Geometric Action Along Orthogonal Eigenvectors

Seeing a Symmetric Matrix Act in Perpendicular Directions

A real symmetric matrix can be understood by separating a vector into components along mutually perpendicular directions. Those directions are eigenvectors: applying the matrix to one of them changes its length by a scalar factor, without turning it toward another direction. Once a vector is expressed in an orthonormal eigenbasis, its image is found by scaling each component separately. This gives a geometric interpretation of matrix multiplication and a direct way to compute images.

Here, \(\mathbf{A}\in\mathbb{R}^{n\times n}\) is symmetric, meaning \(\mathbf{A}^{\top}=\mathbf{A}\). An eigenvector \(\mathbf{q}\) is a nonzero vector satisfying \(\mathbf{A}\mathbf{q}=\lambda\mathbf{q}\), where \(\lambda\in\mathbb{R}\) is its eigenvalue. A list \(\mathbf{q}_1,\dots,\mathbf{q}_n\) is an orthonormal basis if its vectors span \(\mathbb{R}^n\), each has length \(1\), and distinct vectors in the list are perpendicular.

Write the columns of \(\mathbf{Q}\) as \(\mathbf{q}_1,\dots,\mathbf{q}_n\), and write the diagonal entries of \(\boldsymbol{\Lambda}\) as \(\lambda_1,\dots,\lambda_n\). Then \(\mathbf{A}\mathbf{q}_i=\lambda_i\mathbf{q}_i\). Because these vectors are an orthonormal basis, any \(\mathbf{x}\in\mathbb{R}^n\) has a unique expansion \(\mathbf{x}=c_1\mathbf{q}_1+\cdots+c_n\mathbf{q}_n\), where \(c_i=\mathbf{q}_i^{\top}\mathbf{x}\). These \(c_i\) are the coordinates of \(\mathbf{x}\) in the eigenbasis.

Proof. By linearity of matrix multiplication and the eigenvector equations,

This proves the formula for every vector, including \(\mathbf{x}=\mathbf{0}\), for which all \(c_i=0\) and the image is \(\mathbf{0}\). \(\square\)

What the Eigenvalues Do Geometrically

The theorem says to follow each component separately. The component \(c_i\mathbf{q}_i\) becomes \(\lambda_i c_i\mathbf{q}_i\). If \(\lambda_i>0\), its direction is unchanged and its length is multiplied by \(\lambda_i\). If \(\lambda_i<0\), its length is multiplied by \(|\lambda_i|\), and its direction is reversed along that eigenvector line. If \(\lambda_i=0\), that component is sent to \(\mathbf{0}\). Because the basis directions are perpendicular, the output components remain perpendicular to one another.

Thus the matrix need not preserve the length of a general vector, and it need not preserve angles between different vectors. It does, however, act without mixing the coordinates in this particular basis: each input coordinate is multiplied by its corresponding eigenvalue. If several orthogonal directions have the same eigenvalue, every vector in their span is scaled by that same factor. In particular, when an eigenvalue is repeated, the geometric action on its eigenspace is uniform, even though there may be many choices of orthonormal directions within that space.

Lengths and Distances After Applying the Matrix

Orthonormal coordinates also make the length of an image easy to calculate. The squared length of a vector expressed in an orthonormal basis is the sum of the squares of its coordinates. Since the output coordinates are \(\lambda_i c_i\), the squared length after applying \(\mathbf{A}\) is \(\sum_{i=1}^n\lambda_i^2c_i^2\). The sign of an eigenvalue affects the direction, but not the contribution to this squared length.

Proof. The action theorem gives \(\mathbf{A}\mathbf{x}=\sum_{i=1}^n\lambda_i c_i\mathbf{q}_i\). For any vector expressed in an orthonormal basis, its squared length is the sum of the squares of its coordinates: the cross terms vanish because \(\mathbf{q}_i^{\top}\mathbf{q}_j=0\) for \(i\ne j\), and \(\mathbf{q}_i^{\top}\mathbf{q}_i=1\). Therefore \(\|\mathbf{A}\mathbf{x}\|^2=\sum_{i=1}^n(\lambda_i c_i)^2=\sum_{i=1}^n\lambda_i^2c_i^2\). If \(\mathbf{x}=\mathbf{0}\), all terms are zero and the formula still holds. \(\square\)

The same reasoning compares two vectors. If \(\mathbf{x}=\sum c_i\mathbf{q}_i\) and \(\mathbf{y}=\sum d_i\mathbf{q}_i\), their images have coordinates \(\lambda_i c_i\) and \(\lambda_i d_i\). Their inner product is \(\sum_{i=1}^n\lambda_i^2c_id_i\). Taking \(\mathbf{x}-\mathbf{y}\) in the length formula gives the distance formula \(\|\mathbf{A}\mathbf{x}-\mathbf{A}\mathbf{y}\|^2=\sum_{i=1}^n\lambda_i^2(c_i-d_i)^2\). These formulas show that the action on lengths and distances depends on the magnitudes of the eigenvalues.

Using an Orthonormal Eigenbasis in Practice

A reliable calculation separates finding coordinates from applying the matrix. If the eigenvectors are the columns of \(\mathbf{Q}\), then the coordinate vector is \(\mathbf{Q}^{\top}\mathbf{x}\); after multiplying each coordinate by its eigenvalue, convert back to standard coordinates by multiplying by \(\mathbf{Q}\). In matrix form, this is \(\mathbf{A}\mathbf{x}=\mathbf{Q}\boldsymbol{\Lambda}\mathbf{Q}^{\top}\mathbf{x}\). The order matters: first move to eigenbasis coordinates, then scale, then return to standard coordinates.

Common Mistake: Confusing Eigenbasis Coordinates with Vector Entries

The numbers \(c_i=\mathbf{q}_i^{\top}\mathbf{x}\) are coordinates relative to the eigenbasis, not generally the entries of \(\mathbf{x}\) in standard coordinates. A frequent error is to scale those numbers by the eigenvalues and then report the resulting list as the standard-coordinate image without converting back. In the first example, the eigenbasis coordinates of \(\mathbf{x}\) are \((\sqrt{2},\sqrt{2})\), and scaling them gives \((4\sqrt{2},2\sqrt{2})\). These are still eigenbasis coordinates; they are not the entries of the image. The correct conversion is \(4\sqrt{2}\mathbf{q}_1+2\sqrt{2}\mathbf{q}_2=\begin{bmatrix}6\\2\end{bmatrix}\).

A related error is to assume that every symmetric matrix simply multiplies every vector by one number. That conclusion holds only when all eigenvalues are equal. For example, in the first example the two perpendicular components are scaled by \(4\) and \(2\), so the total vector does not undergo one uniform scaling. The eigenbasis method avoids both mistakes by keeping the basis coordinates and standard coordinates distinct.

The geometric viewpoint is useful whenever a matrix is given by an orthogonal diagonalization: it turns a matrix-vector product into a sequence of coordinate changes and independent one-dimensional scalings. It also makes length calculations transparent and reveals exactly which components are reversed or lost. The standard-coordinate matrix product remains the final check, but the eigenbasis explains why the result has the shape it does.