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Magazine Chapter 3: Problem 15
Show that the length of \(A x\) equals the length of \(A^{\mathrm{T}} x\) if \(A A^{\mathrm{T}}=A^{\mathrm{T}} A\).
Short Answer
Expert verified
The lengths of \(Ax\) and \(A^{\mathrm{T}}x\) are equal.
Step by step solution
01
Understand the problem statement
We need to show that the magnitude (or norm) of the vector \(Ax\) is equal to that of \(A^{\mathrm{T}}x\), given that \(AA^{\mathrm{T}} = A^{\mathrm{T}}A\). This requires us to compare the lengths of these two vectors.
02
Express the length of vectors
The length (or norm) of a vector \(v\) can be calculated using the formula \(\|v\| = \sqrt{v^{\mathrm{T}} v}\). So, for vector \(Ax\), its length \(\|Ax\| = \sqrt{(Ax)^{\mathrm{T}} (Ax)}\), and for vector \(A^{\mathrm{T}}x\), its length \(\|A^{\mathrm{T}}x\| = \sqrt{(A^{\mathrm{T}}x)^{\mathrm{T}} (A^{\mathrm{T}}x)}\).
03
Calculate \(\|Ax\|^2\)
Calculate the square of the length of \(Ax\) using: \[(Ax)^{\mathrm{T}}(Ax) = x^{\mathrm{T}}A^{\mathrm{T}}Ax\] This formulation uses properties of transposes and matrix multiplication.
04
Calculate \(\|A^{\mathrm{T}}x\|^2\)
Similarly, calculate the square of the length of \(A^{\mathrm{T}}x\) using: \[(A^{\mathrm{T}}x)^{\mathrm{T}}(A^{\mathrm{T}}x) = x^{\mathrm{T}}AA^{\mathrm{T}}x\] This also follows from the properties of transposes and matrix multiplication.
05
Use the given condition
We know from the problem statement that \(AA^{\mathrm{T}} = A^{\mathrm{T}}A\). This implies:\[x^{\mathrm{T}}A^{\mathrm{T}}Ax = x^{\mathrm{T}}AA^{\mathrm{T}}x\] Thus, \(\|Ax\|^2 = \|A^{\mathrm{T}}x\|^2\).
06
Conclude by equating lengths
Since \(\|Ax\|^2 = \|A^{\mathrm{T}}x\|^2\), taking the square root on both sides shows \(\|Ax\| = \|A^{\mathrm{T}}x\|\). Hence, the lengths are equal.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Transpose
A matrix transpose is a simple transformation where you "flip" a matrix over its diagonal. Imagine you have a matrix \( A \) with rows and columns. When you transpose it to become \( A^{\mathrm{T}} \), each row of the original matrix becomes a column.
- For example, if the original matrix \( A \) has elements \( a_{ij} \), the elements of \( A^{\mathrm{T}} \) will be \( a_{ji} \).
- This operation is essential in various numerical and algebraic calculations.
Matrix Multiplication
Matrix multiplication is an operation where you combine two matrices to form a new one.
- The number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be possible.
- For matrices \( A \) and \( B \), the product \( C = AB \) involves calculating each element \( c_{ij} \) as the dot product of the \( i \)-th row of A and the \( j \)-th column of \( B \).
Orthogonal Matrices
Orthogonal matrices are special square matrices where the rows and columns are mutually orthogonal unit vectors.
- If \( Q \) is an orthogonal matrix, then \( Q^{\mathrm{T}} = Q^{-1} \).
- This means when you multiply a matrix by its transpose, you get the identity matrix: \( QQ^{\mathrm{T}} = Q^{\mathrm{T}}Q = I \).
Vector Norms
Understanding vector norms is essential in assessing the 'length' or magnitude of vectors. The most common norm is the Euclidean norm, denoted as \( \|v\| \), calculated using \( \sqrt{v^{\mathrm{T}}v} \).
- Norms help in measuring and comparing the size of vectors which is vital for numerous algorithms and solutions.
- They also provide a way to define the distance between points in space, given by the difference norm.
