly of $A$ with dimension ...
2
votes
1answer
181 views

Direct sums of subspaces

Can someone check the correctness of my proof. Statement. A single subspace $W_1$ is independent. Two subspaces $W_1,W_2$ are independent $\iff$ $W_1\cap W_2=\{0\}$ Two subspaces are said to be ...
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2answers
67 views

How can we prove that the sum of all elements of $A^{\displaystyle-1}$ is $5$?

Let $\displaystyle A_{5\times 5}$ be an invertible matrix whose sum of each rows equals 1. How can we prove that the sum of all elements of $A^{\displaystyle-1}$ is $5$?.
4
votes
1answer
66 views

Epimorpsims preserve generalized eigenspaces

This is most likely trivial, but I don't get it. In Humphrey's Introduction to Lie algebras, page 82, he says: It is clear that, if $\phi \colon L \to L'$ is epimorphism [of finite dim. Lie ...
3
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0answers
175 views

When every matrix is a sum of nilpotent (idempotent or invertible) matrices ??

Let $R$ be a ring with non-zero identity. Consider the following three properties: Every $A \in M_n(R)$ is a sum of nilpotent matrices. Every $A \in M_n(R)$ is a sum of idempotent matrices. Every ...
0
votes
1answer
49 views

Finding $P$ Such that $D=P^TH\overline P$

If $$H=\begin{bmatrix} 1 & i & 2+i \\ -i & 2 & 1-i \\ 2-i & 1+i & 2 \end{bmatrix}$$ is a hermitian matrix, Find a Non-Singular Matrix P such that $D=P^TH\overline P$ I tried ...
1
vote
0answers
103 views

Sufficient conditions for the convergence of countably infinite products of matrices

I'm interested in countably infinite products of matrices of the form $$T_k=\begin{pmatrix}1&0&\cdots &0&0&0\\ 0 &\ddots&0&\cdots&0&0\\ a_{i1}& \cdots& ...
2
votes
1answer
158 views

Linear Algebra - Linear Independence and Span

Suppose $(v_1 ... v_n )$ is linearly independent in a vector space V, and $w \in V$, if $(v_1 + w,.... v_n +w)$is linearly dependent, then $w \in span(v_1 ... v_n)$. I'm still getting the hang of ...
5
votes
3answers
242 views

Short proof that $X^2 = X \Rightarrow X^{100} = X$

Given that a matrix $X$ satisfies $X^2 = X$ it is clear that $X^{100}=X$ by repeated multiplication of $X$. Algebraically, we might write: $$X^{100} = (X^2)^{50}=X^{50}=(X^2)^{25}=X^{25}=X(X^2)^{12} ...
0
votes
1answer
277 views

Given an m-x-n matrix where m!=n, it must be true that dim( NullSpace(A)) != dim(NullSpace(A-transpose)) / T or F?

I think the answer to the question in the subject line is TRUE. But I wanted to confirm this with folks with more skills than I have. My reasoning is: Say we have a matrix A,where ...
1
vote
1answer
95 views

A question regarding a step in power method justification (Writing a vector in terms of the eigenvectors of a matrix)

Let $A$ be a $t \times t$ matrix. Can we present any $t \times 1$ vector, as a linear combination of eigenvectors of $A$? I think this should not be the case unless all eigenvectors of $A$ happened to ...
1
vote
1answer
107 views

Proving the dimension of basis of given subspace

Prove or disprove the following statement : If $B = {(b_1 , b_2 , b_3 , b_4 , b_5) }$ is a basis for $\mathbb{R}^5$ and $V$ is a two-dimensional ...
3
votes
2answers
44 views

Showing $T(\mathbf{v})$ is an eigenvector of S

Let $V$ be a finite dimensional vector space over a field $\mathbb{F}$, $S,T\in{\mathscr{L}(V)}$, and assume $ST=TS$. If $\mathbf{v}\in{V}$ is an eigenvector of S with eigenvalue $\lambda$, prove ...
2
votes
1answer
122 views

Study of Matrix Calculus

I need to study matrix calculus such as integration, differentiation, differentiation of functions of determinants and inverse matrices and then also other matrix based calculations such as ...
2
votes
0answers
56 views

Proving an optimization problem has a rational optimum.

