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49 votes
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Why is there not a test for diagonalizability of a matrix

There is no continuous function $f$ in the entries of the matrix s.t. $A$ is diagonalizable iff $f(A) = 0$ or iff $f(A) \neq 0$. (Like the case for invertibility, where $A$ is invertible iff $\det(A) \...
David Gao's user avatar
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27 votes

Why is there not a test for diagonalizability of a matrix

Here is a salvage: there is a continuous function $f(A)$ of a matrix $A$, even a polynomial function, which checks whether the eigenvalues of $A$ are distinct (over an algebraic closure). Any such ...
Qiaochu Yuan's user avatar
8 votes

Why is there not a test for diagonalizability of a matrix

A matrix is diagonalizable if and only if its minimal polynomial has distinct roots and a real matrix is diagonalizable over $\mathbb{R}$ iff additionally all the roots are real. But given a real or ...
ronno's user avatar
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4 votes

Is every complex linear operator diagonalizable?

You need to distinguish algebraic and geometric multiplicities here. What's true is that $A$ is diagonalizable iff it has $n$ eigenvalues with geometric multiplicity, and over $\mathbb{C}$, every $A$ ...
Qiaochu Yuan's user avatar
2 votes
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Eigenvectors of two commuting diagonalizable matrices when the eigenspaces need not have dimension one

Take $A$ to be the identity matrix and $B = \begin{pmatrix} 1 & 0\\ 0 & 2 \end{pmatrix}$. Then $AB = BA$ and the $1$-eigenspace of $A$ is $2$-dimensional. In this case, $A$ and $B$ have ...
Ethan Kharitonov's user avatar
2 votes
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Verify that a quadratic form is NOT positive definite

Your second approach provides a solution: in view of the second column of $P^TAP$, the second column $v$ of $P$ should satisfy $q(v)=-1$. Let us check: $$q(-2,1,0)=(-2)^2+4(-2)1+3\cdot1^2+2\cdot1\...
Anne Bauval's user avatar
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2 votes
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Exam question about JNF and matrix diagonalization

I am very confused how they are making the jump to considering Jordan blocks instead of the entire $J$ matrix. The Jordan normal form is block diagonal, hence so is every power of it, and a block ...
Qiaochu Yuan's user avatar
2 votes

Congruent diagonalization using row and column operations

Here is how it works out when keeping track of the matrix $P$ in $P^T HP = D$ $$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc ...
Will Jagy's user avatar
  • 141k
1 vote

Verify that a quadratic form is NOT positive definite

One way to check a matrix's definiteness is by checking its eigenvalues. If you don't want to explicity find them, you can rely on this fact $$ \prod_{i=1}^3 \lambda_i = \det(A) = -7. $$ With three ...
CroCo's user avatar
  • 1,244
1 vote

Verify that a quadratic form is NOT positive definite

We may be overreading this. When you form the symmetric matrix, the determinant is easily proved negative ($-7$). You can't have a positive (semi)definite quadratic form when this holds. We also ...
Oscar Lanzi's user avatar
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1 vote

Verify that a quadratic form is NOT positive definite

Inspired by Sylvester's criterion, let's look at the upper-left submatrix $\begin{bmatrix}1 & 2 \\ 2 & 3 \end{bmatrix}$. It already has a negative determinant. We are also lucky that $\begin{...
angryavian's user avatar
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1 vote
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Congruent diagonalization using row and column operations

The algorithm gives you the desired matrix $P$. Note that $$ \begin{pmatrix} 1 & 0 & 0 \cr -2 & 1 & 0 \cr -3 & 0 & 1 \end{pmatrix} \cdot A\cdot \begin{pmatrix} 1 & 0 & ...
Dietrich Burde's user avatar

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