IB Maths AI 1.12 Definitions

This page contains our IB Maths AI definitions for 1.12. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Advanced matrices'.

All Topic 1 Sub-topics 1.1: Scientific Notation 1.2: Arithmetic Sequences 1.3: Geometric Sequences 1.4: Interest 1.5: Introduction To Exponents & Logs 1.6: Handling Numbers 1.7: Use Of Technology (HL)1.8: Advanced exponents & logs (HL)1.9: Introduction to complex numbers (HL)1.10: Advanced complex numbers (HL)1.11: Introduction to matrices (HL)1.12: Advanced matrices

characteristic equation

The equation

\det\left(A-\lambda I\right)=0

that must be satisfied for

A\mathbf{v}=\lambda\mathbf{v}

to have a non-zero solution.

characteristic polynomial

The polynomial expression

\det\left(A-\lambda I\right)

whose roots are the eigenvalues of

A

determinant

For a

2\times2

matrix

A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}

, the scalar

ad-bc

, which equals the product of the eigenvalues and is used in

\det\left(A-\lambda I\right)

diagonal matrix

A square matrix with all off-diagonal entries equal to

0

, so powers are found by raising each diagonal entry to the power, e.g.

D^n=\begin{pmatrix}\lambda_1^n&&0\\0&&\lambda_2^n\end{pmatrix}

diagonalizable

Able to be expressed as

A=PDP^{-1}

for some invertible matrix

P

and diagonal matrix

D

, which occurs for a

2\times2

matrix with distinct real eigenvalues.

diagonalization

Writing a matrix in the form

A=PDP^{-1}

, where

P

has eigenvectors as columns and

D

is diagonal with the corresponding eigenvalues on the diagonal; for

2\times2

matrices this is restricted to distinct real eigenvalues.

eigenvalue

A scalar

\lambda

in the equation

A\mathbf{v}=\lambda\mathbf{v}

that gives the scale factor by which an eigenvector is stretched, shrunk, reversed, or mapped to the zero vector.

eigenvector

A non-zero vector

\mathbf{v}

that satisfies

A\mathbf{v}=\lambda\mathbf{v}

for some scalar

\lambda

, meaning the matrix transformation changes only its size (and possibly reverses it) but not its direction.

powers

Repeated multiplication of a matrix by itself, written

A^n

, which can be computed efficiently using diagonalization via

A^n=PD^nP^{-1}

when

A=PDP^{-1}

trace

For a

2\times2

matrix

A=\begin{pmatrix}a&&b\\c&&d\end{pmatrix}

, the sum of the diagonal entries

a+d

, which equals the sum of the eigenvalues.

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