Ciro Santilli @cirosantilli 37

 Incoming links: Orthogonal matrix

Change of basis between symmetric matrices Updated 2025-07-16

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

N = BM B^{- 1} = OM O^{T}

 Read the full article

Eigendecomposition of a real symmetric matrix Updated 2025-07-16

The general result from eigendecomposition of a matrix:

M = Q D Q^{- 1}

becomes:

M = O D O^{T}

where

O

is an orthogonal matrix, and therefore has

O^{- 1} = O^{T}

.

 Read the full article

The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose Updated 2025-07-16

Let's show that this definition is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

Note that:

x^{T} y = (O x)^{T} (O y) = x^{T} O^{T} O y

and for that to be true for all possible

x

and

y

then we must have:

O^{T} O = I

i.e. the matrix inverse is equal to the transpose.

Conversely, if:

O^{T} O = I

is true, then

(O x)^{T} (O y) = x^{T} (O^{T} O) y = x^{T} y

These matricese are called the orthogonal matrices.

TODO is there any more intuitive way to think about this?

 Read the full article

Unitary matrix Updated 2025-07-16

Complex analogue of orthogonal matrix.

Applications:

in quantum computers programming basically comes down to creating one big unitary matrix as explained at: quantum computing is just matrix multiplication

 Read the full article