No, you are playing in C# min.
The point of the capo is to allow you to use open string chord forms in any key rather than bar chords.
music.stackexchange.com/users/50691/ggcg
When you play an Am fingering with a capo on the 4th fret, the sounding chord is C#m and sound is what counts .
When other people listen to your guitar and sing or play along with it, they don't need to know how you used a capo.
would this technically change the Key to A minor?
No.
If there's another guitarist, you can explain as a technical detail that you're fingering the chords like in A minor, but with capo 4.
music.stackexchange.com/users/51766/piiperi Reinstate Monica
All those chords are diatonic to key Am - when there's no capo.
Wherever the capo goes means you count up that many semitones.
So if it was on the second fret, all the chords would be one whole tone higher.
In this scenario, they'd be Bm D A Em G F&sharp;, respectively.
However, you've gone up another tone, making two tones higher than original no capo chords.
Now, they're C&sharp;m E B F&sharp;m
A G&sharp;.
Putting the key into C&sharp;m .
The chord shapes as quoted make it easy to change the key to whatever you want.
They're basic chrds, and become dependant on where the capo is.
That song, with capo on 4, will come out as in C&sharp;m on a standard-tuned guitar.
music.stackexchange.com/users/2808/Tim
The capo thing can get real confusing and all the capo cheat sheets online look different, so its hard to know what's the best logic to use to figure this stuff out.
The best logic to understand how it functions is: to learn the chromatic scale to learn the names of the strings to understand what tones will sound when you play a certain chord pattern
The best is you write on a stripe of paper the chromatic scales on the neck along the frets:
e.g. referring to the A-string A-A#-B-C-C#-D-D#-E-F-F#-G-G#-A (A = fret 0 or open string)
let's assume you know the string names from 6,5,4,3,2,1 = E,A,D,G,B,E
The chord of Am is built on the triad A,C,E:
so we can use the open strings
E,A and E (6,5,1) Finger:
String: Fret: Tone: 1
--------
B--------1-
---> C (up 1 semitone) 2--------
D--------2----
>
E (up 2 semitones) 3
--------
G--------2-
---
> A (up 2 semitones) and
you play the finger pattern 123: what happens if you push or press down a string? by each fret higher
you shorten the string and the tone is a semitone higher!
this explains why you have to put your fingers on the frets mentioned above to build this tones A,C,E on the strings G,B,
D what happens if you put the capo?
all strings are shortened and the chord is transposed up by each fret you move the capo: also the A-minor chord (and any other chord you learn ... like C, G, Dm, F and E) will be transposed up a semitone by each fret.
so we get moving the a-minor chord with the capo (mind the chromatic scale A,A# B,C,C# ... fret 1 =
>
A#m fret 2 =
>
Bm fret 3 =
>
Cm fret 4 =
>
C#m etc.
if you play the A-minor chord with capo you will transpose up this chord by each fret (+ a semi tone; mind that you have to learn also the enharmonic names of flat notes (the black keys of the keyboard) that are equivalent to the sharps (#) that are described above.
music.stackexchange.com/users/56219/Albrecht Hügli
Here's a quick and dirty rule that you can apply without thinking about it.(Though
it's good to eventually get a real understanding.)
The open strings of a standard guitar are tuned to C major (and A minor).
Imagine the note C (or A) on the piano, count up 4 additional semi-tones, and you'll be on E (or C♯).
The guitar is now tuned to E major (and C# minor) with a capo on the 4th fret.
music.stackexchange.com/users/63527/Ray Butterworth
I don't know what you mean by 'technically' changing the key?
You're playing chord shapes in A minor - THINKING in A minor -
but on an instrument that has been shifted up by 4 semitones.
So an Am chord shape produces a C&sharp;m sound.
No technicalities, just that straightforward fact.
music.stackexchange.com/users/17003/Laurence Payne
The method I use to determine the capoed key is based on which root chord form (C, D, E, A, F, G, etc.)
in first position I'm using to play higher on the neck in the capoed position.
I simply add 1/2 steps represented by the frets to the original chord fingering pattern to arrive at the newly adjusted key.
In order to determine the scale fingering pattern for playing a melody, you can then use the CAGED fingering patterns, selecting the fingering pattern associated with the original uncapoed root chord, but I have to say I don't know any players that actually use a capo to play melody.
Most melody players/ improvisers, simply determine the key and chose their own preferred fingering patterns to play along with.
It's a simple concept made more confusing by the number of root position chords in the first position.
The main thing to know is which root chord
you start to count from.
music.stackexchange.com/users/44636/skinny peacock
I have a song I'm playing on guitar and the original key is C# minor. If I play with a capo on the 4th fret, the chord shapes in the song are Am, C, G, Dm, F and E. So would this technically change the Key to A minor?
Key of a song with capo on the 4th fret
music.stackexchange.com
<theory><guitar><key><capos>
music.stackexchange.com/questions/93976
music.stackexchange.com/users/65563/Kim
When you play an Am fingering with a capo on the 4th fret, the sounding chord is C#m and  sound is what counts. When other people listen to your guitar and sing or play along with it, they don't need to know how you used a capo. The capo thing can get real confusing and all the capo cheat sheets online look different, so its hard to know what's the best logic to use to figure this stuff out. The best logic to understand how it functions is to learn the chromatic scale  to learn the names of the strings and  to understand what tones will sound when you play a certain chord pattern    The best way  to understand how it functions is to learn the chromatic scale and learn the names of the strings. 
When you play an Am fingering with a capo on the 4th fret, the sounding chord is C#m and  sound is what counts. The best logic to understand how it functions is to learn the chromatic scale and learn the names of the strings. The chromatic scales can be written on a stripe of paper on the neck along the frets.