Some years ago we saw how we could use the Jacobi algorithm to find the eigensystem of a real valued symmetric matrix and scalars that satisfy

known as the eigenvectors and the eigenvalues respectively, with the vectors typically restricted to those of unit length in which case we can define its spectral decomposition as the product

where the columns of diagonal element is the eigenvalue associated with the column of

You may recall that this is a particularly convenient representation of the matrix since we can use it to generalise any scalar function to it with

where is the diagonal matrix whose diagonal element is the result of applying diagonal element of

You may also recall that I suggested that there's a more efficient way to find eigensystems and I think that it's high time that we took a look at it.

**M**, which is defined as the set of pairs of non-zero vectors**v**

_{i}

*λ*

_{i}

**M**×

**v**

_{i}=

*λ*

_{i}×

**v**

_{i}

known as the eigenvectors and the eigenvalues respectively, with the vectors typically restricted to those of unit length in which case we can define its spectral decomposition as the product

**M**=

**V**×

**Λ**×

**V**

^{T}

where the columns of

**V**are the unit eigenvectors,**Λ**is a diagonal matrix whose*i*

^{th}

*i*

^{th}

**V**and the T superscript denotes the transpose, in which the rows and columns of the matrix are swapped.You may recall that this is a particularly convenient representation of the matrix since we can use it to generalise any scalar function to it with

*f*(

**M**) =

**V**×

*f*(

**Λ**) ×

**V**

^{T}

where

*f*(

**Λ**)

*i*

^{th}

*f*to the*i*

^{th}

**Λ**.You may also recall that I suggested that there's a more efficient way to find eigensystems and I think that it's high time that we took a look at it.

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