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Matrix transformations

Invert and diagonalise matrices: finding an inverse by row reduction, the rules inverses obey, symmetric matrices, and diagonalisation of $2 \times 2$ matrices.

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Symmetric matrices

Understand how symmetric matrices can always be diagonalised using an orthogonal basis of eigenvectors, leading to real eigenvalues and mutually orthogonal principal directions - crucial properties for applications such as principal component analysis and covariance matrices in machine learning.

$A = A^T.$

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Lesson 03 · Symmetric matrices

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λ_{1} = 3.62, {\hat{\vec{v}}}_{1} = [\begin{matrix} 0.53 \\ 0.85 \end{matrix}]

λ_{2} = 1.38, {\hat{\vec{v}}}_{2} = [\begin{matrix} 0.85 \\ - 0.53 \end{matrix}]

{\hat{\vec{v}}}_{1} \cdot {\hat{\vec{v}}}_{2} = - 0.00

A = [\begin{matrix} 2 & 1 \\ 1 & 3 \end{matrix}]

{\hat{\vec{v}}}_{1}

{\hat{\vec{v}}}_{2}

The blue and red lines show the eigenvector directions of the symmetric matrix $A$ . The arrows on these lines are the orthonormal eigenvectors (unit length). Notice their dot product is always zero.

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The 4 lessons

The 4 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

Inverse of a matrix via row reduction

10 questions

Learn how to determine whether a $2 \times 2$ matrix is invertible and, if so, find its inverse using row reduction, connecting these concepts to solving systems of linear equations in machine learning.

Covers

What is the inverse of a matrix?

Finding the inverse of a $2 \times 2$ matrix via row reduction

When does a matrix not have an inverse?

Key properties and rules of matrix inverses

9 questions

Learn the core rules for finding inverses of matrix products, transposes, scalar multiples, and diagonal matrices, enabling efficient simplification and manipulation of matrix equations.

Covers

Inverse of a matrix product

Inverse of a transpose and scalar multiple

Inverse of a diagonal matrix

Symmetric matrices

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9 questions

Covers

What is a symmetric matrix?

Orthogonal eigenvectors of symmetric matrices

Orthogonal diagonalisation of symmetric matrices

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Diagonalisation of $2 \times 2$ matrices

12 questions

Learn how to determine whether a $2 \times 2$ matrix is diagonalisable by finding its eigenvalues and eigenvectors, construct its diagonalisation if possible, and use this to efficiently compute powers of the matrix.

Covers

What does it mean to diagonalise?

When is a $2 \times 2$ matrix diagonalisable?

How to diagonalise a $2 \times 2$ matrix

Using diagonalisation

Where this unit sits in the curriculum

This unit is part of our Essential Linear Algebra for ML learning path, which contains 61 lessons. Every one of them is drawn below.

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4 lessons in this unit

52 prerequisite lessons across the unit, counting every step back to the start

5 other lessons in the learning path - after this unit, or alongside it

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Builds on: Eigenvalues, eigenvectors and singular value decomposition, Matrices, Matrices and linear systems

Leads to: Eigenvalues, eigenvectors and singular value decomposition, Matrices, Vectors in Euclidean space

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More in Linear algebra: Vectors in Euclidean space, Matrices, Matrices and linear systems, Eigenvalues, eigenvectors and singular value decomposition