Change of basis

N = B M B^{- 1}

where:

Change of basis matrix

The change of basis matrix

C

is the matrix that allows us to express the new basis in an old basis:

x_{o l d} = C x_{n e w}

Mnemonic is as follows: consider we have an initial basis

(x_{o l d}, y_{o l d})

. Now, we define the new basis in terms of the old basis, e.g.:

x_{n e w} y_{n e w} = 1 x_{o l d} + 2 y_{o l d} = 3 x_{o l d} + 4 y_{o l d}

which can be written in matrix form as:

[x_{n e w} y_{n e w}] = [1324] [x_{o l d} y_{o l d}]

and so if we set:

M = [1324]

we have:

x_{n e w} = M x_{o l d}

The usual question then is: given a vector in the new basis, how do we represent it in the old basis?

The answer is that we simply have to calculate the matrix inverse of

M

x_{o l d} = M^{- 1} x_{n e w}

That

M^{- 1}

is the matrix inverse.

When we have a symmetric matrix, a change of basis keeps symmetry iff it is done by an orthogonal matrix, in which case:

N = B M B^{- 1} = O M O^{T}

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