Linked Questions

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1answer
4k views

Determinant from eigenvalues [duplicate]

Is it always true that product of eigenvalues is determinant of a matrix ? what if one of the eigenvalues are same and matrix is not diagonalizable ? Is this statement is still true ?
2
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2answers
262 views

Trace and Determinant of a linear map $T$ which is not necessarily diagonalizable [duplicate]

How can I prove that if a linear map $T \ \in \ \text{End}(V)$ with $\dim V < \infty $ has eigenvalues ${\lambda}_{1},{\lambda}_{2}, ..., {\lambda}_{n} $, then $\text{trace}(T)$ $=$ ${\lambda}_{1}+...
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3answers
186 views

Determinants and eigenvectors [duplicate]

Hello, I'm trying to work through this question. I define linearly independent as: $a_1*v_1+a_2*v_2+...+a_n*v_n = 0$ iff every $a_i=0$. I also know that an eigenvector is a vector $v$ such that: $...
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1answer
77 views

Prove that if A has eigenvalues then determinant of A is the product of eigenvalues [duplicate]

How can I prove that if A has eigenvalues then determinant of A is the product of its eigenvalues?
40
votes
10answers
64k views

Is a matrix $A$ with an eigenvalue of $0$ invertible?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of ...
8
votes
1answer
2k views

Jacobian of Fourier Transformation

I am trying to calculate the Jacobian determinate of the Fourier transform which I stumbled upon when studying the Path Integral in Quantum Field Theory. I know the answer should be $1$ but I don't ...
2
votes
1answer
3k views

Is it true that the determinant of symmetric positive definite matrix is the product of the eigenvalues?

I was working with a symmetric positive definite matrix when I encountered upon the following "identity" Let A be symmetric pd $\det(A)$ $= \det(Q\Lambda Q^{-1})$ (all symmetric matrices diag'able) ...
0
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4answers
439 views

How do you show that A has a eigenvalue equal to $0$ $\iff$ A is not invertible?

The only thing I can think of I that suppose A is \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} After subtracted from $0$,...
1
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2answers
201 views

How to tackle this polynomial given as a determinant? [closed]

Let $$p (x) = \begin{vmatrix} 1 & x & x & \dots & x & x \\ x & 1 & x & \dots & x & x \\ x & x & 1 & \dots & x & x \\ \vdots & \vdots &...
2
votes
2answers
669 views

How do we know that the eigenvalues of a matrix are the roots of its characteristic polynomial?

More specifically, if $A$ is a matrix over some algebraically closed field and $P_A(x)$ is its characteristic polynomial, how do we know that the roots of $P_A(x)$ are the eigenvalues of $A$, with the ...
2
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1answer
403 views

Pivots, determinant and eigenvalues

In symmetric matrices, Product of pivots = determinant of that matrix Determinant of the matrix = Product of eigenvalues Therefore the product of eigenvalues = product of pivots. Do any or ...
0
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1answer
300 views

On eigenvalues being a scalar multiple of a volume

I was told in lecture the other day that an eigenvalue represents the multiple of the volume of a vector space. For example, if I pour a jar of liquid from jar A into jar B, and express that in terms ...
1
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0answers
361 views

Rank of a matrix and cross product between eigenvectors

I have two questions about linear algebra. I was doing a exercise that says: Let $A$ be a real symmetric matrix $3 \times 3$ and $\det A = 6 $. Suppose that $u =(4,8,-1)$ and $v=(1,0,4)$ are ...
0
votes
1answer
255 views

True or False Linear Transformation Eigenvalues Question

Let $V$ be a finite dimensional vector space over $\Bbb R$ and let $T :V \to V$ be a linear transformation. Let $A$ be the matrix of $T$ with respect to the standard basis for $V$. For each of the ...
1
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0answers
158 views

Find the characteristic polynomial and the eigenvalues $\lambda_1, \lambda_2,\dots,\lambda_n$ of the matrix $A$ [closed]

Let $\lambda_1, \lambda_2, \dots, \lambda_n$ be eigenvalues of the matrix $A=(a_{ij})_{n\times n}$. Is it true that $|A|=\lambda_a\lambda_2\dots\lambda_n$, and $tr(A)=\displaystyle\sum_i^n\lambda_i$. ...

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