Diagonalisation of symmetric matricesEdexcel International A Level Further Maths: Flashcards

What these 13 flashcards ask

What does it mean for \mathbf{A} to be symmetric?
What is special about eigenvalues of a real symmetric matrix?
What is special about eigenvectors of a symmetric matrix for different eigenvalues?
Define an orthogonal matrix.
What is the inverse of an orthogonal matrix?
What are the columns of an orthogonal matrix like?
What values can the determinant of an orthogonal matrix take?
Which equation expresses diagonalising a symmetric \mathbf{A}?
What goes in the columns of \mathbf{P}?
What goes on the diagonal of \mathbf{D}?
Why must the eigenvectors be normalised?
Write \mathbf{A} in terms of \mathbf{P} and \mathbf{D}.
How can you check your eigenvectors for a symmetric matrix?

Exam questions on Diagonalisation of symmetric matrices

The symmetric matrix $\mathbf{S}=\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$ .
Show that eigenvectors of $\mathbf{S}$ corresponding to its two different eigenvalues are perpendicular.2 marks
The symmetric matrix $\mathbf{T}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 5 & 2 \\ 0 & 2 & 2 \end{pmatrix}$ can be reduced to diagonal form by an orthogonal matrix $\mathbf{P}$ , so that $\mathbf{P}^T\mathbf{T}\mathbf{P}$ is diagonal.
Find a normalised eigenvector of $\mathbf{T}$ corresponding to the eigenvalue $6$ .2 marks
The symmetric matrix $\mathbf{S}=\begin{pmatrix} 1 & 2 \\ 2 & -2 \end{pmatrix}$ .
Find the eigenvalues of $\mathbf{S}$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).