An eigenspace is the subspace made of all vectors that share one eigenvalue, plus the zero vector. In Linear Algebra and Differential Equations, you find it by solving (A - λI)v = 0.
An eigenspace is the set of all vectors that get stretched by the same factor under a matrix, for one specific eigenvalue. In Linear Algebra and Differential Equations, that means you take one eigenvalue, then collect every eigenvector associated with it and include the zero vector too. Because it contains the zero vector and is closed under addition and scalar multiplication, it is a subspace, not just a random list of vectors.
The usual setup is this: if A is a matrix and λ is an eigenvalue, the eigenspace for λ is the null space of A - λI. That is why the equation (A - λI)v = 0 shows up everywhere. You are not just finding one eigenvector, you are finding every solution vector that gets sent to zero by A - λI. Those solution vectors are exactly the directions that stay inside the same eigenvalue behavior.
This is where eigenspaces give the geometry behind eigenvalues. An eigenvalue only tells you the scale factor, but the eigenspace tells you the full family of directions that share that scale factor. If an eigenvalue has many independent eigenvectors, its eigenspace is larger. If it only has one direction, the eigenspace is just a line through the origin.
A quick example makes this clearer. Suppose a 2 by 2 matrix has eigenvalue 3, and solving (A - 3I)v = 0 gives vectors of the form (t, 2t). Then every vector on that line is in the eigenspace for λ = 3. The basis can be written with one nonzero vector, like (1, 2), but the whole eigenspace includes every scalar multiple of that vector and the zero vector.
A common mistake is to think an eigenspace is just the set of eigenvectors. Strictly speaking, it is the set of all eigenvectors for that eigenvalue together with the zero vector, organized as a subspace. Another mix-up is confusing eigenspace dimension with algebraic multiplicity. The dimension of the eigenspace is the geometric multiplicity, and it can be smaller than the algebraic multiplicity of the eigenvalue.
Eigenspaces are the part of eigenvalue work that tells you what a matrix is doing geometrically, not just numerically. If you can identify the eigenspace, you can see which directions stay fixed up to scaling, which is exactly what diagonalization depends on.
That matters because diagonalization turns a hard matrix into an easier one by changing coordinates. You only get a diagonal matrix when you have enough linearly independent eigenvectors, and those vectors come from the eigenspaces. So when you check whether a matrix is diagonalizable, you are really checking whether its eigenspaces give you enough independent directions to build a basis.
In Differential Equations, eigenspaces show up when systems are written in matrix form. If a system has a matrix with clean eigenspaces, you can split the system into simpler pieces that behave like separate exponential solutions. That is why eigenvectors and eigenspaces show up in solution formulas for linear systems, matrix exponentiation, and stability questions.
Eigenspaces also help you organize repeated eigenvalues. A repeated eigenvalue does not automatically give a large eigenspace, so you have to calculate it instead of guessing. That distinction is one of the biggest places students get tripped up, especially when deciding whether a matrix is diagonalizable or defective.
Keep studying Linear Algebra and Differential Equations Unit 5
Visual cheatsheet
view galleryEigenvector
An eigenvector is one nonzero vector inside an eigenspace. The eigenspace is the full set of all such vectors for a given eigenvalue, plus the zero vector. When you solve for eigenvectors, you are really finding a basis for the eigenspace, not just one isolated vector. That makes the eigenspace the bigger geometric object.
Eigenvalue
The eigenvalue is the scale factor that labels the eigenspace. Each eigenspace belongs to exactly one eigenvalue, and changing the eigenvalue changes the whole solution set of (A - λI)v = 0. If you know the eigenvalue but not the eigenspace, you still do not know the directions that keep that scaling behavior.
Diagonalization
Diagonalization depends on collecting enough eigenvectors from all the eigenspaces of a matrix. If the dimensions of the eigenspaces add up to the size of the matrix, you can often build the eigenvector matrix P and write A = PDP^{-1}. If the eigenspaces are too small, diagonalization fails.
Geometric Multiplicity
Geometric multiplicity is the dimension of an eigenspace. It tells you how many linearly independent eigenvectors belong to one eigenvalue. This is different from algebraic multiplicity, which counts how many times the eigenvalue appears as a root of the characteristic polynomial. The two numbers can match, but they do not have to.
A problem set or quiz question will usually ask you to find an eigenspace after you find an eigenvalue. The move is to set up (A - λI)v = 0, row-reduce, and describe the solution set as a span of basis vectors. If the result is only the zero vector, then λ is not an eigenvalue. If the eigenspace has dimension 2 or more, that tells you there are multiple independent eigenvectors for that eigenvalue.
You may also be asked whether a matrix is diagonalizable. Then you use eigenspaces to count independent eigenvectors, not just repeated roots of the characteristic polynomial. In differential equations problems, the eigenspace can show which solution directions go with a certain exponential factor, so you may need to separate the system into simpler components.
An eigenvector is a single nonzero vector. An eigenspace is the whole set of all eigenvectors for one eigenvalue, plus the zero vector, so it is a subspace. If you only write one vector, you have not described the full eigenspace yet.
An eigenspace is the set of all vectors that share one eigenvalue, plus the zero vector.
To find an eigenspace, solve (A - λI)v = 0 and describe the solution set as a subspace.
The dimension of an eigenspace is the geometric multiplicity of that eigenvalue.
Eigenspaces show whether a matrix has enough independent eigenvectors to be diagonalized.
In differential equations, eigenspaces help break a system into simpler solution directions.
An eigenspace is the subspace made of every eigenvector for one eigenvalue, plus the zero vector. You find it by solving (A - λI)v = 0 for a chosen eigenvalue λ. In this course, it is the object that tells you the full geometric picture behind an eigenvalue.
Start with an eigenvalue λ, build the matrix A - λI, and solve the homogeneous system (A - λI)v = 0. The solution set is the eigenspace. Write your answer as a span of basis vectors so you show the whole subspace, not just one eigenvector.
No. An eigenvector is one nonzero vector, while an eigenspace is the full collection of all eigenvectors for a given eigenvalue, together with the zero vector. A single eigenvector can generate the eigenspace, but it does not describe every vector in it by itself.
Diagonalization needs enough linearly independent eigenvectors to form a basis. Those eigenvectors come from eigenspaces, so the size of each eigenspace tells you how much freedom you have. If the eigenspaces do not add up to enough independent directions, the matrix is not diagonalizable.