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Exercises 20.04.2015 - SVDS M Mahbub Alam

That for a gie! matrixXmn" #herenis the !umber of obseratio!s a!$mis the !umberof ariables" the ex%ressio!

1N&1'(xij&xj)(xik&xk)

2( )!$er #hich co!$itio!s ta*i!g $oes the eige! $ecom%ositio! of a matrix +iel$ a!

ortho!ormal basis" a!$ for #hich ector s%ace.

the eige! $ecom%ositio! of a matrix +iel$ a! ortho!ormal basis if it is orthogo!all+

$iago!ali,able.

Annnmatrix with coeficients is orthogonally diagonalizable over the coeficients

i and only i it is normal; a square matrix is orthogonally diagonalizable over R i

and only i it is Hermitian.

( he! a%%l+i!g /ri!ci%al om%o!e!ts A!al+sis" each eige!ector %oi!ts i!to the

$irectio! of o!e the $atasets %ri!ci%al com%o!e!ts a!$ ho# much $o the com%o!e!t

co!tribute to the oerall aria!ce is relate$ to each eige!ectors eige!alue.

Eige!alues a!$ eige!ectors of coaria!ce matrix gies $irectio! a!$ alue of ariatio!

from mea! alues.