1
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1answer
56 views

One definition of the determinant of a matrix

Suppose you define as follows : for $(a,b,c,d)\in \mathbb{R}^4$, $\det \begin{pmatrix} a & b \\ c & d\end{pmatrix} = ad-bc$. for $A$ a square matrix of size $n$, you define $\det A$ ...
0
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1answer
21 views

Transformation-Matrix (Definition & explanation)

I have to do a proof ("Let V be a vector space with basis A, B, C. Show that $T_{AC} = T_{BC}T_{AB}.$ Well, I really don't know what is menat by $"T_{AB}"$ or something like this. I thought about the ...
1
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1answer
41 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.
2
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2answers
23 views

$M=(a_{ij})$, with $1\leq x\leq3$ and $1\leq x \leq 3$, such that $a_{ij}=4i+2j-6$.

My exercise asks me to write the following matrix $M=(a_{ij})$, with $1\leq x\leq3$ and $1\leq x \leq 3$, such that $a_{ij}=4i+2j-6$. What I did not understand, and tried unsuccessfully was what ...
0
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2answers
40 views

Question about $(A - \lambda I_A)\vec{x} = 0$.

Finding a solution to $C\vec{x} = (A - \lambda I_A)\vec{x} = 0$ is the equivalent of considering the determinant of $C$ when it is zero. This means the matrix is linearly dependent and has infinite ...
1
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2answers
45 views

Matrix with Functions as Entries

What do we call a matrix with functions as entries? $$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix} $$
2
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1answer
47 views

Adjoint of a Matrix Definition

Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree ...
1
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0answers
43 views

How to Define an Infinite Anti-Diagonal Matrix

We can define an infinite diagonal matrix with some ease, and then say that a finite diagonal matrix is the top left sub-matrix of our infinite one. Can we define an infinite anti-diagonal matrix? It ...
1
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0answers
49 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
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votes
3answers
203 views

Non-symmetric $A^T=A$

Wikipedia says that symmetric matrices are square ones, which have the property $A^T=A$. This assumes that one can have non-square $A^T=A$ and, because it does not satisfy the first property of ...
1
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1answer
69 views

Is this definition of diagonal matrix correct?

I need to know if the following definition: Let $A:=\|a_{i,j}\|_{\substack{i=1,...,m \\ j=1,...,m}}$ be a square matrix. $A$ is diagonal matrix if $$i\neq j \implies a_{ij}=0, \quad\forall i,j \in ...
0
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1answer
73 views

Meaning of $L_A$?

Let $A$ be an m*n matrix with entries from a field $F$. $L_A: F^n \rightarrow F^m$ defined by $L_A=Ax$. I'm a bit confused about this definition. $L_A$ is a matrix representation of linear ...
2
votes
1answer
220 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
12
votes
2answers
655 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
0
votes
1answer
169 views

Differentiability of a matrix function

To prove the differentiability of a function defined on $M\in M_n$, how should I adapt the definition of differentiability? $h^{-1}[f(x+h)-f(x)-hL(x)]\to 0$  How do I take the reciprocal of a ...
0
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2answers
557 views

Definition: Eigenspace of a matrix

If I am given a matrix and told to find a basis for its eigenspace, does that just mean find the eigenvectors of the matrix? In my understanding, an eigenspace of an eigenvalue $\lambda$ is the set of ...
0
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1answer
335 views

Need help understanding Hessian matrix for direction estimation

Additional context: $H = |δ^2f / δx_iδx_j|$ is the Hessian matrix. $(3)$ From my previous question: What are the functionality of δ symbol and $δr^T$?, I got a few questions: I have read more ...
1
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1answer
60 views

What are the functionality of δ symbol and $δr^T$?

I got two questions here: Does anybody know what is the functionality of the small delta letter δ here? Is it simply the same as the rate of change just like the big delta letter Δ? And for the ...
2
votes
2answers
475 views

Definition of Symplectic Matrix

In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given: $$\left( \begin{array}{ccc} A & B \\ C & D \end{array} \right)$$ is symplectic if and only if: ...