Real Matrices with Real Eigenvalue pre- and Post multiplied by a Diagonal Matrix
Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.
matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition
asked Jan 3 at 22:56
Arthur
47112
add a comment |
Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.
matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition
asked Jan 3 at 22:56
Arthur
47112
add a comment |
Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.
matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition
asked Jan 3 at 22:56
Arthur
47112
Suppose all the eigenvalues of $Ain mathbb{R}^{ntimes n}$ (not necessarily symmetric) are real. Let $Din mathbb{R}^{ntimes n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.
matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition
matrices eigenvalues-eigenvectors matrix-calculus matrix-decomposition
asked Jan 3 at 22:56
Arthur
47112
asked Jan 3 at 22:56
Arthur
47112
asked Jan 3 at 22:56
Arthur
47112
asked Jan 3 at 22:56
Arthur
47112
asked Jan 3 at 22:56
Arthur
47112
47112
add a comment |
add a comment |
1 Answer
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I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.
Take
$$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
and
$$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
Then $DAD$ and $A + D$ has non-real eigenvalues.
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
Thanks for the counterexample.
– Arthur
Jan 4 at 3:10
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.
Take
$$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
and
$$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
Then $DAD$ and $A + D$ has non-real eigenvalues.
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
Thanks for the counterexample.
– Arthur
Jan 4 at 3:10
add a comment |
I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.
Take
$$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
and
$$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
Then $DAD$ and $A + D$ has non-real eigenvalues.
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
Thanks for the counterexample.
– Arthur
Jan 4 at 3:10
add a comment |
I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.
Take
$$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
and
$$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
Then $DAD$ and $A + D$ has non-real eigenvalues.
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.
Take
$$A = begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix}^{-1}begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 3 end{pmatrix}begin{pmatrix} 1 & 3 & 2 \ -1 & 1 & 4 \ 1 & 2 & 7 end{pmatrix},$$
and
$$D = begin{pmatrix} 1 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1end{pmatrix}.$$
Then $DAD$ and $A + D$ has non-real eigenvalues.
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
answered Jan 3 at 23:50
Theo Bendit
16.7k12148
16.7k12148
Thanks for the counterexample.
– Arthur
Jan 4 at 3:10
add a comment |
Thanks for the counterexample.
– Arthur
Jan 4 at 3:10
Thanks for the counterexample.
– Arthur
Jan 4 at 3:10
Thanks for the counterexample.
– Arthur
Jan 4 at 3:10
add a comment |
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