# Thread: Eigen value of skew symmetric matrix

1. ## Eigen value of skew symmetric matrix

We were trying to find the eigen value of a skew symmetric matrix:
so we proceeded as :
AX=kX X!=0 for some k as eigen value
and A=-A
so A+A = 0
operating by X (matrix)
AX+AX = 0
AX+kX=0
operating X
XAX+XkX=0
(AX)X+XkX = 0
(kX)X+XkX=0

here we ran into trouble as to the definition of k.
If we take k=k
we get that k=0 (i.e. eigen value is 0)
But skew symmetric matrix can have 0 as well as imaginary eigen values, which we were unable to show.
Is there some other way of doing it ?
Thanks.

2. ## Re: Eigen value of skew symmetric matrix

Here $\displaystyle k$ is just a constant so

$\displaystyle (kX)' = k X'$

$\displaystyle 2k|X|^2 = 0$ and so $\displaystyle k = 0$

Another thing you can do is to use

$\displaystyle A' = -A$

and just do

$\displaystyle A'X = -A X = -k X$

So the eigenvalues of A are either zero or they come in $\displaystyle \pm$ pairs.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://mathhelpforum.com/advanced-algebra/202369-eigen-value-skew-symmetric-matrix.html", "fetch_time": 1527047846000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2018-22/segments/1526794865411.56/warc/CC-MAIN-20180523024534-20180523044434-00010.warc.gz", "warc_record_offset": 179102413, "warc_record_length": 9538, "token_count": 306, "char_count": 957, "score": 3.859375, "int_score": 4, "crawl": "CC-MAIN-2018-22", "snapshot_type": "latest", "language": "en", "language_score": 0.8751145601272583, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00030-of-00064.parquet"}