True or False? The characteristic polynomial of a square matrix will always be a quadratic .

True. The characteristic polynomial of a square matrix will always be a quadratic in this course.

When solving the characteristic polynomial of a matrix, what possible solutions could there be?

When solving the characteristic polynomial of a matrix, there could be: two real eigenvalues, one real , repeated eigenvalue, or two complex eigenvalues.

True or False? The values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix.

True. The values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix. This can be used as a quick check of your calculated eigenvalues.

What is a diagonal matrix?

A non-zero, square matrix is considered to be diagonal if all elements not along its leading diagonal are zero .

What is one of the main applications of diagonalising a matrix?

One of the main applications of diagonalising a matrix is to make it easy to find powers of the matrix.

IBMathsDPApplications & Interpretation (AI)HLFlashcards1. Number & AlgebraEigenvalues & Eigenvectors

Eigenvalues & Eigenvectors (DP IB Applications & Interpretation (AI))

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FrontEigenvalues & Eigenvectors
What is an eigenvalue?
FrontEigenvalues & Eigenvectors
What is the characteristic polynomial of a matrix?
BackEigenvalues & Eigenvectors
The characteristic polynomial of an $n \times n$ matrix is $p (λ) = \det (λ I - A)$
Where:
$A$ is a square matrix
$λ$ is an eigenvalue of the matrix $A$
$x$ is eigenvector corresponding to the eigenvalue $λ$
This equation is not given in your exam formula booklet.

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Cards in this collection (15)

What is an eigenvalue?
For a square matrix $A$ , if $A x = λ x$ when $x$ is a non-zero vector and $λ$ a constant, then $λ$ is an eigenvalue of the matrix $A$ .
What is an eigenvector?
For a square matrix $A$ , if $A x = λ x$ when $x$ is a non-zero vector and $λ$ a constant, then $x$ is an eigenvector corresponding to the eigenvalue $λ$ .
What is the characteristic polynomial of a matrix?
The characteristic polynomial of an $n \times n$ matrix is $p (λ) = \det (λ I - A)$
Where:
- $A$ is a square matrix
- $λ$ is an eigenvalue of the matrix $A$
- $x$ is eigenvector corresponding to the eigenvalue $λ$
This equation is not given in your exam formula booklet.
What is the first step in finding the characteristic polynomial of a matrix, e.g. $A = (\begin{matrix} 2 & 7 \\ - 3 & 4 \end{matrix})$ ?
The first step in finding the characteristic polynomial of a matrix is to write down $λ I - A$ .
E.g. for the matrix $A = (\begin{matrix} 2 & 7 \\ - 3 & 4 \end{matrix})$ , write down $λ (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) - (\begin{matrix} 2 & 7 \\ - 3 & 4 \end{matrix})$ .
Remember that $I$ must be of the same order as $A$ .
What is the next step in finding the characteristic polynomial of a matrix, after writing down $λ I - A$ ?
The next step in finding the characteristic polynomial of a matrix, after writing down $λ I - A$ , is to find its determinant, $\det (λ I - A)$ .
True or False?
The characteristic polynomial of a square matrix will always be a quadratic.
True.
The characteristic polynomial of a square matrix will always be a quadratic in this course.
When solving the characteristic polynomial of a matrix, what possible solutions could there be?
When solving the characteristic polynomial of a matrix, there could be:
- two real eigenvalues,
- one real, repeated eigenvalue,
- or two complex eigenvalues.
How do you find the eigenvectors of a matrix?
To find the eigenvectors of a matrix:
1. Write $x = (\begin{matrix} x \\ y \end{matrix})$ .
2. Substitute the eigenvalues into the equation $(λ I - A) x = 0$ , and form two equations in terms of $x$ and $y$ .
3. Let one of the variables be equal to $1$ and use that to find the other variable.
True or False?
There are a finite number of eigenvectors that correspond to a particular eigenvalue.
False.
There are an infinite number of eigenvectors that correspond to a particular eigenvalue.
This is because for any specific value of $x$ that satisfies the equation $(λ I - A) x = 0$ , any scalar multiple of $x$ will also satisfy the equation.
True or False?
The values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix.
True.
The values along the leading diagonal of the matrix you are analysing should sum to the total of the eigenvalues for the matrix.
This can be used as a quick check of your calculated eigenvalues.
What is a diagonal matrix?
A non-zero, square matrix is considered to be diagonal if all elements not along its leading diagonal are zero.
How can $P$ be used to diagonalise a matrix $M$ , given that $P = (x_{1} x_{2})$ , where the first column is the eigenvector $x_{1}$ and the second column is the eigenvector $x_{2}$ ?
If the matrix $M$ is pre-multiplied by $P$ and post-multiplied by $P^{- 1}$ , then it will result in the diagonal matrix of eigenvalues, $D = (\begin{matrix} λ_{1} & 0 \\ 0 & λ_{2} \end{matrix})$ .
$D = P^{- 1} M P$
What is one of the main applications of diagonalising a matrix?
One of the main applications of diagonalising a matrix is to make it easy to find powers of the matrix.
How do you find higher powers of a diagonalised square matrix, e.g. find $M^{n}$ , given $M = (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ .
To find higher powers of a diagonalised square matrix, raise each element along the leading diagonal of the matrix to the given power.
E.g. if $M = (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ , then $M^{n} = {(\begin{matrix} a & 0 \\ 0 & b \end{matrix})}^{n} = (\begin{matrix} a^{n} & 0 \\ 0 & b^{n} \end{matrix})$ .
What is the power formula for a matrix?
The power formula for a matrix is $M^{n} = P D^{n} P^{- 1}$
Where:
- $M$ is a matrix
- $P$ is the matrix of eigenvectors
- $D$ is the matrix of eigenvalues
This formula is given in your exam formula booklet.

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