Consider the function $$ J_\gamma(X) = \det\left( I - \tfrac{1}{\gamma^2} (A+BXC)^\mathsf{T}(A+BXC)\right) $$ where $A$, $B$, $C$, $X$ are matrices of real numbers. Further suppose that ...
5
votes
1answer
437 views

When solving a linear differential equation by factoring the operator, how does one guarantee no solutions are lost?

I think the best way to make this question clear is with an example. Lets say we want to solve the differential equation $(\Delta^2 - \lambda^4)\phi=0,$ calculations are greatly simplified if we ...
1
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1answer
126 views

What are common notations for the endomorphism group of a vector space?

Given a vector space $V$, the set of endomorphisms in $V$ can be denoted $$\text{End}(V)=\left\{L:V\rightarrow V:L\text{ is linear}\right\},$$ particularly when one wants to be completely unambiguous. ...
1
vote
2answers
302 views

Non-zero rows give the basis of null space?

Q. Find the nullity and basis of null space of linear transformation $A:R^4\rightarrow R^4$ given by the matrix $$\begin{bmatrix} 0 & 1 & -3 & -1 \\ 1 & 0 & 1 & 1 \\ 3 & 1 ...
2
votes
1answer
137 views

Prove that $|A+B|/|A|=|X^TA^{-1}X||X^TAX|$ when $A$ is positive definite, $B=XX^TA-AXX^T$, and $X^TX=1$.

This is Problem 15 in the "Miscellaneous Exercises" section of Chapter 1 of Matrix Differential Calculus with Applications in Statistics and Econometrics. Let $A$ be positive definite and let ...
0
votes
1answer
46 views

How do i prove invertibility of this linear transformation

Let $T$ be a linear transformation from a vector space $V$ over reals into $V$ such that $T-T^2=I$. Show that $T$ is invertible
3
votes
3answers
3k views

difference between dot product and inner product

I was wondering if a dot product is technically a term used when discussing the product of $2$ vectors is equal to $0$. And would anyone agree that an inner product is a term used when discussing the ...
2
votes
1answer
60 views

How to prove this equation

If $\lambda_1,\lambda_2,\lambda_3$ are the eigen values of a matrix $\begin{bmatrix} 26 & -2 & 2 \\ 2 & 21 & 4 \\ 4 & 2 &28 \end{bmatrix}$ Then Show that ...
5
votes
1answer
152 views

Inner Product on Division Algebras

Here, Wikipedia gives a proof that the only finite dimensional associative division algebras over $\mathbb{R}$ are $\mathbb{R}, \mathbb{C}, \mathbb{H}$. The proof proceeds by taking such a division ...
3
votes
2answers
560 views

$AX = 0$ and $BX = 0 $ implies A and B are row equivalent

I'm trying to prove that if the systems $AX = 0$ and $BX=0$ are equivalent, then the matrices $A$ and $B$ are row equivalent. Proving the converse was very simple, but this one seems harder. I saw a ...
0
votes
1answer
80 views

Adding two rows in an augmented matrix: does this always preserve the set of solutions?

Given a system of linear equations, we construct an augmented matrix, and may modify it using the three elementary row operations: Interchange two rows. Multiply a row by a non-zero constant. Add a ...
1
vote
2answers
75 views

What's the role of the inner product here

Consider the following statement: If $V$ is a finite dimensional vector space over $\mathbb R$ and $T:V \to V$ is linear then there is a basis for $V$ of eigen vectors of $T$ if and only if there is ...
9
votes
3answers
294 views

The integer $c_n$ in $(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write $$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$ where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove or disprove ...
0
votes
2answers
97 views

cross product and determinant

$$\begin{align*}\alpha _1=\left(a_{11},a_{12},a_{13}\right),\alpha _2=\left(a_{21},a_{22},a_{23}\right),\alpha _3=\left(a_{31},a_{32},a_{33}\right)\tag{1}\end{align*}$$ Then cross product(Can you ...
5
votes
2answers
213 views

Number of possibilities of $10\times10$ matrix

If $A$ is a $10\times10$ matrix with entries from the set $\{0, 1, 2, 3\}$ and if $AA^T$ is of the form: $$\begin{pmatrix} 0 & * & * & \cdots & * \\ * & 0 & * & \cdots ...
1
vote
1answer
129 views

Accessing elements of packed symmetric distance matrix

Suppose you have a symmetric distance matrix A. For example A is 4*4 (the numbers above and ...
3
votes
3answers
598 views

