Question

If \(A\) is an \(m \times n\) matrix with orthonormal columns, show that \(A^{T} A=I_{n} .\) [Hint: If \(\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\) are the columns of \(A\), show that column \(j\) of \(A^{T} A\) has entries \(\left.\mathbf{c}_{1} \cdot \mathbf{c}_{j}, \mathbf{c}_{2} \cdot \mathbf{c}_{j}, \ldots, \mathbf{c}_{n} \cdot \mathbf{c}_{j}\right]\)

Short answer

If matrix \(A\) has orthonormal columns, then \(A^T A = I_n\).

Step-by-step solution

Understand the Given Problem

We are given that matrix \(A\) is an \(m \times n\) matrix with orthonormal columns. This means that each column of \(A\) is orthogonal to each other and each column vector has a length of 1 (i.e., unit vectors). We want to show that \(A^T A = I_n\).

Express the Product \(A^T A\)

The product \(A^T A\) is an \(n \times n\) matrix, where \(A^T\) is the transpose of \(A\). If the columns of \(A\) are \(\mathbf{c}_1, \mathbf{c}_2, \ldots, \mathbf{c}_n\), then the entry in the \(i\)-th row and \(j\)-th column of \(A^T A\) is \(\mathbf{c}_i \cdot \mathbf{c}_j\).

Compute Dot Products of Columns

Since the columns of \(A\) are orthonormal, the dot product \(\mathbf{c}_i \cdot \mathbf{c}_j = 0\) if \(i eq j\) (because they are orthogonal), and \(\mathbf{c}_i \cdot \mathbf{c}_j = 1\) if \(i = j\) (because each column is a unit vector).

Formulate Matrix \(A^T A\)

Given step 3, we can see that the matrix \(A^T A\) will have 1s on the diagonal (where \(i = j\)) and 0s elsewhere \((i eq j)\). This forms the identity matrix \(I_n\).

State the Conclusion

Based on the steps above, we verify that the matrix \(A^T A\) is indeed the identity matrix \(I_n\), meaning that \(A^T A = I_n\) holds true for an \(m \times n\) matrix \(A\) with orthonormal columns.

Key concepts

Matrix Transposition

Matrix transposition is a fundamental operation in linear algebra. It involves flipping a matrix over its diagonal. This essentially means swapping the matrix's row and column indices. For instance, if you have a matrix \(X = [a,b,c;d,e,f] \), its transpose, denoted as \(X^T\), will be \([a,d;b,e;c,f]\).

Transposition is vital when dealing with properties of orthonormal matrices. When you transpose a matrix, especially those with orthonormal columns, the process plays a key role in proving the identity matrix property, where \(A^T A = I\).

If you transpose a matrix twice, you end up with the original matrix, since \((A^T)^T = A\). This property is particularly useful in verifying correct results when working through problems and exercises involving transpositions.

Dot Product

The dot product, also known as the scalar product, is a crucial concept in vector algebra. It measures the 'projection' of one vector onto another, giving us a scalar value. If two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are in \(n\)-dimensional space, their dot product is calculated as \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n\).

This operation allows us to determine the orthogonality between vectors. If the dot product of two vectors is zero, then the vectors are orthogonal, meaning they are perpendicular to each other in space.

In the context of orthonormal matrices, the dot product helps identify if the matrix's columns are orthogonal. For an orthonormal set of vectors, the dot product between any two distinct vectors is zero, and a vector's dot product with itself is one, confirming each is a unit vector.

Identity Matrix

The identity matrix is a special type of matrix that acts like the "1" of matrix operations. It is denoted by \(I\) and is square (same number of rows and columns) with 1s on the diagonal and 0s elsewhere. For example, a 3x3 identity matrix looks like this:
\[ I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

When you multiply any matrix \(A\) by the identity matrix \(I\), the result is the original matrix \(A\). This is similar to multiplying a number by one. The identity matrix is crucial in solving linear algebra equations, especially when proving that certain matrices, like those with orthonormal columns, maintain their properties after transpositions and multiplications.

In the exercise provided, we see that \(A^T A = I\), highlighting that orthonormal matrices preserve their structure when transposed and multiplied by themselves.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle, forming the basis of orthogonality in vector spaces. For two vectors \(\mathbf{a}\) and \(\mathbf{b}\) to be orthogonal, their dot product must be zero: \(\mathbf{a} \cdot \mathbf{b} = 0\).

This concept is essential in understanding orthonormal matrices, where columns are orthogonal and unit vectors. These properties simplify matrix calculations and are integral to various applications in mathematics, including computer graphics and statistics.

Orthogonal vectors form the backbone for decomposing matrices into simpler forms, aiding in transformations that preserve angles and lengths. This is why orthonormal matrices are preferred in computations, as they inherently maintain numerical stability and efficiency.