Show that if $Ax = lambda x$ for a normal matrix $A,;$ then $A^{*}x = bar{lambda}x$
Suppose $A in mathbb{C}^{ntimes n}$, and $A$ is normal, show that if $lambda$ is an eigenvalue of $A$, and $x$ is a corresponding eigenvector of $A$ associated with $lambda$, then $bar{lambda}$ is an eigenvalue of $A^*$, and $x$ is an eigenvector of $A^*$ associated with $bar{lambda}$. In other words, if $Ax=lambda x$, and $x neq 0$, then $A^*x=bar{lambda}x$, and $x neq 0$.
Hint: Show that if $A$ is normal, and $mu$ is any complex number, then $A-mu I_n$ is also normal. Now, using the norm generated by standard inner product $left langle .,. right rangle$, show that if $B in mathbb{C}^{ntimes n}$, and $B$ is normal, then $left |Bx right |^2=left |B^*x right |^2$ for all $ mathbb{C}^n$.
matrices
edited Dec 8 at 22:23
Andrews
374317
asked Dec 8 at 17:29
user620787
11
add a comment |
Suppose $A in mathbb{C}^{ntimes n}$, and $A$ is normal, show that if $lambda$ is an eigenvalue of $A$, and $x$ is a corresponding eigenvector of $A$ associated with $lambda$, then $bar{lambda}$ is an eigenvalue of $A^*$, and $x$ is an eigenvector of $A^*$ associated with $bar{lambda}$. In other words, if $Ax=lambda x$, and $x neq 0$, then $A^*x=bar{lambda}x$, and $x neq 0$.
Hint: Show that if $A$ is normal, and $mu$ is any complex number, then $A-mu I_n$ is also normal. Now, using the norm generated by standard inner product $left langle .,. right rangle$, show that if $B in mathbb{C}^{ntimes n}$, and $B$ is normal, then $left |Bx right |^2=left |B^*x right |^2$ for all $ mathbb{C}^n$.
matrices
edited Dec 8 at 22:23
Andrews
374317
asked Dec 8 at 17:29
user620787
11
2
What are your thoughts on the problem? What have you tried? Where are you getting stuck?
– Omnomnomnom
Dec 8 at 17:32
Not sure how to show "A - mu I_n" is normal for a start.
– user620787
Dec 8 at 17:44
Please use MathJax in future. It's a type of $LaTeX$. Just search for "MathJax tutorial Mathematics Stack Exchange" and you'll find instructions on how to use it.
– Shaun
Dec 8 at 17:52
add a comment |
Suppose $A in mathbb{C}^{ntimes n}$, and $A$ is normal, show that if $lambda$ is an eigenvalue of $A$, and $x$ is a corresponding eigenvector of $A$ associated with $lambda$, then $bar{lambda}$ is an eigenvalue of $A^*$, and $x$ is an eigenvector of $A^*$ associated with $bar{lambda}$. In other words, if $Ax=lambda x$, and $x neq 0$, then $A^*x=bar{lambda}x$, and $x neq 0$.
Hint: Show that if $A$ is normal, and $mu$ is any complex number, then $A-mu I_n$ is also normal. Now, using the norm generated by standard inner product $left langle .,. right rangle$, show that if $B in mathbb{C}^{ntimes n}$, and $B$ is normal, then $left |Bx right |^2=left |B^*x right |^2$ for all $ mathbb{C}^n$.
matrices
edited Dec 8 at 22:23
Andrews
374317
asked Dec 8 at 17:29
user620787
11
Suppose $A in mathbb{C}^{ntimes n}$, and $A$ is normal, show that if $lambda$ is an eigenvalue of $A$, and $x$ is a corresponding eigenvector of $A$ associated with $lambda$, then $bar{lambda}$ is an eigenvalue of $A^*$, and $x$ is an eigenvector of $A^*$ associated with $bar{lambda}$. In other words, if $Ax=lambda x$, and $x neq 0$, then $A^*x=bar{lambda}x$, and $x neq 0$.
Hint: Show that if $A$ is normal, and $mu$ is any complex number, then $A-mu I_n$ is also normal. Now, using the norm generated by standard inner product $left langle .,. right rangle$, show that if $B in mathbb{C}^{ntimes n}$, and $B$ is normal, then $left |Bx right |^2=left |B^*x right |^2$ for all $ mathbb{C}^n$.
matrices
matrices
edited Dec 8 at 22:23
Andrews
374317
asked Dec 8 at 17:29
user620787
11
edited Dec 8 at 22:23
Andrews
374317
asked Dec 8 at 17:29
user620787
11
edited Dec 8 at 22:23
Andrews
374317
edited Dec 8 at 22:23
Andrews
374317
edited Dec 8 at 22:23
Andrews
374317
374317
asked Dec 8 at 17:29
user620787
11
asked Dec 8 at 17:29
user620787
11
asked Dec 8 at 17:29
user620787
11
11
2
What are your thoughts on the problem? What have you tried? Where are you getting stuck?
– Omnomnomnom
Dec 8 at 17:32
Not sure how to show "A - mu I_n" is normal for a start.
– user620787
Dec 8 at 17:44
Please use MathJax in future. It's a type of $LaTeX$. Just search for "MathJax tutorial Mathematics Stack Exchange" and you'll find instructions on how to use it.
– Shaun
Dec 8 at 17:52
add a comment |
2
What are your thoughts on the problem? What have you tried? Where are you getting stuck?
– Omnomnomnom
Dec 8 at 17:32
Not sure how to show "A - mu I_n" is normal for a start.
– user620787
Dec 8 at 17:44
Please use MathJax in future. It's a type of $LaTeX$. Just search for "MathJax tutorial Mathematics Stack Exchange" and you'll find instructions on how to use it.
– Shaun
Dec 8 at 17:52
2
2
What are your thoughts on the problem? What have you tried? Where are you getting stuck?
– Omnomnomnom
Dec 8 at 17:32
What are your thoughts on the problem? What have you tried? Where are you getting stuck?
– Omnomnomnom
Dec 8 at 17:32
Not sure how to show "A - mu I_n" is normal for a start.
– user620787
Dec 8 at 17:44
Not sure how to show "A - mu I_n" is normal for a start.
– user620787
Dec 8 at 17:44
Please use MathJax in future. It's a type of $LaTeX$. Just search for "MathJax tutorial Mathematics Stack Exchange" and you'll find instructions on how to use it.
– Shaun
Dec 8 at 17:52
Please use MathJax in future. It's a type of $LaTeX$. Just search for "MathJax tutorial Mathematics Stack Exchange" and you'll find instructions on how to use it.
– Shaun
Dec 8 at 17:52
add a comment |
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What are your thoughts on the problem? What have you tried? Where are you getting stuck?
– Omnomnomnom
Dec 8 at 17:32
Not sure how to show "A - mu I_n" is normal for a start.
– user620787
Dec 8 at 17:44
Please use MathJax in future. It's a type of $LaTeX$. Just search for "MathJax tutorial Mathematics Stack Exchange" and you'll find instructions on how to use it.
– Shaun
Dec 8 at 17:52