Show that $A$ is similar to a diagonal matrix iff $b=c=d=e=f=g=0$

Show that $A$ is similar to a diagonal matrix iff $$b=c=d=e=f=g=0$$ $$A= \left(\begin{array}{cccc}a & b & c & d \\ 0 & a & e & f\\ 0 & 0 & a & g\\ 0 & 0 & ...
1
vote
2answers
707 views

Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P94, Strang P59]

From P59 of Intro to Lin Alg, 4th Ed by Strang & P94-95 of Linear Algebra and its Apps by Lay For relief, I denote all row vectors with superscripts and column with subscripts. Define ...
2
votes
2answers
74 views

Construction of a Linear Transformation question

Let there be a linear transformation going from $\mathbb R^4$ to $\mathbb R^3$, such that $(1,0,1,0)$ , $(2,1,3,0)$ span ker T , and $(0,1,2)$ , $(3,1,2)$ span Im T. If such a transformation possible ...
3
votes
2answers
196 views

Find matrices $X$ such that for any matrix $Y$ we have $\det(X^2 + Y^2) \geq 0$ [duplicate]

What is the characterization of real matrices $X \in \mathbb{R}^{n\times n}$ such that for any real matrix $Y \in \mathbb{R}^{n\times n}$: $$\det(X^2 + Y^2) \geq 0?$$
0
votes
3answers
384 views

A basic question on diagonalizability of a matrix

I am following a book where the "diagonalizability" has been introduced as follows: Consider a basis formed by a linearly independent set of eigen vectors $\{v_1,v_2,\dots,v_n\}$. Then it is claimed ...
2
votes
2answers
65 views

When is the linear map of plugging $m$ numbers in a polynomial is surjective?

Let $n$, $m$ be positive integers and $V_n$ be the vector space of the polynomials of degree less than or equal to $n$ whose coefficients are complex numbers. For $m$ complex numbers $a_1,\dots,a_m$, ...
1
vote
4answers
93 views

Simultaneous Linear Equation Problem

I am in the eighth standard. I have an examination on linear equations tomorrow. I am stuck in the following problem. $$ \begin{cases} 2x - 5y = 4,\\ 3x - 2y = -16.\end{cases} $$ Find $x$ and $y$. Any ...
1
vote
1answer
85 views

differentiation of vector norm

what would be the differentiation of this equation :- $$f(A) = \sum_i \left \| Y_{i} - AB_{i} \right \|^2 + \left \| A - A_\text{constant} \right \|^2$$ wrt to $A$. where $Y$ is a column vector and ...
2
votes
1answer
121 views

Effect of change in a column of a matrix in its determinant and eigenvalue

I have matrix $R$ which based on it, another matrix $H$ is computed using $$ H(R) =\frac { (R R^T+ sI)^{1/2} } { trace(R R^T+ sI)^{1/2}} $$ $s$ is a small value. Now, one column of the matrix $R$ ...
0
votes
1answer
102 views

derivation of vector norm

what would be the differentiation of this equation :- $F(A) = \sum_{i} \left \| Y_{i} - AB_{i} \right \|^{2} + \lambda \left \| A - C \right \|^{2}$ wrt to A . Y is a column vector and B is column ...
-3
votes
1answer
76 views

A basic question on linear independence of eigen vectors

To prove that for any $n \times n$ matrix there are $n$ linearly independent eigen vectors if all the eigen values are distinct, I see that in a book it starts with the following matrix $$ ...
1
vote
1answer
268 views

Prove $\exists T\in\mathfrak{L}(V,W)$ s.t. $\text{null}(T)=U$ iff $\text{dim}(U)\geq\text{dim}(V)-\text{dim}(W)$.

The entire problem statement is, Suppose that $V$ and $W$ are finite dimensional and that $U$ is a subspace of $V$. Prove that there exists $T\in\mathfrak{L}(V,W)$ such that $\text{null}(T)=U$ if and ...
2
votes
0answers
38 views

eigen problem for direct scattering method

Consider the KdV equation $$u_{t}+6uu_{x}+u_{xxx}=0$$ with initial condition $$u(x,0)= \begin{cases} 1 &\text{if } x \in [-1,0] ,\\ 0 &\text {if } x \in ...
4
votes
3answers
326 views

A basic question on determinant and rank of a matrix

How to prove that if the determinant of a $n \times n$ matrix is zero then the rank is less than $n$. I can prove the converse. Only a hint is enough. My definition of rank is the maximum number of ...